From VLSM.Lib Require Import Itauto.[Loading ML file ring_plugin.cmxs (using legacy method) ... done]
[Loading ML file zify_plugin.cmxs (using legacy method) ... done]
[Loading ML file micromega_plugin.cmxs (using legacy method) ... done]
[Loading ML file btauto_plugin.cmxs (using legacy method) ... done]
[Loading ML file coq-itauto.plugin ... done]
From Coq Require Import FinFun FunctionalExtensionality.
From stdpp Require Import prelude finite.
From VLSM.Lib Require Import Preamble ListExtras StdppListSet.
From VLSM.Core Require Import VLSM VLSMProjections Composition ProjectionTraces Validator.
From VLSM.Core Require Import SubProjectionTraces Equivocation.
From VLSM.Core Require Import Equivocation.NoEquivocation.
From VLSM.Core Require Import Equivocators.Equivocators Equivocators.EquivocatorsProjections.
From VLSM.Core Require Import Equivocators.EquivocatorsComposition.[Loading ML file extraction_plugin.cmxs (using legacy method) ... done]
[Loading ML file equations_plugin.cmxs (using legacy method) ... done]
From VLSM.Core Require Import Equivocators.MessageProperties.

Core: VLSM Equivocator Composition Projections

Section sec_equivocators_composition_projections.

Context {message : Type}
  `{finite.Finite index}
  (IM : index -> VLSM message)
  `{forall i : index, HasBeenSentCapability (IM i)}
  (FreeE := free_composite_vlsm (equivocator_IM IM))
  (PreFreeE := preloaded_with_all_messages_vlsm FreeE)
  (Free := free_composite_vlsm IM)
  (PreFree := preloaded_with_all_messages_vlsm Free)
  .

Given a transition_item item in the compositions of equivocators of components IM and an equivocator_descriptors, if the descriptors are all valid in the destination of the transition this returns a set of updated descriptors for corresponding positions in the origin state of the transition, and if the transition was an in-place change to an existing alternative it also returns a projected transition item in the plain composition of IM.

Definition equivocators_transition_item_project
  (eqv_descriptors : equivocator_descriptors IM)
  (item : composite_transition_item (equivocator_IM IM))
  : option (option (composite_transition_item IM) * equivocator_descriptors IM)
  :=
  let sx := equivocators_state_project IM eqv_descriptors (destination item) in
  let eqv := projT1 (l item) in
  let deqv := eqv_descriptors eqv in
  match
    equivocator_vlsm_transition_item_project
      (IM eqv)
      (composite_transition_item_projection (equivocator_IM IM) item)
      deqv
      with
  | Some (Some item', deqv') =>
    Some
      (Some (Build_transition_item Free
        (existT eqv (l item'))
        (input item) sx (output item))
      , equivocator_descriptors_update IM eqv_descriptors eqv deqv')
  | Some (None, deqv') => Some (None, equivocator_descriptors_update IM eqv_descriptors eqv deqv')
  | None => None
  end.

Lemma equivocators_transition_item_project_preserves_equivocating_indices
  (descriptors : equivocator_descriptors IM)
  (item : composite_transition_item (equivocator_IM IM))
  oitem idescriptors
  s
  (Hdescriptors : proper_equivocator_descriptors IM descriptors (destination item))
  (Ht : composite_transition (equivocator_IM IM) (l item) (s, input item) =
          (destination item, output item))
  (Hv : composite_valid (equivocator_IM IM) (l item) (s, input item))
  (Hpr : equivocators_transition_item_project descriptors item = Some (oitem, idescriptors))
  :
    set_union
      (equivocating_indices IM (enum index) s)
      (newmachine_descriptors_list IM (enum index) idescriptors)
    ⊆
    set_union
      (equivocating_indices IM (enum index) (destination item))
      (newmachine_descriptors_list IM (enum index) descriptors).message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
descriptors: equivocator_descriptors IM
item: composite_transition_item (equivocator_IM IM)
oitem: option (composite_transition_item IM)
idescriptors: equivocator_descriptors IM
s: composite_state (equivocator_IM IM)
Hdescriptors: proper_equivocator_descriptors IM
  descriptors (destination item)
Ht: composite_transition (equivocator_IM IM) 
  (l item) (s, input item) =
(destination item, output item)
Hv: composite_valid (equivocator_IM IM) 
  (l item) (s, input item)
Hpr: equivocators_transition_item_project descriptors
  item = Some (oitem, idescriptors)
set_union (equivocating_indices IM (enum index) s)
  (newmachine_descriptors_list IM (enum index)
     idescriptors)
⊆ set_union
    (equivocating_indices IM (enum index)
       (destination item))
    (newmachine_descriptors_list IM (enum index)
       descriptors)
Proof.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
descriptors: equivocator_descriptors IM
item: composite_transition_item (equivocator_IM IM)
oitem: option (composite_transition_item IM)
idescriptors: equivocator_descriptors IM
s: composite_state (equivocator_IM IM)
Hdescriptors: proper_equivocator_descriptors IM
  descriptors (destination item)
Ht: composite_transition (equivocator_IM IM) 
  (l item) (s, input item) =
(destination item, output item)
Hv: composite_valid (equivocator_IM IM) 
  (l item) (s, input item)
Hpr: equivocators_transition_item_project descriptors
  item = Some (oitem, idescriptors)
set_union (equivocating_indices IM (enum index) s)
  (newmachine_descriptors_list IM (enum index)
     idescriptors)
⊆ set_union
    (equivocating_indices IM (enum index)
       (destination item))
    (newmachine_descriptors_list IM (enum index)
       descriptors)
  unfold equivocators_transition_item_project
    , composite_transition_item_projection
    , composite_transition_item_projection_from_eq  in Hpr; simpl in Hpr.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
descriptors: equivocator_descriptors IM
item: composite_transition_item (equivocator_IM IM)
oitem: option (composite_transition_item IM)
idescriptors: equivocator_descriptors IM
s: composite_state (equivocator_IM IM)
Hdescriptors: proper_equivocator_descriptors IM
  descriptors (destination item)
Ht: composite_transition (equivocator_IM IM) 
  (l item) (s, input item) =
(destination item, output item)
Hv: composite_valid (equivocator_IM IM) 
  (l item) (s, input item)
Hpr: match
  equivocator_vlsm_transition_item_project
    (IM (projT1 (l item)))
    {|
      l :=
        eq_rect_r
          (λ n : index, EquivocatorLabel (IM n))
          (projT2 (l item)) eq_refl;
      input := input item;
      destination :=
        destination item (projT1 (l item));
      output := output item
    |} (descriptors (projT1 (l item)))
with
| Some (Some item', deqv') =>
    Some
      (Some
         {|
           l :=
             existT (projT1 (l item)) (l item');
           input := input item;
           destination :=
             equivocators_state_project IM
               descriptors 
               (destination item);
           output := output item
         |},
       equivocator_descriptors_update IM
         descriptors (projT1 (l item)) deqv')
| Some (None, deqv') =>
    Some
      (None,
       equivocator_descriptors_update IM
         descriptors (projT1 (l item)) deqv')
| None => None
end = Some (oitem, idescriptors)
set_union (equivocating_indices IM (enum index) s)
  (newmachine_descriptors_list IM (enum index)
     idescriptors)
⊆ set_union
    (equivocating_indices IM (enum index)
       (destination item))
    (newmachine_descriptors_list IM (enum index)
       descriptors)
  unfold eq_rect_r, eq_rect in Hpr; simpl in Hpr.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
descriptors: equivocator_descriptors IM
item: composite_transition_item (equivocator_IM IM)
oitem: option (composite_transition_item IM)
idescriptors: equivocator_descriptors IM
s: composite_state (equivocator_IM IM)
Hdescriptors: proper_equivocator_descriptors IM
  descriptors (destination item)
Ht: composite_transition (equivocator_IM IM) 
  (l item) (s, input item) =
(destination item, output item)
Hv: composite_valid (equivocator_IM IM) 
  (l item) (s, input item)
Hpr: match
  equivocator_vlsm_transition_item_project
    (IM (projT1 (l item)))
    {|
      l := projT2 (l item);
      input := input item;
      destination :=
        destination item (projT1 (l item));
      output := output item
    |} (descriptors (projT1 (l item)))
with
| Some (Some item', deqv') =>
    Some
      (Some
         {|
           l :=
             existT (projT1 (l item)) (l item');
           input := input item;
           destination :=
             equivocators_state_project IM
               descriptors 
               (destination item);
           output := output item
         |},
       equivocator_descriptors_update IM
         descriptors (projT1 (l item)) deqv')
| Some (None, deqv') =>
    Some
      (None,
       equivocator_descriptors_update IM
         descriptors (projT1 (l item)) deqv')
| None => None
end = Some (oitem, idescriptors)
set_union (equivocating_indices IM (enum index) s)
  (newmachine_descriptors_list IM (enum index)
     idescriptors)
⊆ set_union
    (equivocating_indices IM (enum index)
       (destination item))
    (newmachine_descriptors_list IM (enum index)
       descriptors)
  match type of Hpr with
    (match ?exp with _ => _ end = _)
    => destruct exp as [(oitemx, deqv') |] eqn: Hitem_pr; [| by congruence]
  end.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
descriptors: equivocator_descriptors IM
item: composite_transition_item (equivocator_IM IM)
oitem: option (composite_transition_item IM)
idescriptors: equivocator_descriptors IM
s: composite_state (equivocator_IM IM)
Hdescriptors: proper_equivocator_descriptors IM
  descriptors (destination item)
Ht: composite_transition (equivocator_IM IM) 
  (l item) (s, input item) =
(destination item, output item)
Hv: composite_valid (equivocator_IM IM) 
  (l item) (s, input item)
oitemx: option transition_item
deqv': MachineDescriptor (IM (projT1 (l item)))
Hitem_pr: equivocator_vlsm_transition_item_project
  (IM (projT1 (l item)))
  {|
    l := projT2 (l item);
    input := input item;
    destination :=
      destination item (projT1 (l item));
    output := output item
  |} (descriptors (projT1 (l item))) =
Some (oitemx, deqv')
Hpr: match oitemx with
| Some item' =>
    Some
      (Some
         {|
           l :=
             existT (projT1 (l item)) (l item');
           input := input item;
           destination :=
             equivocators_state_project IM
               descriptors 
               (destination item);
           output := output item
         |},
       equivocator_descriptors_update IM
         descriptors (projT1 (l item)) deqv')
| None =>
    Some
      (None,
       equivocator_descriptors_update IM
         descriptors (projT1 (l item)) deqv')
end = Some (oitem, idescriptors)
set_union (equivocating_indices IM (enum index) s)
  (newmachine_descriptors_list IM (enum index)
     idescriptors)
⊆ set_union
    (equivocating_indices IM (enum index)
       (destination item))
    (newmachine_descriptors_list IM (enum index)
       descriptors)
  simpl in Ht.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
descriptors: equivocator_descriptors IM
item: composite_transition_item (equivocator_IM IM)
oitem: option (composite_transition_item IM)
idescriptors: equivocator_descriptors IM
s: composite_state (equivocator_IM IM)
Hdescriptors: proper_equivocator_descriptors IM
  descriptors (destination item)
Ht: (let (i, li) := l item in
 let (si', om') :=
   equivocator_transition 
     (IM i) li (s i, input item) in
 (state_update (equivocator_IM IM) s i si', om')) =
(destination item, output item)
Hv: composite_valid (equivocator_IM IM) 
  (l item) (s, input item)
oitemx: option transition_item
deqv': MachineDescriptor (IM (projT1 (l item)))
Hitem_pr: equivocator_vlsm_transition_item_project
  (IM (projT1 (l item)))
  {|
    l := projT2 (l item);
    input := input item;
    destination :=
      destination item (projT1 (l item));
    output := output item
  |} (descriptors (projT1 (l item))) =
Some (oitemx, deqv')
Hpr: match oitemx with
| Some item' =>
    Some
      (Some
         {|
           l :=
             existT (projT1 (l item)) (l item');
           input := input item;
           destination :=
             equivocators_state_project IM
               descriptors 
               (destination item);
           output := output item
         |},
       equivocator_descriptors_update IM
         descriptors (projT1 (l item)) deqv')
| None =>
    Some
      (None,
       equivocator_descriptors_update IM
         descriptors (projT1 (l item)) deqv')
end = Some (oitem, idescriptors)
set_union (equivocating_indices IM (enum index) s)
  (newmachine_descriptors_list IM (enum index)
     idescriptors)
⊆ set_union
    (equivocating_indices IM (enum index)
       (destination item))
    (newmachine_descriptors_list IM (enum index)
       descriptors)
  destruct item.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
descriptors: equivocator_descriptors IM
l: label (composite_type (equivocator_IM IM))
input: option message
destination: state
  (composite_type (equivocator_IM IM))
output: option message
oitem: option (composite_transition_item IM)
idescriptors: equivocator_descriptors IM
s: composite_state (equivocator_IM IM)
Hdescriptors: proper_equivocator_descriptors IM
  descriptors
  (VLSM.destination
     {|
       l := l;
       input := input;
       destination := destination;
       output := output
     |})
Ht: (let (i, li) :=
   VLSM.l
     {|
       l := l;
       input := input;
       destination := destination;
       output := output
     |} in
 let (si', om') :=
   equivocator_transition 
     (IM i) li
     (s i,
      VLSM.input
        {|
          l := l;
          input := input;
          destination := destination;
          output := output
        |}) in
 (state_update (equivocator_IM IM) s i si', om')) =
(VLSM.destination
   {|
     l := l;
     input := input;
     destination := destination;
     output := output
   |},
 VLSM.output
   {|
     l := l;
     input := input;
     destination := destination;
     output := output
   |})
Hv: composite_valid (equivocator_IM IM)
  (VLSM.l
     {|
       l := l;
       input := input;
       destination := destination;
       output := output
     |})
  (s,
   VLSM.input
     {|
       l := l;
       input := input;
       destination := destination;
       output := output
     |})
oitemx: option transition_item
deqv': MachineDescriptor
  (IM
     (projT1
        (VLSM.l
           {|
             l := l;
             input := input;
             destination := destination;
             output := output
           |})))
Hitem_pr: equivocator_vlsm_transition_item_project
  (IM
     (projT1
        (VLSM.l
           {|
             l := l;
             input := input;
             destination := destination;
             output := output
           |})))
  {|
    l :=
      projT2
        (VLSM.l
           {|
             l := l;
             input := input;
             destination := destination;
             output := output
           |});
    input :=
      VLSM.input
        {|
          l := l;
          input := input;
          destination := destination;
          output := output
        |};
    destination :=
      VLSM.destination
        {|
          l := l;
          input := input;
          destination := destination;
          output := output
        |}
        (projT1
           (VLSM.l
              {|
                l := l;
                input := input;
                destination := destination;
                output := output
              |}));
    output :=
      VLSM.output
        {|
          l := l;
          input := input;
          destination := destination;
          output := output
        |}
  |}
  (descriptors
     (projT1
        (VLSM.l
           {|
             l := l;
             input := input;
             destination := destination;
             output := output
           |}))) = Some (oitemx, deqv')
Hpr: match oitemx with
| Some item' =>
    Some
      (Some
         {|
           l :=
             existT
               (projT1
                  (VLSM.l
                     {|
                      l := l;
                      input := input;
                      destination := destination;
                      output := output
                     |})) 
               (VLSM.l item');
           input :=
             VLSM.input
               {|
                 l := l;
                 input := input;
                 destination := destination;
                 output := output
               |};
           destination :=
             equivocators_state_project IM
               descriptors
               (VLSM.destination
                  {|
                    l := l;
                    input := input;
                    destination := destination;
                    output := output
                  |});
           output :=
             VLSM.output
               {|
                 l := l;
                 input := input;
                 destination := destination;
                 output := output
               |}
         |},
       equivocator_descriptors_update IM
         descriptors
         (projT1
            (VLSM.l
               {|
                 l := l;
                 input := input;
                 destination := destination;
                 output := output
               |})) deqv')
| None =>
    Some
      (None,
       equivocator_descriptors_update IM
         descriptors
         (projT1
            (VLSM.l
               {|
                 l := l;
                 input := input;
                 destination := destination;
                 output := output
               |})) deqv')
end = Some (oitem, idescriptors)
set_union (equivocating_indices IM (enum index) s)
  (newmachine_descriptors_list IM (enum index)
     idescriptors)
⊆ set_union
    (equivocating_indices IM (enum index)
       (VLSM.destination
          {|
            l := l;
            input := input;
            destination := destination;
            output := output
          |}))
    (newmachine_descriptors_list IM (enum index)
       descriptors) simpl in *.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
descriptors: equivocator_descriptors IM
l: composite_label (equivocator_IM IM)
input: option message
destination: composite_state (equivocator_IM IM)
output: option message
oitem: option (composite_transition_item IM)
idescriptors: equivocator_descriptors IM
s: composite_state (equivocator_IM IM)
Hdescriptors: proper_equivocator_descriptors IM
  descriptors destination
Ht: (let (i, li) := l in
 let (si', om') :=
   equivocator_transition (IM i) li (s i, input) in
 (state_update (equivocator_IM IM) s i si', om')) =
(destination, output)
Hv: let (i, li) := l in
equivocator_valid (IM i) li (s i, input)
oitemx: option transition_item
deqv': MachineDescriptor (IM (projT1 l))
Hitem_pr: equivocator_vlsm_transition_item_project
  (IM (projT1 l))
  {|
    l := projT2 l;
    input := input;
    destination := destination (projT1 l);
    output := output
  |} (descriptors (projT1 l)) =
Some (oitemx, deqv')
Hpr: match oitemx with
| Some item' =>
    Some
      (Some
         {|
           l := existT (projT1 l) (VLSM.l item');
           input := input;
           destination :=
             equivocators_state_project IM
               descriptors destination;
           output := output
         |},
       equivocator_descriptors_update IM
         descriptors (projT1 l) deqv')
| None =>
    Some
      (None,
       equivocator_descriptors_update IM
         descriptors (projT1 l) deqv')
end = Some (oitem, idescriptors)
set_union (equivocating_indices IM (enum index) s)
  (newmachine_descriptors_list IM (enum index)
     idescriptors)
⊆ set_union
    (equivocating_indices IM (enum index) destination)
    (newmachine_descriptors_list IM (enum index)
       descriptors) destruct l as (i, li).message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
descriptors: equivocator_descriptors IM
i: index
li: label (equivocator_IM IM i)
input: option message
destination: composite_state (equivocator_IM IM)
output: option message
oitem: option (composite_transition_item IM)
idescriptors: equivocator_descriptors IM
s: composite_state (equivocator_IM IM)
Hdescriptors: proper_equivocator_descriptors IM
  descriptors destination
Ht: (let (si', om') :=
   equivocator_transition (IM i) li (s i, input) in
 (state_update (equivocator_IM IM) s i si', om')) =
(destination, output)
Hv: equivocator_valid (IM i) li (s i, input)
oitemx: option transition_item
deqv': MachineDescriptor (IM (projT1 (existT i li)))
Hitem_pr: equivocator_vlsm_transition_item_project
  (IM (projT1 (existT i li)))
  {|
    l := projT2 (existT i li);
    input := input;
    destination :=
      destination (projT1 (existT i li));
    output := output
  |} (descriptors (projT1 (existT i li))) =
Some (oitemx, deqv')
Hpr: match oitemx with
| Some item' =>
    Some
      (Some
         {|
           l :=
             existT (projT1 (existT i li))
               (l item');
           input := input;
           destination :=
             equivocators_state_project IM
               descriptors destination;
           output := output
         |},
       equivocator_descriptors_update IM
         descriptors (projT1 (existT i li))
         deqv')
| None =>
    Some
      (None,
       equivocator_descriptors_update IM
         descriptors (projT1 (existT i li))
         deqv')
end = Some (oitem, idescriptors)
set_union (equivocating_indices IM (enum index) s)
  (newmachine_descriptors_list IM (enum index)
     idescriptors)
⊆ set_union
    (equivocating_indices IM (enum index) destination)
    (newmachine_descriptors_list IM (enum index)
       descriptors) simpl in *.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
descriptors: equivocator_descriptors IM
i: index
li: EquivocatorLabel (IM i)
input: option message
destination: composite_state (equivocator_IM IM)
output: option message
oitem: option (composite_transition_item IM)
idescriptors: equivocator_descriptors IM
s: composite_state (equivocator_IM IM)
Hdescriptors: proper_equivocator_descriptors IM
  descriptors destination
Ht: (let (si', om') :=
   equivocator_transition (IM i) li (s i, input) in
 (state_update (equivocator_IM IM) s i si', om')) =
(destination, output)
Hv: equivocator_valid (IM i) li (s i, input)
oitemx: option transition_item
deqv': MachineDescriptor (IM i)
Hitem_pr: equivocator_vlsm_transition_item_project
  (IM i)
  {|
    l := li;
    input := input;
    destination := destination i;
    output := output
  |} (descriptors i) = 
Some (oitemx, deqv')
Hpr: match oitemx with
| Some item' =>
    Some
      (Some
         {|
           l := existT i (l item');
           input := input;
           destination :=
             equivocators_state_project IM
               descriptors destination;
           output := output
         |},
       equivocator_descriptors_update IM
         descriptors i deqv')
| None =>
    Some
      (None,
       equivocator_descriptors_update IM
         descriptors i deqv')
end = Some (oitem, idescriptors)
set_union (equivocating_indices IM (enum index) s)
  (newmachine_descriptors_list IM (enum index)
     idescriptors)
⊆ set_union
    (equivocating_indices IM (enum index) destination)
    (newmachine_descriptors_list IM (enum index)
       descriptors)
  match type of Ht with
  | (let '(si', om') := ?t in _) = _ => destruct t as [si' om] eqn: Htei
  end.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
descriptors: equivocator_descriptors IM
i: index
li: EquivocatorLabel (IM i)
input: option message
destination: composite_state (equivocator_IM IM)
output: option message
oitem: option (composite_transition_item IM)
idescriptors: equivocator_descriptors IM
s: composite_state (equivocator_IM IM)
Hdescriptors: proper_equivocator_descriptors IM
  descriptors destination
si': equivocator_state (IM i)
om: option message
Htei: equivocator_transition (IM i) li (s i, input) =
(si', om)
Ht: (state_update (equivocator_IM IM) s i si', om) =
(destination, output)
Hv: equivocator_valid (IM i) li (s i, input)
oitemx: option transition_item
deqv': MachineDescriptor (IM i)
Hitem_pr: equivocator_vlsm_transition_item_project
  (IM i)
  {|
    l := li;
    input := input;
    destination := destination i;
    output := output
  |} (descriptors i) = 
Some (oitemx, deqv')
Hpr: match oitemx with
| Some item' =>
    Some
      (Some
         {|
           l := existT i (l item');
           input := input;
           destination :=
             equivocators_state_project IM
               descriptors destination;
           output := output
         |},
       equivocator_descriptors_update IM
         descriptors i deqv')
| None =>
    Some
      (None,
       equivocator_descriptors_update IM
         descriptors i deqv')
end = Some (oitem, idescriptors)
set_union (equivocating_indices IM (enum index) s)
  (newmachine_descriptors_list IM (enum index)
     idescriptors)
⊆ set_union
    (equivocating_indices IM (enum index) destination)
    (newmachine_descriptors_list IM (enum index)
       descriptors)
  inversion Ht; subst; clear Ht.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
descriptors: equivocator_descriptors IM
i: index
li: EquivocatorLabel (IM i)
input, output: option message
oitem: option (composite_transition_item IM)
idescriptors: equivocator_descriptors IM
s: composite_state (equivocator_IM IM)
si': equivocator_state (IM i)
Hdescriptors: proper_equivocator_descriptors IM
  descriptors
  (state_update (equivocator_IM IM) s i
     si')
Htei: equivocator_transition (IM i) li (s i, input) =
(si', output)
Hv: equivocator_valid (IM i) li (s i, input)
oitemx: option transition_item
deqv': MachineDescriptor (IM i)
Hpr: match oitemx with
| Some item' =>
    Some
      (Some
         {|
           l := existT i (l item');
           input := input;
           destination :=
             equivocators_state_project IM
               descriptors
               (state_update 
                  (equivocator_IM IM) s i si');
           output := output
         |},
       equivocator_descriptors_update IM
         descriptors i deqv')
| None =>
    Some
      (None,
       equivocator_descriptors_update IM
         descriptors i deqv')
end = Some (oitem, idescriptors)
Hitem_pr: equivocator_vlsm_transition_item_project
  (IM i)
  {|
    l := li;
    input := input;
    destination :=
      state_update (equivocator_IM IM) s i
        si' i;
    output := output
  |} (descriptors i) = 
Some (oitemx, deqv')
set_union (equivocating_indices IM (enum index) s)
  (newmachine_descriptors_list IM (enum index)
     idescriptors)
⊆ set_union
    (equivocating_indices IM (enum index)
       (state_update (equivocator_IM IM) s i si'))
    (newmachine_descriptors_list IM (enum index)
       descriptors)
  replace idescriptors with (equivocator_descriptors_update IM descriptors i deqv')
    by (destruct oitemx; congruence); clear oitem Hpr.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
descriptors: equivocator_descriptors IM
i: index
li: EquivocatorLabel (IM i)
input, output: option message
idescriptors: equivocator_descriptors IM
s: composite_state (equivocator_IM IM)
si': equivocator_state (IM i)
Hdescriptors: proper_equivocator_descriptors IM
  descriptors
  (state_update (equivocator_IM IM) s i
     si')
Htei: equivocator_transition (IM i) li (s i, input) =
(si', output)
Hv: equivocator_valid (IM i) li (s i, input)
oitemx: option transition_item
deqv': MachineDescriptor (IM i)
Hitem_pr: equivocator_vlsm_transition_item_project
  (IM i)
  {|
    l := li;
    input := input;
    destination :=
      state_update (equivocator_IM IM) s i
        si' i;
    output := output
  |} (descriptors i) = 
Some (oitemx, deqv')
set_union (equivocating_indices IM (enum index) s)
  (newmachine_descriptors_list IM (enum index)
     (equivocator_descriptors_update IM descriptors i
        deqv'))
⊆ set_union
    (equivocating_indices IM (enum index)
       (state_update (equivocator_IM IM) s i si'))
    (newmachine_descriptors_list IM (enum index)
       descriptors)
  intros eqv Heqv.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
descriptors: equivocator_descriptors IM
i: index
li: EquivocatorLabel (IM i)
input, output: option message
idescriptors: equivocator_descriptors IM
s: composite_state (equivocator_IM IM)
si': equivocator_state (IM i)
Hdescriptors: proper_equivocator_descriptors IM
  descriptors
  (state_update (equivocator_IM IM) s i
     si')
Htei: equivocator_transition (IM i) li (s i, input) =
(si', output)
Hv: equivocator_valid (IM i) li (s i, input)
oitemx: option transition_item
deqv': MachineDescriptor (IM i)
Hitem_pr: equivocator_vlsm_transition_item_project
  (IM i)
  {|
    l := li;
    input := input;
    destination :=
      state_update (equivocator_IM IM) s i
        si' i;
    output := output
  |} (descriptors i) = 
Some (oitemx, deqv')
eqv: index
Heqv: eqv
∈ set_union
    (equivocating_indices IM (enum index) s)
    (newmachine_descriptors_list IM
       (enum index)
       (equivocator_descriptors_update IM
          descriptors i deqv'))
eqv
∈ set_union
    (equivocating_indices IM (enum index)
       (state_update (equivocator_IM IM) s i si'))
    (newmachine_descriptors_list IM (enum index)
       descriptors) apply set_union_iff in Heqv.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
descriptors: equivocator_descriptors IM
i: index
li: EquivocatorLabel (IM i)
input, output: option message
idescriptors: equivocator_descriptors IM
s: composite_state (equivocator_IM IM)
si': equivocator_state (IM i)
Hdescriptors: proper_equivocator_descriptors IM
  descriptors
  (state_update (equivocator_IM IM) s i
     si')
Htei: equivocator_transition (IM i) li (s i, input) =
(si', output)
Hv: equivocator_valid (IM i) li (s i, input)
oitemx: option transition_item
deqv': MachineDescriptor (IM i)
Hitem_pr: equivocator_vlsm_transition_item_project
  (IM i)
  {|
    l := li;
    input := input;
    destination :=
      state_update (equivocator_IM IM) s i
        si' i;
    output := output
  |} (descriptors i) = 
Some (oitemx, deqv')
eqv: index
Heqv: eqv ∈ equivocating_indices IM (enum index) s
∨ eqv
  ∈ newmachine_descriptors_list IM 
      (enum index)
      (equivocator_descriptors_update IM
         descriptors i deqv')
eqv
∈ set_union
    (equivocating_indices IM (enum index)
       (state_update (equivocator_IM IM) s i si'))
    (newmachine_descriptors_list IM (enum index)
       descriptors) apply set_union_iff.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
descriptors: equivocator_descriptors IM
i: index
li: EquivocatorLabel (IM i)
input, output: option message
idescriptors: equivocator_descriptors IM
s: composite_state (equivocator_IM IM)
si': equivocator_state (IM i)
Hdescriptors: proper_equivocator_descriptors IM
  descriptors
  (state_update (equivocator_IM IM) s i
     si')
Htei: equivocator_transition (IM i) li (s i, input) =
(si', output)
Hv: equivocator_valid (IM i) li (s i, input)
oitemx: option transition_item
deqv': MachineDescriptor (IM i)
Hitem_pr: equivocator_vlsm_transition_item_project
  (IM i)
  {|
    l := li;
    input := input;
    destination :=
      state_update (equivocator_IM IM) s i
        si' i;
    output := output
  |} (descriptors i) = 
Some (oitemx, deqv')
eqv: index
Heqv: eqv ∈ equivocating_indices IM (enum index) s
∨ eqv
  ∈ newmachine_descriptors_list IM 
      (enum index)
      (equivocator_descriptors_update IM
         descriptors i deqv')
eqv
∈ equivocating_indices IM (enum index)
    (state_update (equivocator_IM IM) s i si')
∨ eqv
  ∈ newmachine_descriptors_list IM (enum index)
      descriptors
  destruct (decide (eqv = i)).message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
descriptors: equivocator_descriptors IM
i: index
li: EquivocatorLabel (IM i)
input, output: option message
idescriptors: equivocator_descriptors IM
s: composite_state (equivocator_IM IM)
si': equivocator_state (IM i)
Hdescriptors: proper_equivocator_descriptors IM
  descriptors
  (state_update (equivocator_IM IM) s i
     si')
Htei: equivocator_transition (IM i) li (s i, input) =
(si', output)
Hv: equivocator_valid (IM i) li (s i, input)
oitemx: option transition_item
deqv': MachineDescriptor (IM i)
Hitem_pr: equivocator_vlsm_transition_item_project
  (IM i)
  {|
    l := li;
    input := input;
    destination :=
      state_update (equivocator_IM IM) s i
        si' i;
    output := output
  |} (descriptors i) = 
Some (oitemx, deqv')
eqv: index
Heqv: eqv ∈ equivocating_indices IM (enum index) s
∨ eqv
  ∈ newmachine_descriptors_list IM 
      (enum index)
      (equivocator_descriptors_update IM
         descriptors i deqv')
e: eqv = i
eqv
∈ equivocating_indices IM (enum index)
    (state_update (equivocator_IM IM) s i si')
∨ eqv
  ∈ newmachine_descriptors_list IM (enum index)
      descriptors
message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
descriptors: equivocator_descriptors IM
i: index
li: EquivocatorLabel (IM i)
input, output: option message
idescriptors: equivocator_descriptors IM
s: composite_state (equivocator_IM IM)
si': equivocator_state (IM i)
Hdescriptors: proper_equivocator_descriptors IM
  descriptors
  (state_update (equivocator_IM IM) s i
     si')
Htei: equivocator_transition (IM i) li (s i, input) =
(si', output)
Hv: equivocator_valid (IM i) li (s i, input)
oitemx: option transition_item
deqv': MachineDescriptor (IM i)
Hitem_pr: equivocator_vlsm_transition_item_project
  (IM i)
  {|
    l := li;
    input := input;
    destination :=
      state_update (equivocator_IM IM) s i
        si' i;
    output := output
  |} (descriptors i) = 
Some (oitemx, deqv')
eqv: index
Heqv: eqv ∈ equivocating_indices IM (enum index) s
∨ eqv
  ∈ newmachine_descriptors_list IM 
      (enum index)
      (equivocator_descriptors_update IM
         descriptors i deqv')
n: eqv ≠ i
eqv
∈ equivocating_indices IM 
    (enum index)
    (state_update (equivocator_IM IM) s i si')
∨ eqv
  ∈ newmachine_descriptors_list IM 
      (enum index) descriptors
  -message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
descriptors: equivocator_descriptors IM
i: index
li: EquivocatorLabel (IM i)
input, output: option message
idescriptors: equivocator_descriptors IM
s: composite_state (equivocator_IM IM)
si': equivocator_state (IM i)
Hdescriptors: proper_equivocator_descriptors IM
  descriptors
  (state_update (equivocator_IM IM) s i
     si')
Htei: equivocator_transition (IM i) li (s i, input) =
(si', output)
Hv: equivocator_valid (IM i) li (s i, input)
oitemx: option transition_item
deqv': MachineDescriptor (IM i)
Hitem_pr: equivocator_vlsm_transition_item_project
  (IM i)
  {|
    l := li;
    input := input;
    destination :=
      state_update (equivocator_IM IM) s i
        si' i;
    output := output
  |} (descriptors i) = 
Some (oitemx, deqv')
eqv: index
Heqv: eqv ∈ equivocating_indices IM (enum index) s
∨ eqv
  ∈ newmachine_descriptors_list IM 
      (enum index)
      (equivocator_descriptors_update IM
         descriptors i deqv')
e: eqv = i
eqv
∈ equivocating_indices IM (enum index)
    (state_update (equivocator_IM IM) s i si')
∨ eqv
  ∈ newmachine_descriptors_list IM (enum index)
      descriptors subst i.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
descriptors: equivocator_descriptors IM
eqv: index
s: composite_state (equivocator_IM IM)
deqv': MachineDescriptor (IM eqv)
Heqv: eqv ∈ equivocating_indices IM (enum index) s
∨ eqv
  ∈ newmachine_descriptors_list IM 
      (enum index)
      (equivocator_descriptors_update IM
         descriptors eqv deqv')
li: EquivocatorLabel (IM eqv)
input, output: option message
idescriptors: equivocator_descriptors IM
oitemx: option transition_item
si': equivocator_state (IM eqv)
Hitem_pr: equivocator_vlsm_transition_item_project
  (IM eqv)
  {|
    l := li;
    input := input;
    destination :=
      state_update (equivocator_IM IM) s
        eqv si' eqv;
    output := output
  |} (descriptors eqv) =
Some (oitemx, deqv')
Hv: equivocator_valid (IM eqv) li (s eqv, input)
Htei: equivocator_transition 
  (IM eqv) li (s eqv, input) = (
si', output)
Hdescriptors: proper_equivocator_descriptors IM
  descriptors
  (state_update (equivocator_IM IM) s
     eqv si')
eqv
∈ equivocating_indices IM (enum index)
    (state_update (equivocator_IM IM) s eqv si')
∨ eqv
  ∈ newmachine_descriptors_list IM (enum index)
      descriptors
    unfold equivocating_indices in *.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
descriptors: equivocator_descriptors IM
eqv: index
s: composite_state (equivocator_IM IM)
deqv': MachineDescriptor (IM eqv)
Heqv: eqv
∈ filter
    (λ i : index,
       is_equivocating_state (IM i) (s i))
    (enum index)
∨ eqv
  ∈ newmachine_descriptors_list IM 
      (enum index)
      (equivocator_descriptors_update IM
         descriptors eqv deqv')
li: EquivocatorLabel (IM eqv)
input, output: option message
idescriptors: equivocator_descriptors IM
oitemx: option transition_item
si': equivocator_state (IM eqv)
Hitem_pr: equivocator_vlsm_transition_item_project
  (IM eqv)
  {|
    l := li;
    input := input;
    destination :=
      state_update (equivocator_IM IM) s
        eqv si' eqv;
    output := output
  |} (descriptors eqv) =
Some (oitemx, deqv')
Hv: equivocator_valid (IM eqv) li (s eqv, input)
Htei: equivocator_transition 
  (IM eqv) li (s eqv, input) = (
si', output)
Hdescriptors: proper_equivocator_descriptors IM
  descriptors
  (state_update (equivocator_IM IM) s
     eqv si')
eqv
∈ filter
    (λ i : index,
       is_equivocating_state (IM i)
         (state_update (equivocator_IM IM) s eqv si' i))
    (enum index)
∨ eqv
  ∈ newmachine_descriptors_list IM (enum index)
      descriptors
    unfold newmachine_descriptors_list in *.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
descriptors: equivocator_descriptors IM
eqv: index
s: composite_state (equivocator_IM IM)
deqv': MachineDescriptor (IM eqv)
Heqv: eqv
∈ filter
    (λ i : index,
       is_equivocating_state (IM i) (s i))
    (enum index)
∨ eqv
  ∈ filter
      (λ i : index,
         is_newmachine_descriptor 
           (IM i)
           (equivocator_descriptors_update IM
              descriptors eqv deqv' i))
      (enum index)
li: EquivocatorLabel (IM eqv)
input, output: option message
idescriptors: equivocator_descriptors IM
oitemx: option transition_item
si': equivocator_state (IM eqv)
Hitem_pr: equivocator_vlsm_transition_item_project
  (IM eqv)
  {|
    l := li;
    input := input;
    destination :=
      state_update (equivocator_IM IM) s
        eqv si' eqv;
    output := output
  |} (descriptors eqv) =
Some (oitemx, deqv')
Hv: equivocator_valid (IM eqv) li (s eqv, input)
Htei: equivocator_transition 
  (IM eqv) li (s eqv, input) = (
si', output)
Hdescriptors: proper_equivocator_descriptors IM
  descriptors
  (state_update (equivocator_IM IM) s
     eqv si')
eqv
∈ filter
    (λ i : index,
       is_equivocating_state (IM i)
         (state_update (equivocator_IM IM) s eqv si' i))
    (enum index)
∨ eqv
  ∈ filter
      (λ i : index,
         is_newmachine_descriptor (IM i)
           (descriptors i)) (enum index)
    rewrite! elem_of_list_filter in *.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
descriptors: equivocator_descriptors IM
eqv: index
s: composite_state (equivocator_IM IM)
deqv': MachineDescriptor (IM eqv)
Heqv: is_equivocating_state (IM eqv) (s eqv)
∧ eqv ∈ enum index
∨ is_newmachine_descriptor 
    (IM eqv)
    (equivocator_descriptors_update IM
       descriptors eqv deqv' eqv)
  ∧ eqv ∈ enum index
li: EquivocatorLabel (IM eqv)
input, output: option message
idescriptors: equivocator_descriptors IM
oitemx: option transition_item
si': equivocator_state (IM eqv)
Hitem_pr: equivocator_vlsm_transition_item_project
  (IM eqv)
  {|
    l := li;
    input := input;
    destination :=
      state_update (equivocator_IM IM) s
        eqv si' eqv;
    output := output
  |} (descriptors eqv) =
Some (oitemx, deqv')
Hv: equivocator_valid (IM eqv) li (s eqv, input)
Htei: equivocator_transition 
  (IM eqv) li (s eqv, input) = (
si', output)
Hdescriptors: proper_equivocator_descriptors IM
  descriptors
  (state_update (equivocator_IM IM) s
     eqv si')
is_equivocating_state (IM eqv)
  (state_update (equivocator_IM IM) s eqv si' eqv)
∧ eqv ∈ enum index
∨ is_newmachine_descriptor (IM eqv) (descriptors eqv)
  ∧ eqv ∈ enum index
    specialize (Hdescriptors eqv).message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
descriptors: equivocator_descriptors IM
eqv: index
s: composite_state (equivocator_IM IM)
deqv': MachineDescriptor (IM eqv)
Heqv: is_equivocating_state (IM eqv) (s eqv)
∧ eqv ∈ enum index
∨ is_newmachine_descriptor 
    (IM eqv)
    (equivocator_descriptors_update IM
       descriptors eqv deqv' eqv)
  ∧ eqv ∈ enum index
li: EquivocatorLabel (IM eqv)
input, output: option message
idescriptors: equivocator_descriptors IM
oitemx: option transition_item
si': equivocator_state (IM eqv)
Hitem_pr: equivocator_vlsm_transition_item_project
  (IM eqv)
  {|
    l := li;
    input := input;
    destination :=
      state_update (equivocator_IM IM) s
        eqv si' eqv;
    output := output
  |} (descriptors eqv) =
Some (oitemx, deqv')
Hv: equivocator_valid (IM eqv) li (s eqv, input)
Htei: equivocator_transition 
  (IM eqv) li (s eqv, input) = (
si', output)
Hdescriptors: proper_descriptor (IM eqv)
  (descriptors eqv)
  (state_update (equivocator_IM IM) s
     eqv si' eqv)
is_equivocating_state (IM eqv)
  (state_update (equivocator_IM IM) s eqv si' eqv)
∧ eqv ∈ enum index
∨ is_newmachine_descriptor (IM eqv) (descriptors eqv)
  ∧ eqv ∈ enum index
    state_update_simpl.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
descriptors: equivocator_descriptors IM
eqv: index
s: composite_state (equivocator_IM IM)
deqv': MachineDescriptor (IM eqv)
Heqv: is_equivocating_state (IM eqv) (s eqv)
∧ eqv ∈ enum index
∨ is_newmachine_descriptor (IM eqv) deqv'
  ∧ eqv ∈ enum index
li: EquivocatorLabel (IM eqv)
input, output: option message
idescriptors: equivocator_descriptors IM
oitemx: option transition_item
si': equivocator_state (IM eqv)
Hitem_pr: equivocator_vlsm_transition_item_project
  (IM eqv)
  {|
    l := li;
    input := input;
    destination := si';
    output := output
  |} (descriptors eqv) =
Some (oitemx, deqv')
Hv: equivocator_valid (IM eqv) li (s eqv, input)
Htei: equivocator_transition 
  (IM eqv) li (s eqv, input) = (
si', output)
Hdescriptors: proper_descriptor (IM eqv)
  (descriptors eqv) si'
is_equivocating_state (IM eqv) si' ∧ eqv ∈ enum index
∨ is_newmachine_descriptor (IM eqv) (descriptors eqv)
  ∧ eqv ∈ enum index
    cut (is_equivocating_state (IM eqv) si' \/  is_newmachine_descriptor (IM eqv) (descriptors eqv));
      [by itauto |].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
descriptors: equivocator_descriptors IM
eqv: index
s: composite_state (equivocator_IM IM)
deqv': MachineDescriptor (IM eqv)
Heqv: is_equivocating_state (IM eqv) (s eqv)
∧ eqv ∈ enum index
∨ is_newmachine_descriptor (IM eqv) deqv'
  ∧ eqv ∈ enum index
li: EquivocatorLabel (IM eqv)
input, output: option message
idescriptors: equivocator_descriptors IM
oitemx: option transition_item
si': equivocator_state (IM eqv)
Hitem_pr: equivocator_vlsm_transition_item_project
  (IM eqv)
  {|
    l := li;
    input := input;
    destination := si';
    output := output
  |} (descriptors eqv) =
Some (oitemx, deqv')
Hv: equivocator_valid (IM eqv) li (s eqv, input)
Htei: equivocator_transition 
  (IM eqv) li (s eqv, input) = (
si', output)
Hdescriptors: proper_descriptor (IM eqv)
  (descriptors eqv) si'
is_equivocating_state (IM eqv) si'
∨ is_newmachine_descriptor (IM eqv) (descriptors eqv)
    eapply (equivocator_transition_item_project_preserves_equivocating_indices)
      in Hitem_pr; [done.. |].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
descriptors: equivocator_descriptors IM
eqv: index
s: composite_state (equivocator_IM IM)
deqv': MachineDescriptor (IM eqv)
Heqv: is_equivocating_state (IM eqv) (s eqv)
∧ eqv ∈ enum index
∨ is_newmachine_descriptor (IM eqv) deqv'
  ∧ eqv ∈ enum index
li: EquivocatorLabel (IM eqv)
input, output: option message
idescriptors: equivocator_descriptors IM
oitemx: option transition_item
si': equivocator_state (IM eqv)
Hitem_pr: equivocator_vlsm_transition_item_project
  (IM eqv)
  {|
    l := li;
    input := input;
    destination := si';
    output := output
  |} (descriptors eqv) =
Some (oitemx, deqv')
Hv: equivocator_valid (IM eqv) li (s eqv, input)
Htei: equivocator_transition 
  (IM eqv) li (s eqv, input) = (
si', output)
Hdescriptors: proper_descriptor (IM eqv)
  (descriptors eqv) si'
is_equivocating_state (IM eqv) (s eqv)
∨ is_newmachine_descriptor (IM eqv) deqv'
    by itauto.
  -message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
descriptors: equivocator_descriptors IM
i: index
li: EquivocatorLabel (IM i)
input, output: option message
idescriptors: equivocator_descriptors IM
s: composite_state (equivocator_IM IM)
si': equivocator_state (IM i)
Hdescriptors: proper_equivocator_descriptors IM
  descriptors
  (state_update (equivocator_IM IM) s i
     si')
Htei: equivocator_transition (IM i) li (s i, input) =
(si', output)
Hv: equivocator_valid (IM i) li (s i, input)
oitemx: option transition_item
deqv': MachineDescriptor (IM i)
Hitem_pr: equivocator_vlsm_transition_item_project
  (IM i)
  {|
    l := li;
    input := input;
    destination :=
      state_update (equivocator_IM IM) s i
        si' i;
    output := output
  |} (descriptors i) = 
Some (oitemx, deqv')
eqv: index
Heqv: eqv ∈ equivocating_indices IM (enum index) s
∨ eqv
  ∈ newmachine_descriptors_list IM 
      (enum index)
      (equivocator_descriptors_update IM
         descriptors i deqv')
n: eqv ≠ i
eqv
∈ equivocating_indices IM (enum index)
    (state_update (equivocator_IM IM) s i si')
∨ eqv
  ∈ newmachine_descriptors_list IM (enum index)
      descriptors destruct Heqv as [Heqv | Heqv]
    ; apply elem_of_list_filter in Heqv as [Heqv Hin].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
descriptors: equivocator_descriptors IM
i: index
li: EquivocatorLabel (IM i)
input, output: option message
idescriptors: equivocator_descriptors IM
s: composite_state (equivocator_IM IM)
si': equivocator_state (IM i)
Hdescriptors: proper_equivocator_descriptors IM
  descriptors
  (state_update (equivocator_IM IM) s i
     si')
Htei: equivocator_transition (IM i) li (s i, input) =
(si', output)
Hv: equivocator_valid (IM i) li (s i, input)
oitemx: option transition_item
deqv': MachineDescriptor (IM i)
Hitem_pr: equivocator_vlsm_transition_item_project
  (IM i)
  {|
    l := li;
    input := input;
    destination :=
      state_update (equivocator_IM IM) s i
        si' i;
    output := output
  |} (descriptors i) = 
Some (oitemx, deqv')
eqv: index
Heqv: is_equivocating_state (IM eqv) (s eqv)
Hin: eqv ∈ enum index
n: eqv ≠ i
eqv
∈ equivocating_indices IM (enum index)
    (state_update (equivocator_IM IM) s i si')
∨ eqv
  ∈ newmachine_descriptors_list IM (enum index)
      descriptors
message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
descriptors: equivocator_descriptors IM
i: index
li: EquivocatorLabel (IM i)
input, output: option message
idescriptors: equivocator_descriptors IM
s: composite_state (equivocator_IM IM)
si': equivocator_state (IM i)
Hdescriptors: proper_equivocator_descriptors IM
  descriptors
  (state_update (equivocator_IM IM) s i
     si')
Htei: equivocator_transition (IM i) li (s i, input) =
(si', output)
Hv: equivocator_valid (IM i) li (s i, input)
oitemx: option transition_item
deqv': MachineDescriptor (IM i)
Hitem_pr: equivocator_vlsm_transition_item_project
  (IM i)
  {|
    l := li;
    input := input;
    destination :=
      state_update (equivocator_IM IM) s i
        si' i;
    output := output
  |} (descriptors i) = 
Some (oitemx, deqv')
eqv: index
Heqv: is_newmachine_descriptor 
  (IM eqv)
  (equivocator_descriptors_update IM
     descriptors i deqv' eqv)
Hin: eqv ∈ enum index
n: eqv ≠ i
eqv
∈ equivocating_indices IM 
    (enum index)
    (state_update (equivocator_IM IM) s i si')
∨ eqv
  ∈ newmachine_descriptors_list IM 
      (enum index) descriptors
    +message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
descriptors: equivocator_descriptors IM
i: index
li: EquivocatorLabel (IM i)
input, output: option message
idescriptors: equivocator_descriptors IM
s: composite_state (equivocator_IM IM)
si': equivocator_state (IM i)
Hdescriptors: proper_equivocator_descriptors IM
  descriptors
  (state_update (equivocator_IM IM) s i
     si')
Htei: equivocator_transition (IM i) li (s i, input) =
(si', output)
Hv: equivocator_valid (IM i) li (s i, input)
oitemx: option transition_item
deqv': MachineDescriptor (IM i)
Hitem_pr: equivocator_vlsm_transition_item_project
  (IM i)
  {|
    l := li;
    input := input;
    destination :=
      state_update (equivocator_IM IM) s i
        si' i;
    output := output
  |} (descriptors i) = 
Some (oitemx, deqv')
eqv: index
Heqv: is_equivocating_state (IM eqv) (s eqv)
Hin: eqv ∈ enum index
n: eqv ≠ i
eqv
∈ equivocating_indices IM (enum index)
    (state_update (equivocator_IM IM) s i si')
∨ eqv
  ∈ newmachine_descriptors_list IM (enum index)
      descriptors left.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
descriptors: equivocator_descriptors IM
i: index
li: EquivocatorLabel (IM i)
input, output: option message
idescriptors: equivocator_descriptors IM
s: composite_state (equivocator_IM IM)
si': equivocator_state (IM i)
Hdescriptors: proper_equivocator_descriptors IM
  descriptors
  (state_update (equivocator_IM IM) s i
     si')
Htei: equivocator_transition (IM i) li (s i, input) =
(si', output)
Hv: equivocator_valid (IM i) li (s i, input)
oitemx: option transition_item
deqv': MachineDescriptor (IM i)
Hitem_pr: equivocator_vlsm_transition_item_project
  (IM i)
  {|
    l := li;
    input := input;
    destination :=
      state_update (equivocator_IM IM) s i
        si' i;
    output := output
  |} (descriptors i) = 
Some (oitemx, deqv')
eqv: index
Heqv: is_equivocating_state (IM eqv) (s eqv)
Hin: eqv ∈ enum index
n: eqv ≠ i
eqv
∈ equivocating_indices IM (enum index)
    (state_update (equivocator_IM IM) s i si')
      apply elem_of_list_filter.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
descriptors: equivocator_descriptors IM
i: index
li: EquivocatorLabel (IM i)
input, output: option message
idescriptors: equivocator_descriptors IM
s: composite_state (equivocator_IM IM)
si': equivocator_state (IM i)
Hdescriptors: proper_equivocator_descriptors IM
  descriptors
  (state_update (equivocator_IM IM) s i
     si')
Htei: equivocator_transition (IM i) li (s i, input) =
(si', output)
Hv: equivocator_valid (IM i) li (s i, input)
oitemx: option transition_item
deqv': MachineDescriptor (IM i)
Hitem_pr: equivocator_vlsm_transition_item_project
  (IM i)
  {|
    l := li;
    input := input;
    destination :=
      state_update (equivocator_IM IM) s i
        si' i;
    output := output
  |} (descriptors i) = 
Some (oitemx, deqv')
eqv: index
Heqv: is_equivocating_state (IM eqv) (s eqv)
Hin: eqv ∈ enum index
n: eqv ≠ i
is_equivocating_state (IM eqv)
  (state_update (equivocator_IM IM) s i si' eqv)
∧ eqv ∈ enum index
      by state_update_simpl.
    +message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
descriptors: equivocator_descriptors IM
i: index
li: EquivocatorLabel (IM i)
input, output: option message
idescriptors: equivocator_descriptors IM
s: composite_state (equivocator_IM IM)
si': equivocator_state (IM i)
Hdescriptors: proper_equivocator_descriptors IM
  descriptors
  (state_update (equivocator_IM IM) s i
     si')
Htei: equivocator_transition (IM i) li (s i, input) =
(si', output)
Hv: equivocator_valid (IM i) li (s i, input)
oitemx: option transition_item
deqv': MachineDescriptor (IM i)
Hitem_pr: equivocator_vlsm_transition_item_project
  (IM i)
  {|
    l := li;
    input := input;
    destination :=
      state_update (equivocator_IM IM) s i
        si' i;
    output := output
  |} (descriptors i) = 
Some (oitemx, deqv')
eqv: index
Heqv: is_newmachine_descriptor 
  (IM eqv)
  (equivocator_descriptors_update IM
     descriptors i deqv' eqv)
Hin: eqv ∈ enum index
n: eqv ≠ i
eqv
∈ equivocating_indices IM (enum index)
    (state_update (equivocator_IM IM) s i si')
∨ eqv
  ∈ newmachine_descriptors_list IM (enum index)
      descriptors right.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
descriptors: equivocator_descriptors IM
i: index
li: EquivocatorLabel (IM i)
input, output: option message
idescriptors: equivocator_descriptors IM
s: composite_state (equivocator_IM IM)
si': equivocator_state (IM i)
Hdescriptors: proper_equivocator_descriptors IM
  descriptors
  (state_update (equivocator_IM IM) s i
     si')
Htei: equivocator_transition (IM i) li (s i, input) =
(si', output)
Hv: equivocator_valid (IM i) li (s i, input)
oitemx: option transition_item
deqv': MachineDescriptor (IM i)
Hitem_pr: equivocator_vlsm_transition_item_project
  (IM i)
  {|
    l := li;
    input := input;
    destination :=
      state_update (equivocator_IM IM) s i
        si' i;
    output := output
  |} (descriptors i) = 
Some (oitemx, deqv')
eqv: index
Heqv: is_newmachine_descriptor 
  (IM eqv)
  (equivocator_descriptors_update IM
     descriptors i deqv' eqv)
Hin: eqv ∈ enum index
n: eqv ≠ i
eqv
∈ newmachine_descriptors_list IM (enum index)
    descriptors
      apply elem_of_list_filter.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
descriptors: equivocator_descriptors IM
i: index
li: EquivocatorLabel (IM i)
input, output: option message
idescriptors: equivocator_descriptors IM
s: composite_state (equivocator_IM IM)
si': equivocator_state (IM i)
Hdescriptors: proper_equivocator_descriptors IM
  descriptors
  (state_update (equivocator_IM IM) s i
     si')
Htei: equivocator_transition (IM i) li (s i, input) =
(si', output)
Hv: equivocator_valid (IM i) li (s i, input)
oitemx: option transition_item
deqv': MachineDescriptor (IM i)
Hitem_pr: equivocator_vlsm_transition_item_project
  (IM i)
  {|
    l := li;
    input := input;
    destination :=
      state_update (equivocator_IM IM) s i
        si' i;
    output := output
  |} (descriptors i) = 
Some (oitemx, deqv')
eqv: index
Heqv: is_newmachine_descriptor 
  (IM eqv)
  (equivocator_descriptors_update IM
     descriptors i deqv' eqv)
Hin: eqv ∈ enum index
n: eqv ≠ i
is_newmachine_descriptor (IM eqv) (descriptors eqv)
∧ eqv ∈ enum index
      by state_update_simpl.
Qed.

zero_descriptors are preserved when projecting transition_items of the composition of equivocators.

Lemma equivocators_transition_item_project_preserves_zero_descriptors
  (descriptors : equivocator_descriptors IM)
  (item : composite_transition_item (equivocator_IM IM))
  oitem idescriptors
  s
  (Ht : composite_transition (equivocator_IM IM) (l item) (s, input item) =
          (destination item, output item))
  (Hv : composite_valid (equivocator_IM IM) (l item) (s, input item))
  (Hpr : equivocators_transition_item_project descriptors item = Some (oitem, idescriptors))
  : forall i, descriptors i = Existing 0 -> idescriptors i = Existing 0.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
descriptors: equivocator_descriptors IM
item: composite_transition_item (equivocator_IM IM)
oitem: option (composite_transition_item IM)
idescriptors: equivocator_descriptors IM
s: composite_state (equivocator_IM IM)
Ht: composite_transition (equivocator_IM IM) 
  (l item) (s, input item) =
(destination item, output item)
Hv: composite_valid (equivocator_IM IM) 
  (l item) (s, input item)
Hpr: equivocators_transition_item_project descriptors
  item = Some (oitem, idescriptors)
∀ i : index,
  descriptors i = Existing 0
  → idescriptors i = Existing 0
Proof.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
descriptors: equivocator_descriptors IM
item: composite_transition_item (equivocator_IM IM)
oitem: option (composite_transition_item IM)
idescriptors: equivocator_descriptors IM
s: composite_state (equivocator_IM IM)
Ht: composite_transition (equivocator_IM IM) 
  (l item) (s, input item) =
(destination item, output item)
Hv: composite_valid (equivocator_IM IM) 
  (l item) (s, input item)
Hpr: equivocators_transition_item_project descriptors
  item = Some (oitem, idescriptors)
∀ i : index,
  descriptors i = Existing 0
  → idescriptors i = Existing 0
  intros i Hi.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
descriptors: equivocator_descriptors IM
item: composite_transition_item (equivocator_IM IM)
oitem: option (composite_transition_item IM)
idescriptors: equivocator_descriptors IM
s: composite_state (equivocator_IM IM)
Ht: composite_transition (equivocator_IM IM) 
  (l item) (s, input item) =
(destination item, output item)
Hv: composite_valid (equivocator_IM IM) 
  (l item) (s, input item)
Hpr: equivocators_transition_item_project descriptors
  item = Some (oitem, idescriptors)
i: index
Hi: descriptors i = Existing 0
idescriptors i = Existing 0
  unfold equivocators_transition_item_project in Hpr.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
descriptors: equivocator_descriptors IM
item: composite_transition_item (equivocator_IM IM)
oitem: option (composite_transition_item IM)
idescriptors: equivocator_descriptors IM
s: composite_state (equivocator_IM IM)
Ht: composite_transition (equivocator_IM IM) 
  (l item) (s, input item) =
(destination item, output item)
Hv: composite_valid (equivocator_IM IM) 
  (l item) (s, input item)
Hpr: match
  equivocator_vlsm_transition_item_project
    (IM (projT1 (l item)))
    (composite_transition_item_projection
       (equivocator_IM IM) item)
    (descriptors (projT1 (l item)))
with
| Some (Some item', deqv') =>
    Some
      (Some
         {|
           l :=
             existT (projT1 (l item)) (l item');
           input := input item;
           destination :=
             equivocators_state_project IM
               descriptors 
               (destination item);
           output := output item
         |},
       equivocator_descriptors_update IM
         descriptors (projT1 (l item)) deqv')
| Some (None, deqv') =>
    Some
      (None,
       equivocator_descriptors_update IM
         descriptors (projT1 (l item)) deqv')
| None => None
end = Some (oitem, idescriptors)
i: index
Hi: descriptors i = Existing 0
idescriptors i = Existing 0
  destruct (decide (i = projT1 (l item))).message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
descriptors: equivocator_descriptors IM
item: composite_transition_item (equivocator_IM IM)
oitem: option (composite_transition_item IM)
idescriptors: equivocator_descriptors IM
s: composite_state (equivocator_IM IM)
Ht: composite_transition (equivocator_IM IM) 
  (l item) (s, input item) =
(destination item, output item)
Hv: composite_valid (equivocator_IM IM) 
  (l item) (s, input item)
Hpr: match
  equivocator_vlsm_transition_item_project
    (IM (projT1 (l item)))
    (composite_transition_item_projection
       (equivocator_IM IM) item)
    (descriptors (projT1 (l item)))
with
| Some (Some item', deqv') =>
    Some
      (Some
         {|
           l :=
             existT (projT1 (l item)) (l item');
           input := input item;
           destination :=
             equivocators_state_project IM
               descriptors 
               (destination item);
           output := output item
         |},
       equivocator_descriptors_update IM
         descriptors (projT1 (l item)) deqv')
| Some (None, deqv') =>
    Some
      (None,
       equivocator_descriptors_update IM
         descriptors (projT1 (l item)) deqv')
| None => None
end = Some (oitem, idescriptors)
i: index
Hi: descriptors i = Existing 0
e: i = projT1 (l item)
idescriptors i = Existing 0
message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
descriptors: equivocator_descriptors IM
item: composite_transition_item (equivocator_IM IM)
oitem: option (composite_transition_item IM)
idescriptors: equivocator_descriptors IM
s: composite_state (equivocator_IM IM)
Ht: composite_transition (equivocator_IM IM) 
  (l item) (s, input item) =
(destination item, output item)
Hv: composite_valid (equivocator_IM IM) 
  (l item) (s, input item)
Hpr: match
  equivocator_vlsm_transition_item_project
    (IM (projT1 (l item)))
    (composite_transition_item_projection
       (equivocator_IM IM) item)
    (descriptors (projT1 (l item)))
with
| Some (Some item', deqv') =>
    Some
      (Some
         {|
           l :=
             existT (projT1 (l item)) (l item');
           input := input item;
           destination :=
             equivocators_state_project IM
               descriptors 
               (destination item);
           output := output item
         |},
       equivocator_descriptors_update IM
         descriptors (projT1 (l item)) deqv')
| Some (None, deqv') =>
    Some
      (None,
       equivocator_descriptors_update IM
         descriptors (projT1 (l item)) deqv')
| None => None
end = Some (oitem, idescriptors)
i: index
Hi: descriptors i = Existing 0
n: i ≠ projT1 (l item)
idescriptors i = Existing 0
  -message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
descriptors: equivocator_descriptors IM
item: composite_transition_item (equivocator_IM IM)
oitem: option (composite_transition_item IM)
idescriptors: equivocator_descriptors IM
s: composite_state (equivocator_IM IM)
Ht: composite_transition (equivocator_IM IM) 
  (l item) (s, input item) =
(destination item, output item)
Hv: composite_valid (equivocator_IM IM) 
  (l item) (s, input item)
Hpr: match
  equivocator_vlsm_transition_item_project
    (IM (projT1 (l item)))
    (composite_transition_item_projection
       (equivocator_IM IM) item)
    (descriptors (projT1 (l item)))
with
| Some (Some item', deqv') =>
    Some
      (Some
         {|
           l :=
             existT (projT1 (l item)) (l item');
           input := input item;
           destination :=
             equivocators_state_project IM
               descriptors 
               (destination item);
           output := output item
         |},
       equivocator_descriptors_update IM
         descriptors (projT1 (l item)) deqv')
| Some (None, deqv') =>
    Some
      (None,
       equivocator_descriptors_update IM
         descriptors (projT1 (l item)) deqv')
| None => None
end = Some (oitem, idescriptors)
i: index
Hi: descriptors i = Existing 0
e: i = projT1 (l item)
idescriptors i = Existing 0 subst i.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
descriptors: equivocator_descriptors IM
item: composite_transition_item (equivocator_IM IM)
oitem: option (composite_transition_item IM)
idescriptors: equivocator_descriptors IM
s: composite_state (equivocator_IM IM)
Ht: composite_transition (equivocator_IM IM) 
  (l item) (s, input item) =
(destination item, output item)
Hv: composite_valid (equivocator_IM IM) 
  (l item) (s, input item)
Hpr: match
  equivocator_vlsm_transition_item_project
    (IM (projT1 (l item)))
    (composite_transition_item_projection
       (equivocator_IM IM) item)
    (descriptors (projT1 (l item)))
with
| Some (Some item', deqv') =>
    Some
      (Some
         {|
           l :=
             existT (projT1 (l item)) (l item');
           input := input item;
           destination :=
             equivocators_state_project IM
               descriptors 
               (destination item);
           output := output item
         |},
       equivocator_descriptors_update IM
         descriptors (projT1 (l item)) deqv')
| Some (None, deqv') =>
    Some
      (None,
       equivocator_descriptors_update IM
         descriptors (projT1 (l item)) deqv')
| None => None
end = Some (oitem, idescriptors)
Hi: descriptors (projT1 (l item)) = Existing 0
idescriptors (projT1 (l item)) = Existing 0 rewrite Hi in Hpr.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
descriptors: equivocator_descriptors IM
item: composite_transition_item (equivocator_IM IM)
oitem: option (composite_transition_item IM)
idescriptors: equivocator_descriptors IM
s: composite_state (equivocator_IM IM)
Ht: composite_transition (equivocator_IM IM) 
  (l item) (s, input item) =
(destination item, output item)
Hv: composite_valid (equivocator_IM IM) 
  (l item) (s, input item)
Hpr: match
  equivocator_vlsm_transition_item_project
    (IM (projT1 (l item)))
    (composite_transition_item_projection
       (equivocator_IM IM) item) 
    (Existing 0)
with
| Some (Some item', deqv') =>
    Some
      (Some
         {|
           l :=
             existT (projT1 (l item)) (l item');
           input := input item;
           destination :=
             equivocators_state_project IM
               descriptors 
               (destination item);
           output := output item
         |},
       equivocator_descriptors_update IM
         descriptors (projT1 (l item)) deqv')
| Some (None, deqv') =>
    Some
      (None,
       equivocator_descriptors_update IM
         descriptors (projT1 (l item)) deqv')
| None => None
end = Some (oitem, idescriptors)
Hi: descriptors (projT1 (l item)) = Existing 0
idescriptors (projT1 (l item)) = Existing 0
    specialize
      (equivocators_vlsm_transition_item_project_zero_descriptor (IM (projT1 (l item)))
        (composite_transition_item_projection (equivocator_IM IM) item)
        (s (projT1 (l item))))
      as Hpr_item.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
descriptors: equivocator_descriptors IM
item: composite_transition_item (equivocator_IM IM)
oitem: option (composite_transition_item IM)
idescriptors: equivocator_descriptors IM
s: composite_state (equivocator_IM IM)
Ht: composite_transition (equivocator_IM IM) 
  (l item) (s, input item) =
(destination item, output item)
Hv: composite_valid (equivocator_IM IM) 
  (l item) (s, input item)
Hpr: match
  equivocator_vlsm_transition_item_project
    (IM (projT1 (l item)))
    (composite_transition_item_projection
       (equivocator_IM IM) item) 
    (Existing 0)
with
| Some (Some item', deqv') =>
    Some
      (Some
         {|
           l :=
             existT (projT1 (l item)) (l item');
           input := input item;
           destination :=
             equivocators_state_project IM
               descriptors 
               (destination item);
           output := output item
         |},
       equivocator_descriptors_update IM
         descriptors (projT1 (l item)) deqv')
| Some (None, deqv') =>
    Some
      (None,
       equivocator_descriptors_update IM
         descriptors (projT1 (l item)) deqv')
| None => None
end = Some (oitem, idescriptors)
Hi: descriptors (projT1 (l item)) = Existing 0
Hpr_item: transition
  (l
     (composite_transition_item_projection
        (equivocator_IM IM) item))
  (s (projT1 (l item)),
   input
     (composite_transition_item_projection
        (equivocator_IM IM) item)) =
(destination
   (composite_transition_item_projection
      (equivocator_IM IM) item),
 output
   (composite_transition_item_projection
      (equivocator_IM IM) item))
→ valid
    (l
       (composite_transition_item_projection
          (equivocator_IM IM) item))
    (s (projT1 (l item)),
     input
       (composite_transition_item_projection
          (equivocator_IM IM) item))
  → ∃ oitem : option transition_item,
      equivocator_vlsm_transition_item_project
        (IM (projT1 (l item)))
        (composite_transition_item_projection
           (equivocator_IM IM) item)
        (Existing 0) =
      Some (oitem, Existing 0)
idescriptors (projT1 (l item)) = Existing 0
    remember (composite_transition_item_projection (equivocator_IM IM) item) as pr_item.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
descriptors: equivocator_descriptors IM
item: composite_transition_item (equivocator_IM IM)
oitem: option (composite_transition_item IM)
idescriptors: equivocator_descriptors IM
s: composite_state (equivocator_IM IM)
Ht: composite_transition (equivocator_IM IM) 
  (l item) (s, input item) =
(destination item, output item)
Hv: composite_valid (equivocator_IM IM) 
  (l item) (s, input item)
pr_item: let i := projT1 (l item) in transition_item
Heqpr_item: pr_item =
composite_transition_item_projection
  (equivocator_IM IM) item
Hpr: match
  equivocator_vlsm_transition_item_project
    (IM (projT1 (l item))) pr_item 
    (Existing 0)
with
| Some (Some item', deqv') =>
    Some
      (Some
         {|
           l :=
             existT (projT1 (l item)) (l item');
           input := input item;
           destination :=
             equivocators_state_project IM
               descriptors 
               (destination item);
           output := output item
         |},
       equivocator_descriptors_update IM
         descriptors (projT1 (l item)) deqv')
| Some (None, deqv') =>
    Some
      (None,
       equivocator_descriptors_update IM
         descriptors (projT1 (l item)) deqv')
| None => None
end = Some (oitem, idescriptors)
Hi: descriptors (projT1 (l item)) = Existing 0
Hpr_item: transition (l pr_item)
  (s (projT1 (l item)), input pr_item) =
(destination pr_item, output pr_item)
→ valid (l pr_item)
    (s (projT1 (l item)), input pr_item)
  → ∃ oitem : option transition_item,
      equivocator_vlsm_transition_item_project
        (IM (projT1 (l item))) pr_item
        (Existing 0) =
      Some (oitem, Existing 0)
idescriptors (projT1 (l item)) = Existing 0
    spec Hpr_item.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
descriptors: equivocator_descriptors IM
item: composite_transition_item (equivocator_IM IM)
oitem: option (composite_transition_item IM)
idescriptors: equivocator_descriptors IM
s: composite_state (equivocator_IM IM)
Ht: composite_transition (equivocator_IM IM) 
  (l item) (s, input item) =
(destination item, output item)
Hv: composite_valid (equivocator_IM IM) 
  (l item) (s, input item)
pr_item: let i := projT1 (l item) in transition_item
Heqpr_item: pr_item =
composite_transition_item_projection
  (equivocator_IM IM) item
Hpr: match
  equivocator_vlsm_transition_item_project
    (IM (projT1 (l item))) pr_item 
    (Existing 0)
with
| Some (Some item', deqv') =>
    Some
      (Some
         {|
           l :=
             existT (projT1 (l item)) (l item');
           input := input item;
           destination :=
             equivocators_state_project IM
               descriptors 
               (destination item);
           output := output item
         |},
       equivocator_descriptors_update IM
         descriptors (projT1 (l item)) deqv')
| Some (None, deqv') =>
    Some
      (None,
       equivocator_descriptors_update IM
         descriptors (projT1 (l item)) deqv')
| None => None
end = Some (oitem, idescriptors)
Hi: descriptors (projT1 (l item)) = Existing 0
Hpr_item: transition (l pr_item)
  (s (projT1 (l item)), input pr_item) =
(destination pr_item, output pr_item)
→ valid (l pr_item)
    (s (projT1 (l item)), input pr_item)
  → ∃ oitem : option transition_item,
      equivocator_vlsm_transition_item_project
        (IM (projT1 (l item))) pr_item
        (Existing 0) =
      Some (oitem, Existing 0)
transition (l pr_item)
  (s (projT1 (l item)), input pr_item) =
(destination pr_item, output pr_item)
message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
descriptors: equivocator_descriptors IM
item: composite_transition_item (equivocator_IM IM)
oitem: option (composite_transition_item IM)
idescriptors: equivocator_descriptors IM
s: composite_state (equivocator_IM IM)
Ht: composite_transition (equivocator_IM IM) 
  (l item) (s, input item) =
(destination item, output item)
Hv: composite_valid (equivocator_IM IM) 
  (l item) (s, input item)
pr_item: let i := projT1 (l item) in transition_item
Heqpr_item: pr_item =
composite_transition_item_projection
  (equivocator_IM IM) item
Hpr: match
  equivocator_vlsm_transition_item_project
    (IM (projT1 (l item))) pr_item 
    (Existing 0)
with
| Some (Some item', deqv') =>
    Some
      (Some
         {|
           l :=
             existT (projT1 (l item)) (l item');
           input := input item;
           destination :=
             equivocators_state_project IM
               descriptors 
               (destination item);
           output := output item
         |},
       equivocator_descriptors_update IM
         descriptors (projT1 (l item)) deqv')
| Some (None, deqv') =>
    Some
      (None,
       equivocator_descriptors_update IM
         descriptors (projT1 (l item)) deqv')
| None => None
end = Some (oitem, idescriptors)
Hi: descriptors (projT1 (l item)) = Existing 0
Hpr_item: valid (l pr_item)
  (s (projT1 (l item)), input pr_item)
→ ∃ oitem : option transition_item,
    equivocator_vlsm_transition_item_project
      (IM (projT1 (l item))) pr_item
      (Existing 0) =
    Some (oitem, Existing 0)
idescriptors (projT1 (l item)) = Existing 0
    {message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
descriptors: equivocator_descriptors IM
item: composite_transition_item (equivocator_IM IM)
oitem: option (composite_transition_item IM)
idescriptors: equivocator_descriptors IM
s: composite_state (equivocator_IM IM)
Ht: composite_transition (equivocator_IM IM) 
  (l item) (s, input item) =
(destination item, output item)
Hv: composite_valid (equivocator_IM IM) 
  (l item) (s, input item)
pr_item: let i := projT1 (l item) in transition_item
Heqpr_item: pr_item =
composite_transition_item_projection
  (equivocator_IM IM) item
Hpr: match
  equivocator_vlsm_transition_item_project
    (IM (projT1 (l item))) pr_item 
    (Existing 0)
with
| Some (Some item', deqv') =>
    Some
      (Some
         {|
           l :=
             existT (projT1 (l item)) (l item');
           input := input item;
           destination :=
             equivocators_state_project IM
               descriptors 
               (destination item);
           output := output item
         |},
       equivocator_descriptors_update IM
         descriptors (projT1 (l item)) deqv')
| Some (None, deqv') =>
    Some
      (None,
       equivocator_descriptors_update IM
         descriptors (projT1 (l item)) deqv')
| None => None
end = Some (oitem, idescriptors)
Hi: descriptors (projT1 (l item)) = Existing 0
Hpr_item: transition (l pr_item)
  (s (projT1 (l item)), input pr_item) =
(destination pr_item, output pr_item)
→ valid (l pr_item)
    (s (projT1 (l item)), input pr_item)
  → ∃ oitem : option transition_item,
      equivocator_vlsm_transition_item_project
        (IM (projT1 (l item))) pr_item
        (Existing 0) =
      Some (oitem, Existing 0)
transition (l pr_item)
  (s (projT1 (l item)), input pr_item) =
(destination pr_item, output pr_item)
      clear -Ht Heqpr_item.message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
item: composite_transition_item (equivocator_IM IM)
s: composite_state (equivocator_IM IM)
Ht: composite_transition (equivocator_IM IM) 
  (l item) (s, input item) =
(destination item, output item)
pr_item: let i := projT1 (l item) in transition_item
Heqpr_item: pr_item =
composite_transition_item_projection
  (equivocator_IM IM) item
transition (l pr_item)
  (s (projT1 (l item)), input pr_item) =
(destination pr_item, output pr_item)
      destruct item.message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
l: label (composite_type (equivocator_IM IM))
input: option message
destination: state
  (composite_type (equivocator_IM IM))
output: option message
s: composite_state (equivocator_IM IM)
Ht: composite_transition (equivocator_IM IM)
  (VLSM.l
     {|
       l := l;
       input := input;
       destination := destination;
       output := output
     |})
  (s,
   VLSM.input
     {|
       l := l;
       input := input;
       destination := destination;
       output := output
     |}) =
(VLSM.destination
   {|
     l := l;
     input := input;
     destination := destination;
     output := output
   |},
 VLSM.output
   {|
     l := l;
     input := input;
     destination := destination;
     output := output
   |})
pr_item: let i :=
  projT1
    (VLSM.l
       {|
         l := l;
         input := input;
         destination := destination;
         output := output
       |}) in
transition_item
Heqpr_item: pr_item =
composite_transition_item_projection
  (equivocator_IM IM)
  {|
    l := l;
    input := input;
    destination := destination;
    output := output
  |}
transition (VLSM.l pr_item)
  (s
     (projT1
        (VLSM.l
           {|
             l := l;
             input := input;
             destination := destination;
             output := output
           |})), VLSM.input pr_item) =
(VLSM.destination pr_item, VLSM.output pr_item) simpl in *.message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
l: composite_label (equivocator_IM IM)
input: option message
destination: composite_state (equivocator_IM IM)
output: option message
s: composite_state (equivocator_IM IM)
Ht: (let (i, li) := l in
 let (si', om') :=
   equivocator_transition (IM i) li (s i, input) in
 (state_update (equivocator_IM IM) s i si', om')) =
(destination, output)
pr_item: transition_item
Heqpr_item: pr_item =
composite_transition_item_projection
  (equivocator_IM IM)
  {|
    l := l;
    input := input;
    destination := destination;
    output := output
  |}
equivocator_transition (IM (projT1 l))
  (VLSM.l pr_item) (s (projT1 l), VLSM.input pr_item) =
(VLSM.destination pr_item, VLSM.output pr_item)
      destruct l as (i, li).message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
i: index
li: label (equivocator_IM IM i)
input: option message
destination: composite_state (equivocator_IM IM)
output: option message
s: composite_state (equivocator_IM IM)
Ht: (let (si', om') :=
   equivocator_transition (IM i) li (s i, input) in
 (state_update (equivocator_IM IM) s i si', om')) =
(destination, output)
pr_item: transition_item
Heqpr_item: pr_item =
composite_transition_item_projection
  (equivocator_IM IM)
  {|
    l := existT i li;
    input := input;
    destination := destination;
    output := output
  |}
equivocator_transition (IM (projT1 (existT i li)))
  (l pr_item)
  (s (projT1 (existT i li)), VLSM.input pr_item) =
(VLSM.destination pr_item, VLSM.output pr_item)
      unfold projT1 .message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
i: index
li: label (equivocator_IM IM i)
input: option message
destination: composite_state (equivocator_IM IM)
output: option message
s: composite_state (equivocator_IM IM)
Ht: (let (si', om') :=
   equivocator_transition (IM i) li (s i, input) in
 (state_update (equivocator_IM IM) s i si', om')) =
(destination, output)
pr_item: transition_item
Heqpr_item: pr_item =
composite_transition_item_projection
  (equivocator_IM IM)
  {|
    l := existT i li;
    input := input;
    destination := destination;
    output := output
  |}
equivocator_transition (IM i) (l pr_item)
  (s i, VLSM.input pr_item) =
(VLSM.destination pr_item, VLSM.output pr_item)
      case_match.message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
i: index
li: label (equivocator_IM IM i)
input: option message
destination: composite_state (equivocator_IM IM)
output: option message
s: composite_state (equivocator_IM IM)
e: equivocator_state (IM i)
o: option message
H: equivocator_transition (IM i) li (s i, input) =
(e, o)
Ht: (state_update (equivocator_IM IM) s i e, o) =
(destination, output)
pr_item: transition_item
Heqpr_item: pr_item =
composite_transition_item_projection
  (equivocator_IM IM)
  {|
    l := existT i li;
    input := input;
    destination := destination;
    output := output
  |}
equivocator_transition (IM i) (l pr_item)
  (s i, VLSM.input pr_item) =
(VLSM.destination pr_item, VLSM.output pr_item)
      inversion Ht; subst; cbn.message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
i: index
li: label (equivocator_IM IM i)
input, output: option message
s: composite_state (equivocator_IM IM)
e: equivocator_state (IM i)
Ht: (state_update (equivocator_IM IM) s i e, output) =
(state_update (equivocator_IM IM) s i e, output)
H: equivocator_transition (IM i) li (s i, input) =
(e, output)
equivocator_transition (IM i) li (s i, input) =
(state_update (equivocator_IM IM) s i e i, output)
      by state_update_simpl.
    }message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
descriptors: equivocator_descriptors IM
item: composite_transition_item (equivocator_IM IM)
oitem: option (composite_transition_item IM)
idescriptors: equivocator_descriptors IM
s: composite_state (equivocator_IM IM)
Ht: composite_transition (equivocator_IM IM) 
  (l item) (s, input item) =
(destination item, output item)
Hv: composite_valid (equivocator_IM IM) 
  (l item) (s, input item)
pr_item: let i := projT1 (l item) in transition_item
Heqpr_item: pr_item =
composite_transition_item_projection
  (equivocator_IM IM) item
Hpr: match
  equivocator_vlsm_transition_item_project
    (IM (projT1 (l item))) pr_item 
    (Existing 0)
with
| Some (Some item', deqv') =>
    Some
      (Some
         {|
           l :=
             existT (projT1 (l item)) (l item');
           input := input item;
           destination :=
             equivocators_state_project IM
               descriptors 
               (destination item);
           output := output item
         |},
       equivocator_descriptors_update IM
         descriptors (projT1 (l item)) deqv')
| Some (None, deqv') =>
    Some
      (None,
       equivocator_descriptors_update IM
         descriptors (projT1 (l item)) deqv')
| None => None
end = Some (oitem, idescriptors)
Hi: descriptors (projT1 (l item)) = Existing 0
Hpr_item: valid (l pr_item)
  (s (projT1 (l item)), input pr_item)
→ ∃ oitem : option transition_item,
    equivocator_vlsm_transition_item_project
      (IM (projT1 (l item))) pr_item
      (Existing 0) =
    Some (oitem, Existing 0)
idescriptors (projT1 (l item)) = Existing 0
    spec Hpr_item.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
descriptors: equivocator_descriptors IM
item: composite_transition_item (equivocator_IM IM)
oitem: option (composite_transition_item IM)
idescriptors: equivocator_descriptors IM
s: composite_state (equivocator_IM IM)
Ht: composite_transition (equivocator_IM IM) 
  (l item) (s, input item) =
(destination item, output item)
Hv: composite_valid (equivocator_IM IM) 
  (l item) (s, input item)
pr_item: let i := projT1 (l item) in transition_item
Heqpr_item: pr_item =
composite_transition_item_projection
  (equivocator_IM IM) item
Hpr: match
  equivocator_vlsm_transition_item_project
    (IM (projT1 (l item))) pr_item 
    (Existing 0)
with
| Some (Some item', deqv') =>
    Some
      (Some
         {|
           l :=
             existT (projT1 (l item)) (l item');
           input := input item;
           destination :=
             equivocators_state_project IM
               descriptors 
               (destination item);
           output := output item
         |},
       equivocator_descriptors_update IM
         descriptors (projT1 (l item)) deqv')
| Some (None, deqv') =>
    Some
      (None,
       equivocator_descriptors_update IM
         descriptors (projT1 (l item)) deqv')
| None => None
end = Some (oitem, idescriptors)
Hi: descriptors (projT1 (l item)) = Existing 0
Hpr_item: valid (l pr_item)
  (s (projT1 (l item)), input pr_item)
→ ∃ oitem : option transition_item,
    equivocator_vlsm_transition_item_project
      (IM (projT1 (l item))) pr_item
      (Existing 0) =
    Some (oitem, Existing 0)
valid (l pr_item) (s (projT1 (l item)), input pr_item)
message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
descriptors: equivocator_descriptors IM
item: composite_transition_item (equivocator_IM IM)
oitem: option (composite_transition_item IM)
idescriptors: equivocator_descriptors IM
s: composite_state (equivocator_IM IM)
Ht: composite_transition (equivocator_IM IM) 
  (l item) (s, input item) =
(destination item, output item)
Hv: composite_valid (equivocator_IM IM) 
  (l item) (s, input item)
pr_item: let i := projT1 (l item) in transition_item
Heqpr_item: pr_item =
composite_transition_item_projection
  (equivocator_IM IM) item
Hpr: match
  equivocator_vlsm_transition_item_project
    (IM (projT1 (l item))) pr_item 
    (Existing 0)
with
| Some (Some item', deqv') =>
    Some
      (Some
         {|
           l :=
             existT (projT1 (l item)) (l item');
           input := input item;
           destination :=
             equivocators_state_project IM
               descriptors 
               (destination item);
           output := output item
         |},
       equivocator_descriptors_update IM
         descriptors (projT1 (l item)) deqv')
| Some (None, deqv') =>
    Some
      (None,
       equivocator_descriptors_update IM
         descriptors (projT1 (l item)) deqv')
| None => None
end = Some (oitem, idescriptors)
Hi: descriptors (projT1 (l item)) = Existing 0
Hpr_item: ∃ oitem : option transition_item,
  equivocator_vlsm_transition_item_project
    (IM (projT1 (l item))) pr_item
    (Existing 0) = Some (oitem, Existing 0)
idescriptors (projT1 (l item)) = Existing 0
    {message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
descriptors: equivocator_descriptors IM
item: composite_transition_item (equivocator_IM IM)
oitem: option (composite_transition_item IM)
idescriptors: equivocator_descriptors IM
s: composite_state (equivocator_IM IM)
Ht: composite_transition (equivocator_IM IM) 
  (l item) (s, input item) =
(destination item, output item)
Hv: composite_valid (equivocator_IM IM) 
  (l item) (s, input item)
pr_item: let i := projT1 (l item) in transition_item
Heqpr_item: pr_item =
composite_transition_item_projection
  (equivocator_IM IM) item
Hpr: match
  equivocator_vlsm_transition_item_project
    (IM (projT1 (l item))) pr_item 
    (Existing 0)
with
| Some (Some item', deqv') =>
    Some
      (Some
         {|
           l :=
             existT (projT1 (l item)) (l item');
           input := input item;
           destination :=
             equivocators_state_project IM
               descriptors 
               (destination item);
           output := output item
         |},
       equivocator_descriptors_update IM
         descriptors (projT1 (l item)) deqv')
| Some (None, deqv') =>
    Some
      (None,
       equivocator_descriptors_update IM
         descriptors (projT1 (l item)) deqv')
| None => None
end = Some (oitem, idescriptors)
Hi: descriptors (projT1 (l item)) = Existing 0
Hpr_item: valid (l pr_item)
  (s (projT1 (l item)), input pr_item)
→ ∃ oitem : option transition_item,
    equivocator_vlsm_transition_item_project
      (IM (projT1 (l item))) pr_item
      (Existing 0) =
    Some (oitem, Existing 0)
valid (l pr_item) (s (projT1 (l item)), input pr_item)
      clear -Hv Heqpr_item.message, index: Type
IM: index → VLSM message
item: composite_transition_item (equivocator_IM IM)
s: composite_state (equivocator_IM IM)
Hv: composite_valid (equivocator_IM IM) 
  (l item) (s, input item)
pr_item: let i := projT1 (l item) in transition_item
Heqpr_item: pr_item =
composite_transition_item_projection
  (equivocator_IM IM) item
valid (l pr_item) (s (projT1 (l item)), input pr_item)
      destruct item.message, index: Type
IM: index → VLSM message
l: label (composite_type (equivocator_IM IM))
input: option message
destination: state
  (composite_type (equivocator_IM IM))
output: option message
s: composite_state (equivocator_IM IM)
Hv: composite_valid (equivocator_IM IM)
  (VLSM.l
     {|
       l := l;
       input := input;
       destination := destination;
       output := output
     |})
  (s,
   VLSM.input
     {|
       l := l;
       input := input;
       destination := destination;
       output := output
     |})
pr_item: let i :=
  projT1
    (VLSM.l
       {|
         l := l;
         input := input;
         destination := destination;
         output := output
       |}) in
transition_item
Heqpr_item: pr_item =
composite_transition_item_projection
  (equivocator_IM IM)
  {|
    l := l;
    input := input;
    destination := destination;
    output := output
  |}
valid (VLSM.l pr_item)
  (s
     (projT1
        (VLSM.l
           {|
             l := l;
             input := input;
             destination := destination;
             output := output
           |})), VLSM.input pr_item) simpl in *.message, index: Type
IM: index → VLSM message
l: composite_label (equivocator_IM IM)
input: option message
destination: composite_state (equivocator_IM IM)
output: option message
s: composite_state (equivocator_IM IM)
Hv: let (i, li) := l in
equivocator_valid (IM i) li (s i, input)
pr_item: transition_item
Heqpr_item: pr_item =
composite_transition_item_projection
  (equivocator_IM IM)
  {|
    l := l;
    input := input;
    destination := destination;
    output := output
  |}
equivocator_valid (IM (projT1 l)) (VLSM.l pr_item)
  (s (projT1 l), VLSM.input pr_item)
      by destruct l as [i li]; subst.
    }message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
descriptors: equivocator_descriptors IM
item: composite_transition_item (equivocator_IM IM)
oitem: option (composite_transition_item IM)
idescriptors: equivocator_descriptors IM
s: composite_state (equivocator_IM IM)
Ht: composite_transition (equivocator_IM IM) 
  (l item) (s, input item) =
(destination item, output item)
Hv: composite_valid (equivocator_IM IM) 
  (l item) (s, input item)
pr_item: let i := projT1 (l item) in transition_item
Heqpr_item: pr_item =
composite_transition_item_projection
  (equivocator_IM IM) item
Hpr: match
  equivocator_vlsm_transition_item_project
    (IM (projT1 (l item))) pr_item 
    (Existing 0)
with
| Some (Some item', deqv') =>
    Some
      (Some
         {|
           l :=
             existT (projT1 (l item)) (l item');
           input := input item;
           destination :=
             equivocators_state_project IM
               descriptors 
               (destination item);
           output := output item
         |},
       equivocator_descriptors_update IM
         descriptors (projT1 (l item)) deqv')
| Some (None, deqv') =>
    Some
      (None,
       equivocator_descriptors_update IM
         descriptors (projT1 (l item)) deqv')
| None => None
end = Some (oitem, idescriptors)
Hi: descriptors (projT1 (l item)) = Existing 0
Hpr_item: ∃ oitem : option transition_item,
  equivocator_vlsm_transition_item_project
    (IM (projT1 (l item))) pr_item
    (Existing 0) = Some (oitem, Existing 0)
idescriptors (projT1 (l item)) = Existing 0
    destruct Hpr_item as [oitem' Hpr_item].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
descriptors: equivocator_descriptors IM
item: composite_transition_item (equivocator_IM IM)
oitem: option (composite_transition_item IM)
idescriptors: equivocator_descriptors IM
s: composite_state (equivocator_IM IM)
Ht: composite_transition (equivocator_IM IM) 
  (l item) (s, input item) =
(destination item, output item)
Hv: composite_valid (equivocator_IM IM) 
  (l item) (s, input item)
pr_item: let i := projT1 (l item) in transition_item
Heqpr_item: pr_item =
composite_transition_item_projection
  (equivocator_IM IM) item
Hpr: match
  equivocator_vlsm_transition_item_project
    (IM (projT1 (l item))) pr_item 
    (Existing 0)
with
| Some (Some item', deqv') =>
    Some
      (Some
         {|
           l :=
             existT (projT1 (l item)) (l item');
           input := input item;
           destination :=
             equivocators_state_project IM
               descriptors 
               (destination item);
           output := output item
         |},
       equivocator_descriptors_update IM
         descriptors (projT1 (l item)) deqv')
| Some (None, deqv') =>
    Some
      (None,
       equivocator_descriptors_update IM
         descriptors (projT1 (l item)) deqv')
| None => None
end = Some (oitem, idescriptors)
Hi: descriptors (projT1 (l item)) = Existing 0
oitem': option transition_item
Hpr_item: equivocator_vlsm_transition_item_project
  (IM (projT1 (l item))) pr_item
  (Existing 0) = Some (oitem', Existing 0)
idescriptors (projT1 (l item)) = Existing 0
    rewrite Hpr_item in Hpr.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
descriptors: equivocator_descriptors IM
item: composite_transition_item (equivocator_IM IM)
oitem: option (composite_transition_item IM)
idescriptors: equivocator_descriptors IM
s: composite_state (equivocator_IM IM)
Ht: composite_transition (equivocator_IM IM) 
  (l item) (s, input item) =
(destination item, output item)
Hv: composite_valid (equivocator_IM IM) 
  (l item) (s, input item)
pr_item: let i := projT1 (l item) in transition_item
Heqpr_item: pr_item =
composite_transition_item_projection
  (equivocator_IM IM) item
oitem': option transition_item
Hpr: match oitem' with
| Some item' =>
    Some
      (Some
         {|
           l :=
             existT (projT1 (l item)) (l item');
           input := input item;
           destination :=
             equivocators_state_project IM
               descriptors 
               (destination item);
           output := output item
         |},
       equivocator_descriptors_update IM
         descriptors (projT1 (l item))
         (Existing 0))
| None =>
    Some
      (None,
       equivocator_descriptors_update IM
         descriptors (projT1 (l item))
         (Existing 0))
end = Some (oitem, idescriptors)
Hi: descriptors (projT1 (l item)) = Existing 0
Hpr_item: equivocator_vlsm_transition_item_project
  (IM (projT1 (l item))) pr_item
  (Existing 0) = Some (oitem', Existing 0)
idescriptors (projT1 (l item)) = Existing 0
    by destruct oitem'; inversion Hpr; state_update_simpl.
  -message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
descriptors: equivocator_descriptors IM
item: composite_transition_item (equivocator_IM IM)
oitem: option (composite_transition_item IM)
idescriptors: equivocator_descriptors IM
s: composite_state (equivocator_IM IM)
Ht: composite_transition (equivocator_IM IM) 
  (l item) (s, input item) =
(destination item, output item)
Hv: composite_valid (equivocator_IM IM) 
  (l item) (s, input item)
Hpr: match
  equivocator_vlsm_transition_item_project
    (IM (projT1 (l item)))
    (composite_transition_item_projection
       (equivocator_IM IM) item)
    (descriptors (projT1 (l item)))
with
| Some (Some item', deqv') =>
    Some
      (Some
         {|
           l :=
             existT (projT1 (l item)) (l item');
           input := input item;
           destination :=
             equivocators_state_project IM
               descriptors 
               (destination item);
           output := output item
         |},
       equivocator_descriptors_update IM
         descriptors (projT1 (l item)) deqv')
| Some (None, deqv') =>
    Some
      (None,
       equivocator_descriptors_update IM
         descriptors (projT1 (l item)) deqv')
| None => None
end = Some (oitem, idescriptors)
i: index
Hi: descriptors i = Existing 0
n: i ≠ projT1 (l item)
idescriptors i = Existing 0 destruct
    (equivocator_vlsm_transition_item_project (IM (projT1 (l item)))
      (composite_transition_item_projection (equivocator_IM IM) item)
      (descriptors (projT1 (l item))))
    eqn: Hpr'; [| by congruence].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
descriptors: equivocator_descriptors IM
item: composite_transition_item (equivocator_IM IM)
oitem: option (composite_transition_item IM)
idescriptors: equivocator_descriptors IM
s: composite_state (equivocator_IM IM)
Ht: composite_transition (equivocator_IM IM) 
  (l item) (s, input item) =
(destination item, output item)
Hv: composite_valid (equivocator_IM IM) 
  (l item) (s, input item)
p: (option transition_item *
 MachineDescriptor (IM (projT1 (l item))))%type
Hpr': equivocator_vlsm_transition_item_project
  (IM (projT1 (l item)))
  (composite_transition_item_projection
     (equivocator_IM IM) item)
  (descriptors (projT1 (l item))) = 
Some p
Hpr: (let (o, deqv') := p in
 match o with
 | Some item' =>
     Some
       (Some
          {|
            l :=
              existT (projT1 (l item)) (l item');
            input := input item;
            destination :=
              equivocators_state_project IM
                descriptors 
                (destination item);
            output := output item
          |},
        equivocator_descriptors_update IM
          descriptors (projT1 (l item)) deqv')
 | None =>
     Some
       (None,
        equivocator_descriptors_update IM
          descriptors (projT1 (l item)) deqv')
 end) = Some (oitem, idescriptors)
i: index
Hi: descriptors i = Existing 0
n: i ≠ projT1 (l item)
idescriptors i = Existing 0
  by destruct p, o; inversion Hpr; state_update_simpl.
Qed.

Lemma equivocators_transition_item_project_proper_descriptor
  (eqv_descriptors : equivocator_descriptors IM)
  (item : composite_transition_item (equivocator_IM IM))
  (i := projT1 (l item))
  (Hproper : proper_descriptor (IM i) (eqv_descriptors i) (destination item i))
  : is_Some (equivocators_transition_item_project eqv_descriptors item).message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
eqv_descriptors: equivocator_descriptors IM
item: composite_transition_item (equivocator_IM IM)
i:= projT1 (l item): index
Hproper: proper_descriptor (IM i) 
  (eqv_descriptors i) (destination item i)
is_Some
  (equivocators_transition_item_project
     eqv_descriptors item)
Proof.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
eqv_descriptors: equivocator_descriptors IM
item: composite_transition_item (equivocator_IM IM)
i:= projT1 (l item): index
Hproper: proper_descriptor (IM i) 
  (eqv_descriptors i) (destination item i)
is_Some
  (equivocators_transition_item_project
     eqv_descriptors item)
  specialize
    (equivocator_transition_item_project_proper (IM (projT1 (l item)))
      (composite_transition_item_projection (equivocator_IM IM) item)
      (eqv_descriptors (projT1 (l item))) Hproper)
    as [itemx Hpr_item].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
eqv_descriptors: equivocator_descriptors IM
item: composite_transition_item (equivocator_IM IM)
i:= projT1 (l item): index
Hproper: proper_descriptor (IM i) 
  (eqv_descriptors i) (destination item i)
itemx: (option transition_item *
 MachineDescriptor (IM (projT1 (l item))))%type
Hpr_item: equivocator_vlsm_transition_item_project
  (IM (projT1 (l item)))
  (composite_transition_item_projection
     (equivocator_IM IM) item)
  (eqv_descriptors (projT1 (l item))) =
Some itemx
is_Some
  (equivocators_transition_item_project
     eqv_descriptors item)
  unfold equivocators_transition_item_project.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
eqv_descriptors: equivocator_descriptors IM
item: composite_transition_item (equivocator_IM IM)
i:= projT1 (l item): index
Hproper: proper_descriptor (IM i) 
  (eqv_descriptors i) (destination item i)
itemx: (option transition_item *
 MachineDescriptor (IM (projT1 (l item))))%type
Hpr_item: equivocator_vlsm_transition_item_project
  (IM (projT1 (l item)))
  (composite_transition_item_projection
     (equivocator_IM IM) item)
  (eqv_descriptors (projT1 (l item))) =
Some itemx
is_Some
  match
    equivocator_vlsm_transition_item_project
      (IM (projT1 (l item)))
      (composite_transition_item_projection
         (equivocator_IM IM) item)
      (eqv_descriptors (projT1 (l item)))
  with
  | Some (Some item', deqv') =>
      Some
        (Some
           {|
             l := existT (projT1 (l item)) (l item');
             input := input item;
             destination :=
               equivocators_state_project IM
                 eqv_descriptors (destination item);
             output := output item
           |},
         equivocator_descriptors_update IM
           eqv_descriptors (projT1 (l item)) deqv')
  | Some (None, deqv') =>
      Some
        (None,
         equivocator_descriptors_update IM
           eqv_descriptors (projT1 (l item)) deqv')
  | None => None
  end
  rewrite Hpr_item.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
eqv_descriptors: equivocator_descriptors IM
item: composite_transition_item (equivocator_IM IM)
i:= projT1 (l item): index
Hproper: proper_descriptor (IM i) 
  (eqv_descriptors i) (destination item i)
itemx: (option transition_item *
 MachineDescriptor (IM (projT1 (l item))))%type
Hpr_item: equivocator_vlsm_transition_item_project
  (IM (projT1 (l item)))
  (composite_transition_item_projection
     (equivocator_IM IM) item)
  (eqv_descriptors (projT1 (l item))) =
Some itemx
is_Some
  (let (o, deqv') := itemx in
   match o with
   | Some item' =>
       Some
         (Some
            {|
              l := existT (projT1 (l item)) (l item');
              input := input item;
              destination :=
                equivocators_state_project IM
                  eqv_descriptors (destination item);
              output := output item
            |},
          equivocator_descriptors_update IM
            eqv_descriptors (projT1 (l item)) deqv')
   | None =>
       Some
         (None,
          equivocator_descriptors_update IM
            eqv_descriptors (projT1 (l item)) deqv')
   end)
  by destruct itemx, o; eexists.
Qed.

Lemma equivocators_transition_item_project_proper
  (eqv_descriptors : equivocator_descriptors IM)
  (item : composite_transition_item (equivocator_IM IM))
  (Hproper : proper_equivocator_descriptors IM eqv_descriptors (destination item))
  : is_Some (equivocators_transition_item_project eqv_descriptors item).message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
eqv_descriptors: equivocator_descriptors IM
item: composite_transition_item (equivocator_IM IM)
Hproper: proper_equivocator_descriptors IM
  eqv_descriptors (destination item)
is_Some
  (equivocators_transition_item_project
     eqv_descriptors item)
Proof.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
eqv_descriptors: equivocator_descriptors IM
item: composite_transition_item (equivocator_IM IM)
Hproper: proper_equivocator_descriptors IM
  eqv_descriptors (destination item)
is_Some
  (equivocators_transition_item_project
     eqv_descriptors item)
  apply equivocators_transition_item_project_proper_descriptor.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
eqv_descriptors: equivocator_descriptors IM
item: composite_transition_item (equivocator_IM IM)
Hproper: proper_equivocator_descriptors IM
  eqv_descriptors (destination item)
proper_descriptor (IM (projT1 (l item)))
  (eqv_descriptors (projT1 (l item)))
  (destination item (projT1 (l item)))
  by apply Hproper.
Qed.

A generalization of no_equivocating_equivocator_transition_item_project to the composition of equivocators.

Lemma no_equivocating_equivocators_transition_item_project
  (eqv_descriptors : equivocator_descriptors IM)
  (item : composite_transition_item (equivocator_IM IM))
  (i := projT1 (l item))
  (Hzero : (eqv_descriptors i) = Existing 0)
  (Hdest_i : is_singleton_state (IM i) (destination item i))
  (s : composite_state (equivocator_IM IM))
  (Hv : composite_valid (equivocator_IM IM) (l item) (s, input item))
  (Ht : composite_transition (equivocator_IM IM) (l item) (s, input item) =
          (destination item, output item))
  : exists (Hex : existing_equivocator_label _ (projT2 (l item)))
    (lx : composite_label IM :=
      existT i (existing_equivocator_label_extract _ (projT2 (l item)) Hex)),
  equivocators_transition_item_project eqv_descriptors item =
    Some (Some
      {| l := lx; input := input item; output := output item;
        destination := equivocators_state_project IM eqv_descriptors (destination item) |},
      eqv_descriptors).message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
eqv_descriptors: equivocator_descriptors IM
item: composite_transition_item (equivocator_IM IM)
i:= projT1 (l item): index
Hzero: eqv_descriptors i = Existing 0
Hdest_i: is_singleton_state (IM i)
  (destination item i)
s: composite_state (equivocator_IM IM)
Hv: composite_valid (equivocator_IM IM) 
  (l item) (s, input item)
Ht: composite_transition (equivocator_IM IM) 
  (l item) (s, input item) =
(destination item, output item)
∃ Hex : existing_equivocator_label
          (IM (projT1 (l item))) (projT2 (l item)),
  let lx :=
    existT i
      (existing_equivocator_label_extract
         (IM (projT1 (l item))) (projT2 (l item)) Hex)
    in
  equivocators_transition_item_project eqv_descriptors
    item =
  Some
    (Some
       {|
         l := lx;
         input := input item;
         destination :=
           equivocators_state_project IM
             eqv_descriptors (destination item);
         output := output item
       |}, eqv_descriptors)
Proof.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
eqv_descriptors: equivocator_descriptors IM
item: composite_transition_item (equivocator_IM IM)
i:= projT1 (l item): index
Hzero: eqv_descriptors i = Existing 0
Hdest_i: is_singleton_state (IM i)
  (destination item i)
s: composite_state (equivocator_IM IM)
Hv: composite_valid (equivocator_IM IM) 
  (l item) (s, input item)
Ht: composite_transition (equivocator_IM IM) 
  (l item) (s, input item) =
(destination item, output item)
∃ Hex : existing_equivocator_label
          (IM (projT1 (l item))) (projT2 (l item)),
  let lx :=
    existT i
      (existing_equivocator_label_extract
         (IM (projT1 (l item))) (projT2 (l item)) Hex)
    in
  equivocators_transition_item_project eqv_descriptors
    item =
  Some
    (Some
       {|
         l := lx;
         input := input item;
         destination :=
           equivocators_state_project IM
             eqv_descriptors (destination item);
         output := output item
       |}, eqv_descriptors)
  specialize
    (no_equivocating_equivocator_transition_item_project (IM i)
      (composite_transition_item_projection (equivocator_IM IM) item)
      Hdest_i
      (s i))
    as Heqv_pr.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
eqv_descriptors: equivocator_descriptors IM
item: composite_transition_item (equivocator_IM IM)
i:= projT1 (l item): index
Hzero: eqv_descriptors i = Existing 0
Hdest_i: is_singleton_state (IM i)
  (destination item i)
s: composite_state (equivocator_IM IM)
Hv: composite_valid (equivocator_IM IM) 
  (l item) (s, input item)
Ht: composite_transition (equivocator_IM IM) 
  (l item) (s, input item) =
(destination item, output item)
Heqv_pr: valid
  (l
     (composite_transition_item_projection
        (equivocator_IM IM) item))
  (s i,
   input
     (composite_transition_item_projection
        (equivocator_IM IM) item))
→ transition
    (l
       (composite_transition_item_projection
          (equivocator_IM IM) item))
    (s i,
     input
       (composite_transition_item_projection
          (equivocator_IM IM) item)) =
  (destination
     (composite_transition_item_projection
        (equivocator_IM IM) item),
   output
     (composite_transition_item_projection
        (equivocator_IM IM) item))
  → ∃ Hex : existing_equivocator_label
              (IM i)
              (l
                 (composite_transition_item_projection
                    (equivocator_IM IM) item)),
      equivocator_vlsm_transition_item_project
        (IM i)
        (composite_transition_item_projection
           (equivocator_IM IM) item)
        (Existing 0) =
      Some
        (Some
           {|
             l :=
               existing_equivocator_label_extract
                 (IM i)
                 (l
                    (composite_transition_item_projection
                      (equivocator_IM IM)
                      item)) Hex;
             input :=
               input
                 (composite_transition_item_projection
                    (equivocator_IM IM) item);
             destination :=
               equivocator_state_descriptor_project
                 (destination
                    (composite_transition_item_projection
                      (equivocator_IM IM)
                      item)) 
                 (Existing 0);
             output :=
               output
                 (composite_transition_item_projection
                    (equivocator_IM IM) item)
           |}, Existing 0)
∃ Hex : existing_equivocator_label
          (IM (projT1 (l item))) (projT2 (l item)),
  let lx :=
    existT i
      (existing_equivocator_label_extract
         (IM (projT1 (l item))) (projT2 (l item)) Hex)
    in
  equivocators_transition_item_project eqv_descriptors
    item =
  Some
    (Some
       {|
         l := lx;
         input := input item;
         destination :=
           equivocators_state_project IM
             eqv_descriptors (destination item);
         output := output item
       |}, eqv_descriptors)
  destruct item, l as [x l].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
eqv_descriptors: equivocator_descriptors IM
x: index
l: label (equivocator_IM IM x)
input: option message
destination: state
  (composite_type (equivocator_IM IM))
output: option message
i:= projT1
  (VLSM.l
     {|
       l := existT x l;
       input := input;
       destination := destination;
       output := output
     |}): index
Hzero: eqv_descriptors i = Existing 0
Hdest_i: is_singleton_state (IM i)
  (VLSM.destination
     {|
       l := existT x l;
       input := input;
       destination := destination;
       output := output
     |} i)
s: composite_state (equivocator_IM IM)
Hv: composite_valid (equivocator_IM IM)
  (VLSM.l
     {|
       l := existT x l;
       input := input;
       destination := destination;
       output := output
     |})
  (s,
   VLSM.input
     {|
       l := existT x l;
       input := input;
       destination := destination;
       output := output
     |})
Ht: composite_transition (equivocator_IM IM)
  (VLSM.l
     {|
       l := existT x l;
       input := input;
       destination := destination;
       output := output
     |})
  (s,
   VLSM.input
     {|
       l := existT x l;
       input := input;
       destination := destination;
       output := output
     |}) =
(VLSM.destination
   {|
     l := existT x l;
     input := input;
     destination := destination;
     output := output
   |},
 VLSM.output
   {|
     l := existT x l;
     input := input;
     destination := destination;
     output := output
   |})
Heqv_pr: valid
  (VLSM.l
     (composite_transition_item_projection
        (equivocator_IM IM)
        {|
          l := existT x l;
          input := input;
          destination := destination;
          output := output
        |}))
  (s i,
   VLSM.input
     (composite_transition_item_projection
        (equivocator_IM IM)
        {|
          l := existT x l;
          input := input;
          destination := destination;
          output := output
        |}))
→ transition
    (VLSM.l
       (composite_transition_item_projection
          (equivocator_IM IM)
          {|
            l := existT x l;
            input := input;
            destination := destination;
            output := output
          |}))
    (s i,
     VLSM.input
       (composite_transition_item_projection
          (equivocator_IM IM)
          {|
            l := existT x l;
            input := input;
            destination := destination;
            output := output
          |})) =
  (VLSM.destination
     (composite_transition_item_projection
        (equivocator_IM IM)
        {|
          l := existT x l;
          input := input;
          destination := destination;
          output := output
        |}),
   VLSM.output
     (composite_transition_item_projection
        (equivocator_IM IM)
        {|
          l := existT x l;
          input := input;
          destination := destination;
          output := output
        |}))
  → ∃ Hex : existing_equivocator_label
              (IM i)
              (VLSM.l
                 (composite_transition_item_projection
                    (equivocator_IM IM)
                    {|
                      l := existT x l;
                      input := input;
                      destination :=
                      destination;
                      output := output
                    |})),
      equivocator_vlsm_transition_item_project
        (IM i)
        (composite_transition_item_projection
           (equivocator_IM IM)
           {|
             l := existT x l;
             input := input;
             destination := destination;
             output := output
           |}) (Existing 0) =
      Some
        (Some
           {|
             l :=
               existing_equivocator_label_extract
                 (IM i)
                 (VLSM.l
                    (composite_transition_item_projection
                      (equivocator_IM IM)
                      {|
                      l := existT x l;
                      input := input;
                      destination :=
                      destination;
                      output := output
                      |})) Hex;
             input :=
               VLSM.input
                 (composite_transition_item_projection
                    (equivocator_IM IM)
                    {|
                      l := existT x l;
                      input := input;
                      destination :=
                      destination;
                      output := output
                    |});
             destination :=
               equivocator_state_descriptor_project
                 (VLSM.destination
                    (composite_transition_item_projection
                      (equivocator_IM IM)
                      {|
                      l := existT x l;
                      input := input;
                      destination :=
                      destination;
                      output := output
                      |})) 
                 (Existing 0);
             output :=
               VLSM.output
                 (composite_transition_item_projection
                    (equivocator_IM IM)
                    {|
                      l := existT x l;
                      input := input;
                      destination :=
                      destination;
                      output := output
                    |})
           |}, Existing 0)
∃ Hex : existing_equivocator_label
          (IM
             (projT1
                (VLSM.l
                   {|
                     l := existT x l;
                     input := input;
                     destination := destination;
                     output := output
                   |})))
          (projT2
             (VLSM.l
                {|
                  l := existT x l;
                  input := input;
                  destination := destination;
                  output := output
                |})),
  let lx :=
    existT i
      (existing_equivocator_label_extract
         (IM
            (projT1
               (VLSM.l
                  {|
                    l := existT x l;
                    input := input;
                    destination := destination;
                    output := output
                  |})))
         (projT2
            (VLSM.l
               {|
                 l := existT x l;
                 input := input;
                 destination := destination;
                 output := output
               |})) Hex) in
  equivocators_transition_item_project eqv_descriptors
    {|
      l := existT x l;
      input := input;
      destination := destination;
      output := output
    |} =
  Some
    (Some
       {|
         l := lx;
         input :=
           VLSM.input
             {|
               l := existT x l;
               input := input;
               destination := destination;
               output := output
             |};
         destination :=
           equivocators_state_project IM
             eqv_descriptors
             (VLSM.destination
                {|
                  l := existT x l;
                  input := input;
                  destination := destination;
                  output := output
                |});
         output :=
           VLSM.output
             {|
               l := existT x l;
               input := input;
               destination := destination;
               output := output
             |}
       |}, eqv_descriptors) simpl in Ht, Hv.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
eqv_descriptors: equivocator_descriptors IM
x: index
l: label (equivocator_IM IM x)
input: option message
destination: state
  (composite_type (equivocator_IM IM))
output: option message
i:= projT1
  (VLSM.l
     {|
       l := existT x l;
       input := input;
       destination := destination;
       output := output
     |}): index
Hzero: eqv_descriptors i = Existing 0
Hdest_i: is_singleton_state (IM i)
  (VLSM.destination
     {|
       l := existT x l;
       input := input;
       destination := destination;
       output := output
     |} i)
s: composite_state (equivocator_IM IM)
Hv: equivocator_valid (IM x) l (s x, input)
Ht: (let (si', om') :=
   equivocator_transition (IM x) l (s x, input) in
 (state_update (equivocator_IM IM) s x si', om')) =
(destination, output)
Heqv_pr: valid
  (VLSM.l
     (composite_transition_item_projection
        (equivocator_IM IM)
        {|
          l := existT x l;
          input := input;
          destination := destination;
          output := output
        |}))
  (s i,
   VLSM.input
     (composite_transition_item_projection
        (equivocator_IM IM)
        {|
          l := existT x l;
          input := input;
          destination := destination;
          output := output
        |}))
→ transition
    (VLSM.l
       (composite_transition_item_projection
          (equivocator_IM IM)
          {|
            l := existT x l;
            input := input;
            destination := destination;
            output := output
          |}))
    (s i,
     VLSM.input
       (composite_transition_item_projection
          (equivocator_IM IM)
          {|
            l := existT x l;
            input := input;
            destination := destination;
            output := output
          |})) =
  (VLSM.destination
     (composite_transition_item_projection
        (equivocator_IM IM)
        {|
          l := existT x l;
          input := input;
          destination := destination;
          output := output
        |}),
   VLSM.output
     (composite_transition_item_projection
        (equivocator_IM IM)
        {|
          l := existT x l;
          input := input;
          destination := destination;
          output := output
        |}))
  → ∃ Hex : existing_equivocator_label
              (IM i)
              (VLSM.l
                 (composite_transition_item_projection
                    (equivocator_IM IM)
                    {|
                      l := existT x l;
                      input := input;
                      destination :=
                      destination;
                      output := output
                    |})),
      equivocator_vlsm_transition_item_project
        (IM i)
        (composite_transition_item_projection
           (equivocator_IM IM)
           {|
             l := existT x l;
             input := input;
             destination := destination;
             output := output
           |}) (Existing 0) =
      Some
        (Some
           {|
             l :=
               existing_equivocator_label_extract
                 (IM i)
                 (VLSM.l
                    (composite_transition_item_projection
                      (equivocator_IM IM)
                      {|
                      l := existT x l;
                      input := input;
                      destination :=
                      destination;
                      output := output
                      |})) Hex;
             input :=
               VLSM.input
                 (composite_transition_item_projection
                    (equivocator_IM IM)
                    {|
                      l := existT x l;
                      input := input;
                      destination :=
                      destination;
                      output := output
                    |});
             destination :=
               equivocator_state_descriptor_project
                 (VLSM.destination
                    (composite_transition_item_projection
                      (equivocator_IM IM)
                      {|
                      l := existT x l;
                      input := input;
                      destination :=
                      destination;
                      output := output
                      |})) 
                 (Existing 0);
             output :=
               VLSM.output
                 (composite_transition_item_projection
                    (equivocator_IM IM)
                    {|
                      l := existT x l;
                      input := input;
                      destination :=
                      destination;
                      output := output
                    |})
           |}, Existing 0)
∃ Hex : existing_equivocator_label
          (IM
             (projT1
                (VLSM.l
                   {|
                     l := existT x l;
                     input := input;
                     destination := destination;
                     output := output
                   |})))
          (projT2
             (VLSM.l
                {|
                  l := existT x l;
                  input := input;
                  destination := destination;
                  output := output
                |})),
  let lx :=
    existT i
      (existing_equivocator_label_extract
         (IM
            (projT1
               (VLSM.l
                  {|
                    l := existT x l;
                    input := input;
                    destination := destination;
                    output := output
                  |})))
         (projT2
            (VLSM.l
               {|
                 l := existT x l;
                 input := input;
                 destination := destination;
                 output := output
               |})) Hex) in
  equivocators_transition_item_project eqv_descriptors
    {|
      l := existT x l;
      input := input;
      destination := destination;
      output := output
    |} =
  Some
    (Some
       {|
         l := lx;
         input :=
           VLSM.input
             {|
               l := existT x l;
               input := input;
               destination := destination;
               output := output
             |};
         destination :=
           equivocators_state_project IM
             eqv_descriptors
             (VLSM.destination
                {|
                  l := existT x l;
                  input := input;
                  destination := destination;
                  output := output
                |});
         output :=
           VLSM.output
             {|
               l := existT x l;
               input := input;
               destination := destination;
               output := output
             |}
       |}, eqv_descriptors) simpl in i.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
eqv_descriptors: equivocator_descriptors IM
x: index
l: label (equivocator_IM IM x)
input: option message
destination: state
  (composite_type (equivocator_IM IM))
output: option message
i:= x: index
Hzero: eqv_descriptors i = Existing 0
Hdest_i: is_singleton_state (IM i)
  (VLSM.destination
     {|
       l := existT x l;
       input := input;
       destination := destination;
       output := output
     |} i)
s: composite_state (equivocator_IM IM)
Hv: equivocator_valid (IM x) l (s x, input)
Ht: (let (si', om') :=
   equivocator_transition (IM x) l (s x, input) in
 (state_update (equivocator_IM IM) s x si', om')) =
(destination, output)
Heqv_pr: valid
  (VLSM.l
     (composite_transition_item_projection
        (equivocator_IM IM)
        {|
          l := existT x l;
          input := input;
          destination := destination;
          output := output
        |}))
  (s i,
   VLSM.input
     (composite_transition_item_projection
        (equivocator_IM IM)
        {|
          l := existT x l;
          input := input;
          destination := destination;
          output := output
        |}))
→ transition
    (VLSM.l
       (composite_transition_item_projection
          (equivocator_IM IM)
          {|
            l := existT x l;
            input := input;
            destination := destination;
            output := output
          |}))
    (s i,
     VLSM.input
       (composite_transition_item_projection
          (equivocator_IM IM)
          {|
            l := existT x l;
            input := input;
            destination := destination;
            output := output
          |})) =
  (VLSM.destination
     (composite_transition_item_projection
        (equivocator_IM IM)
        {|
          l := existT x l;
          input := input;
          destination := destination;
          output := output
        |}),
   VLSM.output
     (composite_transition_item_projection
        (equivocator_IM IM)
        {|
          l := existT x l;
          input := input;
          destination := destination;
          output := output
        |}))
  → ∃ Hex : existing_equivocator_label
              (IM i)
              (VLSM.l
                 (composite_transition_item_projection
                    (equivocator_IM IM)
                    {|
                      l := existT x l;
                      input := input;
                      destination :=
                      destination;
                      output := output
                    |})),
      equivocator_vlsm_transition_item_project
        (IM i)
        (composite_transition_item_projection
           (equivocator_IM IM)
           {|
             l := existT x l;
             input := input;
             destination := destination;
             output := output
           |}) (Existing 0) =
      Some
        (Some
           {|
             l :=
               existing_equivocator_label_extract
                 (IM i)
                 (VLSM.l
                    (composite_transition_item_projection
                      (equivocator_IM IM)
                      {|
                      l := existT x l;
                      input := input;
                      destination :=
                      destination;
                      output := output
                      |})) Hex;
             input :=
               VLSM.input
                 (composite_transition_item_projection
                    (equivocator_IM IM)
                    {|
                      l := existT x l;
                      input := input;
                      destination :=
                      destination;
                      output := output
                    |});
             destination :=
               equivocator_state_descriptor_project
                 (VLSM.destination
                    (composite_transition_item_projection
                      (equivocator_IM IM)
                      {|
                      l := existT x l;
                      input := input;
                      destination :=
                      destination;
                      output := output
                      |})) 
                 (Existing 0);
             output :=
               VLSM.output
                 (composite_transition_item_projection
                    (equivocator_IM IM)
                    {|
                      l := existT x l;
                      input := input;
                      destination :=
                      destination;
                      output := output
                    |})
           |}, Existing 0)
∃ Hex : existing_equivocator_label
          (IM
             (projT1
                (VLSM.l
                   {|
                     l := existT x l;
                     input := input;
                     destination := destination;
                     output := output
                   |})))
          (projT2
             (VLSM.l
                {|
                  l := existT x l;
                  input := input;
                  destination := destination;
                  output := output
                |})),
  let lx :=
    existT i
      (existing_equivocator_label_extract
         (IM
            (projT1
               (VLSM.l
                  {|
                    l := existT x l;
                    input := input;
                    destination := destination;
                    output := output
                  |})))
         (projT2
            (VLSM.l
               {|
                 l := existT x l;
                 input := input;
                 destination := destination;
                 output := output
               |})) Hex) in
  equivocators_transition_item_project eqv_descriptors
    {|
      l := existT x l;
      input := input;
      destination := destination;
      output := output
    |} =
  Some
    (Some
       {|
         l := lx;
         input :=
           VLSM.input
             {|
               l := existT x l;
               input := input;
               destination := destination;
               output := output
             |};
         destination :=
           equivocators_state_project IM
             eqv_descriptors
             (VLSM.destination
                {|
                  l := existT x l;
                  input := input;
                  destination := destination;
                  output := output
                |});
         output :=
           VLSM.output
             {|
               l := existT x l;
               input := input;
               destination := destination;
               output := output
             |}
       |}, eqv_descriptors) subst i.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
eqv_descriptors: equivocator_descriptors IM
x: index
l: label (equivocator_IM IM x)
input: option message
destination: state
  (composite_type (equivocator_IM IM))
output: option message
Hzero: eqv_descriptors x = Existing 0
Hdest_i: is_singleton_state (IM x)
  (VLSM.destination
     {|
       l := existT x l;
       input := input;
       destination := destination;
       output := output
     |} x)
s: composite_state (equivocator_IM IM)
Hv: equivocator_valid (IM x) l (s x, input)
Ht: (let (si', om') :=
   equivocator_transition (IM x) l (s x, input) in
 (state_update (equivocator_IM IM) s x si', om')) =
(destination, output)
Heqv_pr: valid
  (VLSM.l
     (composite_transition_item_projection
        (equivocator_IM IM)
        {|
          l := existT x l;
          input := input;
          destination := destination;
          output := output
        |}))
  (s x,
   VLSM.input
     (composite_transition_item_projection
        (equivocator_IM IM)
        {|
          l := existT x l;
          input := input;
          destination := destination;
          output := output
        |}))
→ transition
    (VLSM.l
       (composite_transition_item_projection
          (equivocator_IM IM)
          {|
            l := existT x l;
            input := input;
            destination := destination;
            output := output
          |}))
    (s x,
     VLSM.input
       (composite_transition_item_projection
          (equivocator_IM IM)
          {|
            l := existT x l;
            input := input;
            destination := destination;
            output := output
          |})) =
  (VLSM.destination
     (composite_transition_item_projection
        (equivocator_IM IM)
        {|
          l := existT x l;
          input := input;
          destination := destination;
          output := output
        |}),
   VLSM.output
     (composite_transition_item_projection
        (equivocator_IM IM)
        {|
          l := existT x l;
          input := input;
          destination := destination;
          output := output
        |}))
  → ∃ Hex : existing_equivocator_label
              (IM x)
              (VLSM.l
                 (composite_transition_item_projection
                    (equivocator_IM IM)
                    {|
                      l := existT x l;
                      input := input;
                      destination :=
                      destination;
                      output := output
                    |})),
      equivocator_vlsm_transition_item_project
        (IM x)
        (composite_transition_item_projection
           (equivocator_IM IM)
           {|
             l := existT x l;
             input := input;
             destination := destination;
             output := output
           |}) (Existing 0) =
      Some
        (Some
           {|
             l :=
               existing_equivocator_label_extract
                 (IM x)
                 (VLSM.l
                    (composite_transition_item_projection
                      (equivocator_IM IM)
                      {|
                      l := existT x l;
                      input := input;
                      destination :=
                      destination;
                      output := output
                      |})) Hex;
             input :=
               VLSM.input
                 (composite_transition_item_projection
                    (equivocator_IM IM)
                    {|
                      l := existT x l;
                      input := input;
                      destination :=
                      destination;
                      output := output
                    |});
             destination :=
               equivocator_state_descriptor_project
                 (VLSM.destination
                    (composite_transition_item_projection
                      (equivocator_IM IM)
                      {|
                      l := existT x l;
                      input := input;
                      destination :=
                      destination;
                      output := output
                      |})) 
                 (Existing 0);
             output :=
               VLSM.output
                 (composite_transition_item_projection
                    (equivocator_IM IM)
                    {|
                      l := existT x l;
                      input := input;
                      destination :=
                      destination;
                      output := output
                    |})
           |}, Existing 0)
∃ Hex : existing_equivocator_label
          (IM
             (projT1
                (VLSM.l
                   {|
                     l := existT x l;
                     input := input;
                     destination := destination;
                     output := output
                   |})))
          (projT2
             (VLSM.l
                {|
                  l := existT x l;
                  input := input;
                  destination := destination;
                  output := output
                |})),
  let lx :=
    existT x
      (existing_equivocator_label_extract
         (IM
            (projT1
               (VLSM.l
                  {|
                    l := existT x l;
                    input := input;
                    destination := destination;
                    output := output
                  |})))
         (projT2
            (VLSM.l
               {|
                 l := existT x l;
                 input := input;
                 destination := destination;
                 output := output
               |})) Hex) in
  equivocators_transition_item_project eqv_descriptors
    {|
      l := existT x l;
      input := input;
      destination := destination;
      output := output
    |} =
  Some
    (Some
       {|
         l := lx;
         input :=
           VLSM.input
             {|
               l := existT x l;
               input := input;
               destination := destination;
               output := output
             |};
         destination :=
           equivocators_state_project IM
             eqv_descriptors
             (VLSM.destination
                {|
                  l := existT x l;
                  input := input;
                  destination := destination;
                  output := output
                |});
         output :=
           VLSM.output
             {|
               l := existT x l;
               input := input;
               destination := destination;
               output := output
             |}
       |}, eqv_descriptors)
  specialize (Heqv_pr Hv).message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
eqv_descriptors: equivocator_descriptors IM
x: index
l: label (equivocator_IM IM x)
input: option message
destination: state
  (composite_type (equivocator_IM IM))
output: option message
Hzero: eqv_descriptors x = Existing 0
Hdest_i: is_singleton_state (IM x)
  (VLSM.destination
     {|
       l := existT x l;
       input := input;
       destination := destination;
       output := output
     |} x)
s: composite_state (equivocator_IM IM)
Hv: equivocator_valid (IM x) l (s x, input)
Ht: (let (si', om') :=
   equivocator_transition (IM x) l (s x, input) in
 (state_update (equivocator_IM IM) s x si', om')) =
(destination, output)
Heqv_pr: transition
  (VLSM.l
     (composite_transition_item_projection
        (equivocator_IM IM)
        {|
          l := existT x l;
          input := input;
          destination := destination;
          output := output
        |}))
  (s x,
   VLSM.input
     (composite_transition_item_projection
        (equivocator_IM IM)
        {|
          l := existT x l;
          input := input;
          destination := destination;
          output := output
        |})) =
(VLSM.destination
   (composite_transition_item_projection
      (equivocator_IM IM)
      {|
        l := existT x l;
        input := input;
        destination := destination;
        output := output
      |}),
 VLSM.output
   (composite_transition_item_projection
      (equivocator_IM IM)
      {|
        l := existT x l;
        input := input;
        destination := destination;
        output := output
      |}))
→ ∃ Hex : existing_equivocator_label 
            (IM x)
            (VLSM.l
               (composite_transition_item_projection
                  (equivocator_IM IM)
                  {|
                    l := existT x l;
                    input := input;
                    destination :=
                      destination;
                    output := output
                  |})),
    equivocator_vlsm_transition_item_project
      (IM x)
      (composite_transition_item_projection
         (equivocator_IM IM)
         {|
           l := existT x l;
           input := input;
           destination := destination;
           output := output
         |}) (Existing 0) =
    Some
      (Some
         {|
           l :=
             existing_equivocator_label_extract
               (IM x)
               (VLSM.l
                  (composite_transition_item_projection
                     (equivocator_IM IM)
                     {|
                      l := existT x l;
                      input := input;
                      destination :=
                      destination;
                      output := output
                     |})) Hex;
           input :=
             VLSM.input
               (composite_transition_item_projection
                  (equivocator_IM IM)
                  {|
                    l := existT x l;
                    input := input;
                    destination :=
                      destination;
                    output := output
                  |});
           destination :=
             equivocator_state_descriptor_project
               (VLSM.destination
                  (composite_transition_item_projection
                     (equivocator_IM IM)
                     {|
                      l := existT x l;
                      input := input;
                      destination :=
                      destination;
                      output := output
                     |})) 
               (Existing 0);
           output :=
             VLSM.output
               (composite_transition_item_projection
                  (equivocator_IM IM)
                  {|
                    l := existT x l;
                    input := input;
                    destination :=
                      destination;
                    output := output
                  |})
         |}, Existing 0)
∃ Hex : existing_equivocator_label
          (IM
             (projT1
                (VLSM.l
                   {|
                     l := existT x l;
                     input := input;
                     destination := destination;
                     output := output
                   |})))
          (projT2
             (VLSM.l
                {|
                  l := existT x l;
                  input := input;
                  destination := destination;
                  output := output
                |})),
  let lx :=
    existT x
      (existing_equivocator_label_extract
         (IM
            (projT1
               (VLSM.l
                  {|
                    l := existT x l;
                    input := input;
                    destination := destination;
                    output := output
                  |})))
         (projT2
            (VLSM.l
               {|
                 l := existT x l;
                 input := input;
                 destination := destination;
                 output := output
               |})) Hex) in
  equivocators_transition_item_project eqv_descriptors
    {|
      l := existT x l;
      input := input;
      destination := destination;
      output := output
    |} =
  Some
    (Some
       {|
         l := lx;
         input :=
           VLSM.input
             {|
               l := existT x l;
               input := input;
               destination := destination;
               output := output
             |};
         destination :=
           equivocators_state_project IM
             eqv_descriptors
             (VLSM.destination
                {|
                  l := existT x l;
                  input := input;
                  destination := destination;
                  output := output
                |});
         output :=
           VLSM.output
             {|
               l := existT x l;
               input := input;
               destination := destination;
               output := output
             |}
       |}, eqv_descriptors)
  spec Heqv_pr.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
eqv_descriptors: equivocator_descriptors IM
x: index
l: label (equivocator_IM IM x)
input: option message
destination: state
  (composite_type (equivocator_IM IM))
output: option message
Hzero: eqv_descriptors x = Existing 0
Hdest_i: is_singleton_state (IM x)
  (VLSM.destination
     {|
       l := existT x l;
       input := input;
       destination := destination;
       output := output
     |} x)
s: composite_state (equivocator_IM IM)
Hv: equivocator_valid (IM x) l (s x, input)
Ht: (let (si', om') :=
   equivocator_transition (IM x) l (s x, input) in
 (state_update (equivocator_IM IM) s x si', om')) =
(destination, output)
Heqv_pr: transition
  (VLSM.l
     (composite_transition_item_projection
        (equivocator_IM IM)
        {|
          l := existT x l;
          input := input;
          destination := destination;
          output := output
        |}))
  (s x,
   VLSM.input
     (composite_transition_item_projection
        (equivocator_IM IM)
        {|
          l := existT x l;
          input := input;
          destination := destination;
          output := output
        |})) =
(VLSM.destination
   (composite_transition_item_projection
      (equivocator_IM IM)
      {|
        l := existT x l;
        input := input;
        destination := destination;
        output := output
      |}),
 VLSM.output
   (composite_transition_item_projection
      (equivocator_IM IM)
      {|
        l := existT x l;
        input := input;
        destination := destination;
        output := output
      |}))
→ ∃ Hex : existing_equivocator_label 
            (IM x)
            (VLSM.l
               (composite_transition_item_projection
                  (equivocator_IM IM)
                  {|
                    l := existT x l;
                    input := input;
                    destination :=
                      destination;
                    output := output
                  |})),
    equivocator_vlsm_transition_item_project
      (IM x)
      (composite_transition_item_projection
         (equivocator_IM IM)
         {|
           l := existT x l;
           input := input;
           destination := destination;
           output := output
         |}) (Existing 0) =
    Some
      (Some
         {|
           l :=
             existing_equivocator_label_extract
               (IM x)
               (VLSM.l
                  (composite_transition_item_projection
                     (equivocator_IM IM)
                     {|
                      l := existT x l;
                      input := input;
                      destination :=
                      destination;
                      output := output
                     |})) Hex;
           input :=
             VLSM.input
               (composite_transition_item_projection
                  (equivocator_IM IM)
                  {|
                    l := existT x l;
                    input := input;
                    destination :=
                      destination;
                    output := output
                  |});
           destination :=
             equivocator_state_descriptor_project
               (VLSM.destination
                  (composite_transition_item_projection
                     (equivocator_IM IM)
                     {|
                      l := existT x l;
                      input := input;
                      destination :=
                      destination;
                      output := output
                     |})) 
               (Existing 0);
           output :=
             VLSM.output
               (composite_transition_item_projection
                  (equivocator_IM IM)
                  {|
                    l := existT x l;
                    input := input;
                    destination :=
                      destination;
                    output := output
                  |})
         |}, Existing 0)
transition
  (VLSM.l
     (composite_transition_item_projection
        (equivocator_IM IM)
        {|
          l := existT x l;
          input := input;
          destination := destination;
          output := output
        |}))
  (s x,
   VLSM.input
     (composite_transition_item_projection
        (equivocator_IM IM)
        {|
          l := existT x l;
          input := input;
          destination := destination;
          output := output
        |})) =
(VLSM.destination
   (composite_transition_item_projection
      (equivocator_IM IM)
      {|
        l := existT x l;
        input := input;
        destination := destination;
        output := output
      |}),
 VLSM.output
   (composite_transition_item_projection
      (equivocator_IM IM)
      {|
        l := existT x l;
        input := input;
        destination := destination;
        output := output
      |}))
message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
eqv_descriptors: equivocator_descriptors IM
x: index
l: label (equivocator_IM IM x)
input: option message
destination: state
  (composite_type (equivocator_IM IM))
output: option message
Hzero: eqv_descriptors x = Existing 0
Hdest_i: is_singleton_state (IM x)
  (VLSM.destination
     {|
       l := existT x l;
       input := input;
       destination := destination;
       output := output
     |} x)
s: composite_state (equivocator_IM IM)
Hv: equivocator_valid (IM x) l (s x, input)
Ht: (let (si', om') :=
   equivocator_transition (IM x) l (s x, input) in
 (state_update (equivocator_IM IM) s x si', om')) =
(destination, output)
Heqv_pr: ∃ Hex : existing_equivocator_label 
          (IM x)
          (VLSM.l
             (composite_transition_item_projection
                (equivocator_IM IM)
                {|
                  l := existT x l;
                  input := input;
                  destination := destination;
                  output := output
                |})),
  equivocator_vlsm_transition_item_project
    (IM x)
    (composite_transition_item_projection
       (equivocator_IM IM)
       {|
         l := existT x l;
         input := input;
         destination := destination;
         output := output
       |}) (Existing 0) =
  Some
    (Some
       {|
         l :=
           existing_equivocator_label_extract
             (IM x)
             (VLSM.l
                (composite_transition_item_projection
                   (equivocator_IM IM)
                   {|
                     l := existT x l;
                     input := input;
                     destination :=
                      destination;
                     output := output
                   |})) Hex;
         input :=
           VLSM.input
             (composite_transition_item_projection
                (equivocator_IM IM)
                {|
                  l := existT x l;
                  input := input;
                  destination := destination;
                  output := output
                |});
         destination :=
           equivocator_state_descriptor_project
             (VLSM.destination
                (composite_transition_item_projection
                   (equivocator_IM IM)
                   {|
                     l := existT x l;
                     input := input;
                     destination :=
                      destination;
                     output := output
                   |})) 
             (Existing 0);
         output :=
           VLSM.output
             (composite_transition_item_projection
                (equivocator_IM IM)
                {|
                  l := existT x l;
                  input := input;
                  destination := destination;
                  output := output
                |})
       |}, Existing 0)
∃ Hex : existing_equivocator_label
          (IM
             (projT1
                (VLSM.l
                   {|
                     l := existT x l;
                     input := input;
                     destination := destination;
                     output := output
                   |})))
          (projT2
             (VLSM.l
                {|
                  l := existT x l;
                  input := input;
                  destination := destination;
                  output := output
                |})),
  let lx :=
    existT x
      (existing_equivocator_label_extract
         (IM
            (projT1
               (VLSM.l
                  {|
                    l := existT x l;
                    input := input;
                    destination := destination;
                    output := output
                  |})))
         (projT2
            (VLSM.l
               {|
                 l := existT x l;
                 input := input;
                 destination := destination;
                 output := output
               |})) Hex) in
  equivocators_transition_item_project eqv_descriptors
    {|
      l := existT x l;
      input := input;
      destination := destination;
      output := output
    |} =
  Some
    (Some
       {|
         l := lx;
         input :=
           VLSM.input
             {|
               l := existT x l;
               input := input;
               destination := destination;
               output := output
             |};
         destination :=
           equivocators_state_project IM
             eqv_descriptors
             (VLSM.destination
                {|
                  l := existT x l;
                  input := input;
                  destination := destination;
                  output := output
                |});
         output :=
           VLSM.output
             {|
               l := existT x l;
               input := input;
               destination := destination;
               output := output
             |}
       |}, eqv_descriptors)
  {message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
eqv_descriptors: equivocator_descriptors IM
x: index
l: label (equivocator_IM IM x)
input: option message
destination: state
  (composite_type (equivocator_IM IM))
output: option message
Hzero: eqv_descriptors x = Existing 0
Hdest_i: is_singleton_state (IM x)
  (VLSM.destination
     {|
       l := existT x l;
       input := input;
       destination := destination;
       output := output
     |} x)
s: composite_state (equivocator_IM IM)
Hv: equivocator_valid (IM x) l (s x, input)
Ht: (let (si', om') :=
   equivocator_transition (IM x) l (s x, input) in
 (state_update (equivocator_IM IM) s x si', om')) =
(destination, output)
Heqv_pr: transition
  (VLSM.l
     (composite_transition_item_projection
        (equivocator_IM IM)
        {|
          l := existT x l;
          input := input;
          destination := destination;
          output := output
        |}))
  (s x,
   VLSM.input
     (composite_transition_item_projection
        (equivocator_IM IM)
        {|
          l := existT x l;
          input := input;
          destination := destination;
          output := output
        |})) =
(VLSM.destination
   (composite_transition_item_projection
      (equivocator_IM IM)
      {|
        l := existT x l;
        input := input;
        destination := destination;
        output := output
      |}),
 VLSM.output
   (composite_transition_item_projection
      (equivocator_IM IM)
      {|
        l := existT x l;
        input := input;
        destination := destination;
        output := output
      |}))
→ ∃ Hex : existing_equivocator_label 
            (IM x)
            (VLSM.l
               (composite_transition_item_projection
                  (equivocator_IM IM)
                  {|
                    l := existT x l;
                    input := input;
                    destination :=
                      destination;
                    output := output
                  |})),
    equivocator_vlsm_transition_item_project
      (IM x)
      (composite_transition_item_projection
         (equivocator_IM IM)
         {|
           l := existT x l;
           input := input;
           destination := destination;
           output := output
         |}) (Existing 0) =
    Some
      (Some
         {|
           l :=
             existing_equivocator_label_extract
               (IM x)
               (VLSM.l
                  (composite_transition_item_projection
                     (equivocator_IM IM)
                     {|
                      l := existT x l;
                      input := input;
                      destination :=
                      destination;
                      output := output
                     |})) Hex;
           input :=
             VLSM.input
               (composite_transition_item_projection
                  (equivocator_IM IM)
                  {|
                    l := existT x l;
                    input := input;
                    destination :=
                      destination;
                    output := output
                  |});
           destination :=
             equivocator_state_descriptor_project
               (VLSM.destination
                  (composite_transition_item_projection
                     (equivocator_IM IM)
                     {|
                      l := existT x l;
                      input := input;
                      destination :=
                      destination;
                      output := output
                     |})) 
               (Existing 0);
           output :=
             VLSM.output
               (composite_transition_item_projection
                  (equivocator_IM IM)
                  {|
                    l := existT x l;
                    input := input;
                    destination :=
                      destination;
                    output := output
                  |})
         |}, Existing 0)
transition
  (VLSM.l
     (composite_transition_item_projection
        (equivocator_IM IM)
        {|
          l := existT x l;
          input := input;
          destination := destination;
          output := output
        |}))
  (s x,
   VLSM.input
     (composite_transition_item_projection
        (equivocator_IM IM)
        {|
          l := existT x l;
          input := input;
          destination := destination;
          output := output
        |})) =
(VLSM.destination
   (composite_transition_item_projection
      (equivocator_IM IM)
      {|
        l := existT x l;
        input := input;
        destination := destination;
        output := output
      |}),
 VLSM.output
   (composite_transition_item_projection
      (equivocator_IM IM)
      {|
        l := existT x l;
        input := input;
        destination := destination;
        output := output
      |})) simpl.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
eqv_descriptors: equivocator_descriptors IM
x: index
l: label (equivocator_IM IM x)
input: option message
destination: state
  (composite_type (equivocator_IM IM))
output: option message
Hzero: eqv_descriptors x = Existing 0
Hdest_i: is_singleton_state (IM x)
  (VLSM.destination
     {|
       l := existT x l;
       input := input;
       destination := destination;
       output := output
     |} x)
s: composite_state (equivocator_IM IM)
Hv: equivocator_valid (IM x) l (s x, input)
Ht: (let (si', om') :=
   equivocator_transition (IM x) l (s x, input) in
 (state_update (equivocator_IM IM) s x si', om')) =
(destination, output)
Heqv_pr: transition
  (VLSM.l
     (composite_transition_item_projection
        (equivocator_IM IM)
        {|
          l := existT x l;
          input := input;
          destination := destination;
          output := output
        |}))
  (s x,
   VLSM.input
     (composite_transition_item_projection
        (equivocator_IM IM)
        {|
          l := existT x l;
          input := input;
          destination := destination;
          output := output
        |})) =
(VLSM.destination
   (composite_transition_item_projection
      (equivocator_IM IM)
      {|
        l := existT x l;
        input := input;
        destination := destination;
        output := output
      |}),
 VLSM.output
   (composite_transition_item_projection
      (equivocator_IM IM)
      {|
        l := existT x l;
        input := input;
        destination := destination;
        output := output
      |}))
→ ∃ Hex : existing_equivocator_label 
            (IM x)
            (VLSM.l
               (composite_transition_item_projection
                  (equivocator_IM IM)
                  {|
                    l := existT x l;
                    input := input;
                    destination :=
                      destination;
                    output := output
                  |})),
    equivocator_vlsm_transition_item_project
      (IM x)
      (composite_transition_item_projection
         (equivocator_IM IM)
         {|
           l := existT x l;
           input := input;
           destination := destination;
           output := output
         |}) (Existing 0) =
    Some
      (Some
         {|
           l :=
             existing_equivocator_label_extract
               (IM x)
               (VLSM.l
                  (composite_transition_item_projection
                     (equivocator_IM IM)
                     {|
                      l := existT x l;
                      input := input;
                      destination :=
                      destination;
                      output := output
                     |})) Hex;
           input :=
             VLSM.input
               (composite_transition_item_projection
                  (equivocator_IM IM)
                  {|
                    l := existT x l;
                    input := input;
                    destination :=
                      destination;
                    output := output
                  |});
           destination :=
             equivocator_state_descriptor_project
               (VLSM.destination
                  (composite_transition_item_projection
                     (equivocator_IM IM)
                     {|
                      l := existT x l;
                      input := input;
                      destination :=
                      destination;
                      output := output
                     |})) 
               (Existing 0);
           output :=
             VLSM.output
               (composite_transition_item_projection
                  (equivocator_IM IM)
                  {|
                    l := existT x l;
                    input := input;
                    destination :=
                      destination;
                    output := output
                  |})
         |}, Existing 0)
equivocator_transition (IM x)
  (eq_rect_r (λ n : index, EquivocatorLabel (IM n)) l
     eq_refl) (s x, input) = (destination x, output) unfold eq_rect_r.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
eqv_descriptors: equivocator_descriptors IM
x: index
l: label (equivocator_IM IM x)
input: option message
destination: state
  (composite_type (equivocator_IM IM))
output: option message
Hzero: eqv_descriptors x = Existing 0
Hdest_i: is_singleton_state (IM x)
  (VLSM.destination
     {|
       l := existT x l;
       input := input;
       destination := destination;
       output := output
     |} x)
s: composite_state (equivocator_IM IM)
Hv: equivocator_valid (IM x) l (s x, input)
Ht: (let (si', om') :=
   equivocator_transition (IM x) l (s x, input) in
 (state_update (equivocator_IM IM) s x si', om')) =
(destination, output)
Heqv_pr: transition
  (VLSM.l
     (composite_transition_item_projection
        (equivocator_IM IM)
        {|
          l := existT x l;
          input := input;
          destination := destination;
          output := output
        |}))
  (s x,
   VLSM.input
     (composite_transition_item_projection
        (equivocator_IM IM)
        {|
          l := existT x l;
          input := input;
          destination := destination;
          output := output
        |})) =
(VLSM.destination
   (composite_transition_item_projection
      (equivocator_IM IM)
      {|
        l := existT x l;
        input := input;
        destination := destination;
        output := output
      |}),
 VLSM.output
   (composite_transition_item_projection
      (equivocator_IM IM)
      {|
        l := existT x l;
        input := input;
        destination := destination;
        output := output
      |}))
→ ∃ Hex : existing_equivocator_label 
            (IM x)
            (VLSM.l
               (composite_transition_item_projection
                  (equivocator_IM IM)
                  {|
                    l := existT x l;
                    input := input;
                    destination :=
                      destination;
                    output := output
                  |})),
    equivocator_vlsm_transition_item_project
      (IM x)
      (composite_transition_item_projection
         (equivocator_IM IM)
         {|
           l := existT x l;
           input := input;
           destination := destination;
           output := output
         |}) (Existing 0) =
    Some
      (Some
         {|
           l :=
             existing_equivocator_label_extract
               (IM x)
               (VLSM.l
                  (composite_transition_item_projection
                     (equivocator_IM IM)
                     {|
                      l := existT x l;
                      input := input;
                      destination :=
                      destination;
                      output := output
                     |})) Hex;
           input :=
             VLSM.input
               (composite_transition_item_projection
                  (equivocator_IM IM)
                  {|
                    l := existT x l;
                    input := input;
                    destination :=
                      destination;
                    output := output
                  |});
           destination :=
             equivocator_state_descriptor_project
               (VLSM.destination
                  (composite_transition_item_projection
                     (equivocator_IM IM)
                     {|
                      l := existT x l;
                      input := input;
                      destination :=
                      destination;
                      output := output
                     |})) 
               (Existing 0);
           output :=
             VLSM.output
               (composite_transition_item_projection
                  (equivocator_IM IM)
                  {|
                    l := existT x l;
                    input := input;
                    destination :=
                      destination;
                    output := output
                  |})
         |}, Existing 0)
equivocator_transition (IM x)
  (eq_rect x (λ y : index, EquivocatorLabel (IM y)) l
     x (eq_sym eq_refl)) (s x, input) =
(destination x, output) simpl.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
eqv_descriptors: equivocator_descriptors IM
x: index
l: label (equivocator_IM IM x)
input: option message
destination: state
  (composite_type (equivocator_IM IM))
output: option message
Hzero: eqv_descriptors x = Existing 0
Hdest_i: is_singleton_state (IM x)
  (VLSM.destination
     {|
       l := existT x l;
       input := input;
       destination := destination;
       output := output
     |} x)
s: composite_state (equivocator_IM IM)
Hv: equivocator_valid (IM x) l (s x, input)
Ht: (let (si', om') :=
   equivocator_transition (IM x) l (s x, input) in
 (state_update (equivocator_IM IM) s x si', om')) =
(destination, output)
Heqv_pr: transition
  (VLSM.l
     (composite_transition_item_projection
        (equivocator_IM IM)
        {|
          l := existT x l;
          input := input;
          destination := destination;
          output := output
        |}))
  (s x,
   VLSM.input
     (composite_transition_item_projection
        (equivocator_IM IM)
        {|
          l := existT x l;
          input := input;
          destination := destination;
          output := output
        |})) =
(VLSM.destination
   (composite_transition_item_projection
      (equivocator_IM IM)
      {|
        l := existT x l;
        input := input;
        destination := destination;
        output := output
      |}),
 VLSM.output
   (composite_transition_item_projection
      (equivocator_IM IM)
      {|
        l := existT x l;
        input := input;
        destination := destination;
        output := output
      |}))
→ ∃ Hex : existing_equivocator_label 
            (IM x)
            (VLSM.l
               (composite_transition_item_projection
                  (equivocator_IM IM)
                  {|
                    l := existT x l;
                    input := input;
                    destination :=
                      destination;
                    output := output
                  |})),
    equivocator_vlsm_transition_item_project
      (IM x)
      (composite_transition_item_projection
         (equivocator_IM IM)
         {|
           l := existT x l;
           input := input;
           destination := destination;
           output := output
         |}) (Existing 0) =
    Some
      (Some
         {|
           l :=
             existing_equivocator_label_extract
               (IM x)
               (VLSM.l
                  (composite_transition_item_projection
                     (equivocator_IM IM)
                     {|
                      l := existT x l;
                      input := input;
                      destination :=
                      destination;
                      output := output
                     |})) Hex;
           input :=
             VLSM.input
               (composite_transition_item_projection
                  (equivocator_IM IM)
                  {|
                    l := existT x l;
                    input := input;
                    destination :=
                      destination;
                    output := output
                  |});
           destination :=
             equivocator_state_descriptor_project
               (VLSM.destination
                  (composite_transition_item_projection
                     (equivocator_IM IM)
                     {|
                      l := existT x l;
                      input := input;
                      destination :=
                      destination;
                      output := output
                     |})) 
               (Existing 0);
           output :=
             VLSM.output
               (composite_transition_item_projection
                  (equivocator_IM IM)
                  {|
                    l := existT x l;
                    input := input;
                    destination :=
                      destination;
                    output := output
                  |})
         |}, Existing 0)
equivocator_transition (IM x) l (s x, input) =
(destination x, output)
    case_match; clear -Ht; inversion Ht; subst.message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
x: index
output: option message
s: composite_state (equivocator_IM IM)
e: equivocator_state (IM x)
Ht: (state_update (equivocator_IM IM) s x e, output) =
(state_update (equivocator_IM IM) s x e, output)
(e, output) =
(state_update (equivocator_IM IM) s x e x, output)
    by state_update_simpl.
  }message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
eqv_descriptors: equivocator_descriptors IM
x: index
l: label (equivocator_IM IM x)
input: option message
destination: state
  (composite_type (equivocator_IM IM))
output: option message
Hzero: eqv_descriptors x = Existing 0
Hdest_i: is_singleton_state (IM x)
  (VLSM.destination
     {|
       l := existT x l;
       input := input;
       destination := destination;
       output := output
     |} x)
s: composite_state (equivocator_IM IM)
Hv: equivocator_valid (IM x) l (s x, input)
Ht: (let (si', om') :=
   equivocator_transition (IM x) l (s x, input) in
 (state_update (equivocator_IM IM) s x si', om')) =
(destination, output)
Heqv_pr: ∃ Hex : existing_equivocator_label 
          (IM x)
          (VLSM.l
             (composite_transition_item_projection
                (equivocator_IM IM)
                {|
                  l := existT x l;
                  input := input;
                  destination := destination;
                  output := output
                |})),
  equivocator_vlsm_transition_item_project
    (IM x)
    (composite_transition_item_projection
       (equivocator_IM IM)
       {|
         l := existT x l;
         input := input;
         destination := destination;
         output := output
       |}) (Existing 0) =
  Some
    (Some
       {|
         l :=
           existing_equivocator_label_extract
             (IM x)
             (VLSM.l
                (composite_transition_item_projection
                   (equivocator_IM IM)
                   {|
                     l := existT x l;
                     input := input;
                     destination :=
                      destination;
                     output := output
                   |})) Hex;
         input :=
           VLSM.input
             (composite_transition_item_projection
                (equivocator_IM IM)
                {|
                  l := existT x l;
                  input := input;
                  destination := destination;
                  output := output
                |});
         destination :=
           equivocator_state_descriptor_project
             (VLSM.destination
                (composite_transition_item_projection
                   (equivocator_IM IM)
                   {|
                     l := existT x l;
                     input := input;
                     destination :=
                      destination;
                     output := output
                   |})) 
             (Existing 0);
         output :=
           VLSM.output
             (composite_transition_item_projection
                (equivocator_IM IM)
                {|
                  l := existT x l;
                  input := input;
                  destination := destination;
                  output := output
                |})
       |}, Existing 0)
∃ Hex : existing_equivocator_label
          (IM
             (projT1
                (VLSM.l
                   {|
                     l := existT x l;
                     input := input;
                     destination := destination;
                     output := output
                   |})))
          (projT2
             (VLSM.l
                {|
                  l := existT x l;
                  input := input;
                  destination := destination;
                  output := output
                |})),
  let lx :=
    existT x
      (existing_equivocator_label_extract
         (IM
            (projT1
               (VLSM.l
                  {|
                    l := existT x l;
                    input := input;
                    destination := destination;
                    output := output
                  |})))
         (projT2
            (VLSM.l
               {|
                 l := existT x l;
                 input := input;
                 destination := destination;
                 output := output
               |})) Hex) in
  equivocators_transition_item_project eqv_descriptors
    {|
      l := existT x l;
      input := input;
      destination := destination;
      output := output
    |} =
  Some
    (Some
       {|
         l := lx;
         input :=
           VLSM.input
             {|
               l := existT x l;
               input := input;
               destination := destination;
               output := output
             |};
         destination :=
           equivocators_state_project IM
             eqv_descriptors
             (VLSM.destination
                {|
                  l := existT x l;
                  input := input;
                  destination := destination;
                  output := output
                |});
         output :=
           VLSM.output
             {|
               l := existT x l;
               input := input;
               destination := destination;
               output := output
             |}
       |}, eqv_descriptors)
  destruct Heqv_pr as [Hex Heqv_pr].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
eqv_descriptors: equivocator_descriptors IM
x: index
l: label (equivocator_IM IM x)
input: option message
destination: state
  (composite_type (equivocator_IM IM))
output: option message
Hzero: eqv_descriptors x = Existing 0
Hdest_i: is_singleton_state (IM x)
  (VLSM.destination
     {|
       l := existT x l;
       input := input;
       destination := destination;
       output := output
     |} x)
s: composite_state (equivocator_IM IM)
Hv: equivocator_valid (IM x) l (s x, input)
Ht: (let (si', om') :=
   equivocator_transition (IM x) l (s x, input) in
 (state_update (equivocator_IM IM) s x si', om')) =
(destination, output)
Hex: existing_equivocator_label 
  (IM x)
  (VLSM.l
     (composite_transition_item_projection
        (equivocator_IM IM)
        {|
          l := existT x l;
          input := input;
          destination := destination;
          output := output
        |}))
Heqv_pr: equivocator_vlsm_transition_item_project
  (IM x)
  (composite_transition_item_projection
     (equivocator_IM IM)
     {|
       l := existT x l;
       input := input;
       destination := destination;
       output := output
     |}) (Existing 0) =
Some
  (Some
     {|
       l :=
         existing_equivocator_label_extract
           (IM x)
           (VLSM.l
              (composite_transition_item_projection
                 (equivocator_IM IM)
                 {|
                   l := existT x l;
                   input := input;
                   destination :=
                     destination;
                   output := output
                 |})) Hex;
       input :=
         VLSM.input
           (composite_transition_item_projection
              (equivocator_IM IM)
              {|
                l := existT x l;
                input := input;
                destination := destination;
                output := output
              |});
       destination :=
         equivocator_state_descriptor_project
           (VLSM.destination
              (composite_transition_item_projection
                 (equivocator_IM IM)
                 {|
                   l := existT x l;
                   input := input;
                   destination :=
                     destination;
                   output := output
                 |})) (Existing 0);
       output :=
         VLSM.output
           (composite_transition_item_projection
              (equivocator_IM IM)
              {|
                l := existT x l;
                input := input;
                destination := destination;
                output := output
              |})
     |}, Existing 0)
∃ Hex : existing_equivocator_label
          (IM
             (projT1
                (VLSM.l
                   {|
                     l := existT x l;
                     input := input;
                     destination := destination;
                     output := output
                   |})))
          (projT2
             (VLSM.l
                {|
                  l := existT x l;
                  input := input;
                  destination := destination;
                  output := output
                |})),
  let lx :=
    existT x
      (existing_equivocator_label_extract
         (IM
            (projT1
               (VLSM.l
                  {|
                    l := existT x l;
                    input := input;
                    destination := destination;
                    output := output
                  |})))
         (projT2
            (VLSM.l
               {|
                 l := existT x l;
                 input := input;
                 destination := destination;
                 output := output
               |})) Hex) in
  equivocators_transition_item_project eqv_descriptors
    {|
      l := existT x l;
      input := input;
      destination := destination;
      output := output
    |} =
  Some
    (Some
       {|
         l := lx;
         input :=
           VLSM.input
             {|
               l := existT x l;
               input := input;
               destination := destination;
               output := output
             |};
         destination :=
           equivocators_state_project IM
             eqv_descriptors
             (VLSM.destination
                {|
                  l := existT x l;
                  input := input;
                  destination := destination;
                  output := output
                |});
         output :=
           VLSM.output
             {|
               l := existT x l;
               input := input;
               destination := destination;
               output := output
             |}
       |}, eqv_descriptors)
  exists Hex.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
eqv_descriptors: equivocator_descriptors IM
x: index
l: label (equivocator_IM IM x)
input: option message
destination: state
  (composite_type (equivocator_IM IM))
output: option message
Hzero: eqv_descriptors x = Existing 0
Hdest_i: is_singleton_state (IM x)
  (VLSM.destination
     {|
       l := existT x l;
       input := input;
       destination := destination;
       output := output
     |} x)
s: composite_state (equivocator_IM IM)
Hv: equivocator_valid (IM x) l (s x, input)
Ht: (let (si', om') :=
   equivocator_transition (IM x) l (s x, input) in
 (state_update (equivocator_IM IM) s x si', om')) =
(destination, output)
Hex: existing_equivocator_label 
  (IM x)
  (VLSM.l
     (composite_transition_item_projection
        (equivocator_IM IM)
        {|
          l := existT x l;
          input := input;
          destination := destination;
          output := output
        |}))
Heqv_pr: equivocator_vlsm_transition_item_project
  (IM x)
  (composite_transition_item_projection
     (equivocator_IM IM)
     {|
       l := existT x l;
       input := input;
       destination := destination;
       output := output
     |}) (Existing 0) =
Some
  (Some
     {|
       l :=
         existing_equivocator_label_extract
           (IM x)
           (VLSM.l
              (composite_transition_item_projection
                 (equivocator_IM IM)
                 {|
                   l := existT x l;
                   input := input;
                   destination :=
                     destination;
                   output := output
                 |})) Hex;
       input :=
         VLSM.input
           (composite_transition_item_projection
              (equivocator_IM IM)
              {|
                l := existT x l;
                input := input;
                destination := destination;
                output := output
              |});
       destination :=
         equivocator_state_descriptor_project
           (VLSM.destination
              (composite_transition_item_projection
                 (equivocator_IM IM)
                 {|
                   l := existT x l;
                   input := input;
                   destination :=
                     destination;
                   output := output
                 |})) (Existing 0);
       output :=
         VLSM.output
           (composite_transition_item_projection
              (equivocator_IM IM)
              {|
                l := existT x l;
                input := input;
                destination := destination;
                output := output
              |})
     |}, Existing 0)
let lx :=
  existT x
    (existing_equivocator_label_extract
       (IM
          (projT1
             (VLSM.l
                {|
                  l := existT x l;
                  input := input;
                  destination := destination;
                  output := output
                |})))
       (projT2
          (VLSM.l
             {|
               l := existT x l;
               input := input;
               destination := destination;
               output := output
             |})) Hex) in
equivocators_transition_item_project eqv_descriptors
  {|
    l := existT x l;
    input := input;
    destination := destination;
    output := output
  |} =
Some
  (Some
     {|
       l := lx;
       input :=
         VLSM.input
           {|
             l := existT x l;
             input := input;
             destination := destination;
             output := output
           |};
       destination :=
         equivocators_state_project IM eqv_descriptors
           (VLSM.destination
              {|
                l := existT x l;
                input := input;
                destination := destination;
                output := output
              |});
       output :=
         VLSM.output
           {|
             l := existT x l;
             input := input;
             destination := destination;
             output := output
           |}
     |}, eqv_descriptors)
  unfold equivocators_transition_item_project.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
eqv_descriptors: equivocator_descriptors IM
x: index
l: label (equivocator_IM IM x)
input: option message
destination: state
  (composite_type (equivocator_IM IM))
output: option message
Hzero: eqv_descriptors x = Existing 0
Hdest_i: is_singleton_state (IM x)
  (VLSM.destination
     {|
       l := existT x l;
       input := input;
       destination := destination;
       output := output
     |} x)
s: composite_state (equivocator_IM IM)
Hv: equivocator_valid (IM x) l (s x, input)
Ht: (let (si', om') :=
   equivocator_transition (IM x) l (s x, input) in
 (state_update (equivocator_IM IM) s x si', om')) =
(destination, output)
Hex: existing_equivocator_label 
  (IM x)
  (VLSM.l
     (composite_transition_item_projection
        (equivocator_IM IM)
        {|
          l := existT x l;
          input := input;
          destination := destination;
          output := output
        |}))
Heqv_pr: equivocator_vlsm_transition_item_project
  (IM x)
  (composite_transition_item_projection
     (equivocator_IM IM)
     {|
       l := existT x l;
       input := input;
       destination := destination;
       output := output
     |}) (Existing 0) =
Some
  (Some
     {|
       l :=
         existing_equivocator_label_extract
           (IM x)
           (VLSM.l
              (composite_transition_item_projection
                 (equivocator_IM IM)
                 {|
                   l := existT x l;
                   input := input;
                   destination :=
                     destination;
                   output := output
                 |})) Hex;
       input :=
         VLSM.input
           (composite_transition_item_projection
              (equivocator_IM IM)
              {|
                l := existT x l;
                input := input;
                destination := destination;
                output := output
              |});
       destination :=
         equivocator_state_descriptor_project
           (VLSM.destination
              (composite_transition_item_projection
                 (equivocator_IM IM)
                 {|
                   l := existT x l;
                   input := input;
                   destination :=
                     destination;
                   output := output
                 |})) (Existing 0);
       output :=
         VLSM.output
           (composite_transition_item_projection
              (equivocator_IM IM)
              {|
                l := existT x l;
                input := input;
                destination := destination;
                output := output
              |})
     |}, Existing 0)
match
  equivocator_vlsm_transition_item_project
    (IM
       (projT1
          (VLSM.l
             {|
               l := existT x l;
               input := input;
               destination := destination;
               output := output
             |})))
    (composite_transition_item_projection
       (equivocator_IM IM)
       {|
         l := existT x l;
         input := input;
         destination := destination;
         output := output
       |})
    (eqv_descriptors
       (projT1
          (VLSM.l
             {|
               l := existT x l;
               input := input;
               destination := destination;
               output := output
             |})))
with
| Some (Some item', deqv') =>
    Some
      (Some
         {|
           l :=
             existT
               (projT1
                  (VLSM.l
                     {|
                       l := existT x l;
                       input := input;
                       destination := destination;
                       output := output
                     |})) (VLSM.l item');
           input :=
             VLSM.input
               {|
                 l := existT x l;
                 input := input;
                 destination := destination;
                 output := output
               |};
           destination :=
             equivocators_state_project IM
               eqv_descriptors
               (VLSM.destination
                  {|
                    l := existT x l;
                    input := input;
                    destination := destination;
                    output := output
                  |});
           output :=
             VLSM.output
               {|
                 l := existT x l;
                 input := input;
                 destination := destination;
                 output := output
               |}
         |},
       equivocator_descriptors_update IM
         eqv_descriptors
         (projT1
            (VLSM.l
               {|
                 l := existT x l;
                 input := input;
                 destination := destination;
                 output := output
               |})) deqv')
| Some (None, deqv') =>
    Some
      (None,
       equivocator_descriptors_update IM
         eqv_descriptors
         (projT1
            (VLSM.l
               {|
                 l := existT x l;
                 input := input;
                 destination := destination;
                 output := output
               |})) deqv')
| None => None
end =
Some
  (Some
     {|
       l :=
         existT x
           (existing_equivocator_label_extract
              (IM
                 (projT1
                    (VLSM.l
                       {|
                         l := existT x l;
                         input := input;
                         destination := destination;
                         output := output
                       |})))
              (projT2
                 (VLSM.l
                    {|
                      l := existT x l;
                      input := input;
                      destination := destination;
                      output := output
                    |})) Hex);
       input :=
         VLSM.input
           {|
             l := existT x l;
             input := input;
             destination := destination;
             output := output
           |};
       destination :=
         equivocators_state_project IM eqv_descriptors
           (VLSM.destination
              {|
                l := existT x l;
                input := input;
                destination := destination;
                output := output
              |});
       output :=
         VLSM.output
           {|
             l := existT x l;
             input := input;
             destination := destination;
             output := output
           |}
     |}, eqv_descriptors)
  cbn in *.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
eqv_descriptors: equivocator_descriptors IM
x: index
l: EquivocatorLabel (IM x)
input: option message
destination: composite_state (equivocator_IM IM)
output: option message
Hzero: eqv_descriptors x = Existing 0
Hdest_i: is_singleton_state (IM x) (destination x)
s: composite_state (equivocator_IM IM)
Hv: equivocator_valid (IM x) l (s x, input)
Ht: (let (si', om') :=
   equivocator_transition (IM x) l (s x, input) in
 (state_update (equivocator_IM IM) s x si', om')) =
(destination, output)
Hex: existing_equivocator_label (IM x) l
Heqv_pr: match
  equivocator_state_project 
    (destination x) 0
with
| Some sj =>
    match l with
    | Spawn sn =>
        if
         decide
           (0 =
            equivocator_state_last
              (destination x))
        then Some (None, NewMachine sn)
        else Some (None, Existing 0)
    | ContinueWith i lx =>
        if decide (i = 0)
        then
         Some
           (Some
              {|
                l := lx;
                input := input;
                destination := sj;
                output := output
              |}, Existing i)
        else Some (None, Existing 0)
    | ForkWith i lx =>
        if
         decide
           (0 =
            equivocator_state_last
              (destination x))
        then
         Some
           (Some
              {|
                l := lx;
                input := input;
                destination := sj;
                output := output
              |}, Existing i)
        else Some (None, Existing 0)
    end
| None => None
end =
Some
  (Some
     {|
       l :=
         existing_equivocator_label_extract
           (IM x) l Hex;
       input := input;
       destination :=
         default
           (equivocator_state_zero
              (destination x))
           (equivocator_state_project
              (destination x) 0);
       output := output
     |}, Existing 0)
match
  equivocator_vlsm_transition_item_project (IM x)
    (composite_transition_item_projection
       (equivocator_IM IM)
       {|
         l := existT x l;
         input := input;
         destination := destination;
         output := output
       |}) (eqv_descriptors x)
with
| Some (Some item', deqv') =>
    Some
      (Some
         {|
           l := existT x (VLSM.l item');
           input := input;
           destination :=
             equivocators_state_project IM
               eqv_descriptors destination;
           output := output
         |},
       equivocator_descriptors_update IM
         eqv_descriptors x deqv')
| Some (None, deqv') =>
    Some
      (None,
       equivocator_descriptors_update IM
         eqv_descriptors x deqv')
| None => None
end =
Some
  (Some
     {|
       l :=
         existT x
           (existing_equivocator_label_extract (IM x)
              l Hex);
       input := input;
       destination :=
         equivocators_state_project IM eqv_descriptors
           destination;
       output := output
     |}, eqv_descriptors)
  unfold equivocator_vlsm_transition_item_project,
    composite_transition_item_projection; cbn.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
eqv_descriptors: equivocator_descriptors IM
x: index
l: EquivocatorLabel (IM x)
input: option message
destination: composite_state (equivocator_IM IM)
output: option message
Hzero: eqv_descriptors x = Existing 0
Hdest_i: is_singleton_state (IM x) (destination x)
s: composite_state (equivocator_IM IM)
Hv: equivocator_valid (IM x) l (s x, input)
Ht: (let (si', om') :=
   equivocator_transition (IM x) l (s x, input) in
 (state_update (equivocator_IM IM) s x si', om')) =
(destination, output)
Hex: existing_equivocator_label (IM x) l
Heqv_pr: match
  equivocator_state_project 
    (destination x) 0
with
| Some sj =>
    match l with
    | Spawn sn =>
        if
         decide
           (0 =
            equivocator_state_last
              (destination x))
        then Some (None, NewMachine sn)
        else Some (None, Existing 0)
    | ContinueWith i lx =>
        if decide (i = 0)
        then
         Some
           (Some
              {|
                l := lx;
                input := input;
                destination := sj;
                output := output
              |}, Existing i)
        else Some (None, Existing 0)
    | ForkWith i lx =>
        if
         decide
           (0 =
            equivocator_state_last
              (destination x))
        then
         Some
           (Some
              {|
                l := lx;
                input := input;
                destination := sj;
                output := output
              |}, Existing i)
        else Some (None, Existing 0)
    end
| None => None
end =
Some
  (Some
     {|
       l :=
         existing_equivocator_label_extract
           (IM x) l Hex;
       input := input;
       destination :=
         default
           (equivocator_state_zero
              (destination x))
           (equivocator_state_project
              (destination x) 0);
       output := output
     |}, Existing 0)
match
  match eqv_descriptors x with
  | NewMachine _ => Some (None, eqv_descriptors x)
  | Existing j =>
      match
        equivocator_state_project (destination x) j
      with
      | Some sj =>
          match l with
          | Spawn sn =>
              if
               decide
                 (j =
                  equivocator_state_last
                    (destination x))
              then Some (None, NewMachine sn)
              else Some (None, Existing j)
          | ContinueWith i lx =>
              if decide (i = j)
              then
               Some
                 (Some
                    {|
                      l := lx;
                      input := input;
                      destination := sj;
                      output := output
                    |}, Existing i)
              else Some (None, Existing j)
          | ForkWith i lx =>
              if
               decide
                 (j =
                  equivocator_state_last
                    (destination x))
              then
               Some
                 (Some
                    {|
                      l := lx;
                      input := input;
                      destination := sj;
                      output := output
                    |}, Existing i)
              else Some (None, Existing j)
          end
      | None => None
      end
  end
with
| Some (Some item', deqv') =>
    Some
      (Some
         {|
           l := existT x (VLSM.l item');
           input := input;
           destination :=
             equivocators_state_project IM
               eqv_descriptors destination;
           output := output
         |},
       equivocator_descriptors_update IM
         eqv_descriptors x deqv')
| Some (None, deqv') =>
    Some
      (None,
       equivocator_descriptors_update IM
         eqv_descriptors x deqv')
| None => None
end =
Some
  (Some
     {|
       l :=
         existT x
           (existing_equivocator_label_extract (IM x)
              l Hex);
       input := input;
       destination :=
         equivocators_state_project IM eqv_descriptors
           destination;
       output := output
     |}, eqv_descriptors)
  rewrite Hzero, Heqv_pr; cbn; repeat f_equal.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
eqv_descriptors: equivocator_descriptors IM
x: index
l: EquivocatorLabel (IM x)
input: option message
destination: composite_state (equivocator_IM IM)
output: option message
Hzero: eqv_descriptors x = Existing 0
Hdest_i: is_singleton_state (IM x) (destination x)
s: composite_state (equivocator_IM IM)
Hv: equivocator_valid (IM x) l (s x, input)
Ht: (let (si', om') :=
   equivocator_transition (IM x) l (s x, input) in
 (state_update (equivocator_IM IM) s x si', om')) =
(destination, output)
Hex: existing_equivocator_label (IM x) l
Heqv_pr: match
  equivocator_state_project 
    (destination x) 0
with
| Some sj =>
    match l with
    | Spawn sn =>
        if
         decide
           (0 =
            equivocator_state_last
              (destination x))
        then Some (None, NewMachine sn)
        else Some (None, Existing 0)
    | ContinueWith i lx =>
        if decide (i = 0)
        then
         Some
           (Some
              {|
                l := lx;
                input := input;
                destination := sj;
                output := output
              |}, Existing i)
        else Some (None, Existing 0)
    | ForkWith i lx =>
        if
         decide
           (0 =
            equivocator_state_last
              (destination x))
        then
         Some
           (Some
              {|
                l := lx;
                input := input;
                destination := sj;
                output := output
              |}, Existing i)
        else Some (None, Existing 0)
    end
| None => None
end =
Some
  (Some
     {|
       l :=
         existing_equivocator_label_extract
           (IM x) l Hex;
       input := input;
       destination :=
         default
           (equivocator_state_zero
              (destination x))
           (equivocator_state_project
              (destination x) 0);
       output := output
     |}, Existing 0)
equivocator_descriptors_update IM eqv_descriptors x
  (Existing 0) = eqv_descriptors
  by state_update_simpl.
Qed.

Lemma exists_equivocators_transition_item_project
  (item : composite_transition_item (equivocator_IM IM))
  (s : composite_state (equivocator_IM IM))
  (Hs : proper_existing_equivocator_label _ (projT2 (l item)) (s (projT1 (l item))))
  (Hv : composite_valid (equivocator_IM IM) (l item) (s, input item))
  (Ht : composite_transition (equivocator_IM IM) (l item) (s, input item) =
          (destination item, output item))
  : exists equivocators,
      not_equivocating_equivocator_descriptors IM equivocators (destination item)
      /\ exists (equivocators' : equivocator_descriptors IM)
        (lx : composite_label IM := existT (projT1 (l item))
          (existing_equivocator_label_extract _ _ (existing_equivocator_label_forget_proper _ Hs)))
        (sx : composite_state IM := equivocators_state_project IM equivocators (destination item))
      ,
        proper_equivocator_descriptors IM equivocators' s
        /\ equivocators_transition_item_project equivocators item = Some
          (Some ({| l := lx; input := input item; output := output item; destination := sx |}),
            equivocators').message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
item: composite_transition_item (equivocator_IM IM)
s: composite_state (equivocator_IM IM)
Hs: proper_existing_equivocator_label
  (IM (projT1 (l item))) 
  (projT2 (l item)) (s (projT1 (l item)))
Hv: composite_valid (equivocator_IM IM) 
  (l item) (s, input item)
Ht: composite_transition (equivocator_IM IM) 
  (l item) (s, input item) =
(destination item, output item)
∃ equivocators : equivocator_descriptors IM,
  not_equivocating_equivocator_descriptors IM
    equivocators (destination item)
  ∧ (∃ equivocators' : equivocator_descriptors IM,
       let lx :=
         existT (projT1 (l item))
           (existing_equivocator_label_extract
              (IM (projT1 (l item))) (projT2 (l item))
              (existing_equivocator_label_forget_proper
                 (IM (projT1 (l item))) Hs)) in
       let sx :=
         equivocators_state_project IM equivocators
           (destination item) in
       proper_equivocator_descriptors IM equivocators'
         s
       ∧ equivocators_transition_item_project
           equivocators item =
         Some
           (Some
              {|
                l := lx;
                input := input item;
                destination := sx;
                output := output item
              |}, equivocators'))
Proof.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
item: composite_transition_item (equivocator_IM IM)
s: composite_state (equivocator_IM IM)
Hs: proper_existing_equivocator_label
  (IM (projT1 (l item))) 
  (projT2 (l item)) (s (projT1 (l item)))
Hv: composite_valid (equivocator_IM IM) 
  (l item) (s, input item)
Ht: composite_transition (equivocator_IM IM) 
  (l item) (s, input item) =
(destination item, output item)
∃ equivocators : equivocator_descriptors IM,
  not_equivocating_equivocator_descriptors IM
    equivocators (destination item)
  ∧ (∃ equivocators' : equivocator_descriptors IM,
       let lx :=
         existT (projT1 (l item))
           (existing_equivocator_label_extract
              (IM (projT1 (l item))) (projT2 (l item))
              (existing_equivocator_label_forget_proper
                 (IM (projT1 (l item))) Hs)) in
       let sx :=
         equivocators_state_project IM equivocators
           (destination item) in
       proper_equivocator_descriptors IM equivocators'
         s
       ∧ equivocators_transition_item_project
           equivocators item =
         Some
           (Some
              {|
                l := lx;
                input := input item;
                destination := sx;
                output := output item
              |}, equivocators'))
  specialize
    (exists_equivocator_transition_item_project
      (IM (projT1 (l item)))
      (composite_transition_item_projection (equivocator_IM IM) item)
      (s (projT1 (l item)))
      Hs)
    as Hproject.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
item: composite_transition_item (equivocator_IM IM)
s: composite_state (equivocator_IM IM)
Hs: proper_existing_equivocator_label
  (IM (projT1 (l item))) 
  (projT2 (l item)) (s (projT1 (l item)))
Hv: composite_valid (equivocator_IM IM) 
  (l item) (s, input item)
Ht: composite_transition (equivocator_IM IM) 
  (l item) (s, input item) =
(destination item, output item)
Hproject: valid
  (l
     (composite_transition_item_projection
        (equivocator_IM IM) item))
  (s (projT1 (l item)),
   input
     (composite_transition_item_projection
        (equivocator_IM IM) item))
→ transition
    (l
       (composite_transition_item_projection
          (equivocator_IM IM) item))
    (s (projT1 (l item)),
     input
       (composite_transition_item_projection
          (equivocator_IM IM) item)) =
  (destination
     (composite_transition_item_projection
        (equivocator_IM IM) item),
   output
     (composite_transition_item_projection
        (equivocator_IM IM) item))
  → proper_equivocator_label
      (IM (projT1 (l item)))
      (l
         (composite_transition_item_projection
            (equivocator_IM IM) item))
      (s (projT1 (l item)))
    ∧ (∃ dest_eqv : MachineDescriptor
                      (IM (projT1 (l item))),
         existing_descriptor
           (IM (projT1 (l item))) dest_eqv
           (destination
              (composite_transition_item_projection
                 (equivocator_IM IM) item))
         ∧ equivocator_vlsm_transition_item_project
             (IM (projT1 (l item)))
             (composite_transition_item_projection
                (equivocator_IM IM) item)
             dest_eqv =
           Some
             (Some
                {|
                  l :=
                    existing_equivocator_label_extract
                      (IM (projT1 (l item)))
                      (l
                      (composite_transition_item_projection
                      (equivocator_IM IM)
                      item))
                      (existing_equivocator_label_forget_proper
                      (IM (projT1 (l item)))
                      Hs);
                  input :=
                    input
                      (composite_transition_item_projection
                      (equivocator_IM IM)
                      item);
                  destination :=
                    equivocator_state_descriptor_project
                      (destination
                      (composite_transition_item_projection
                      (equivocator_IM IM)
                      item)) dest_eqv;
                  output :=
                    output
                      (composite_transition_item_projection
                      (equivocator_IM IM)
                      item)
                |},
              equivocator_label_descriptor
                (l
                   (composite_transition_item_projection
                      (equivocator_IM IM)
                      item))))
∃ equivocators : equivocator_descriptors IM,
  not_equivocating_equivocator_descriptors IM
    equivocators (destination item)
  ∧ (∃ equivocators' : equivocator_descriptors IM,
       let lx :=
         existT (projT1 (l item))
           (existing_equivocator_label_extract
              (IM (projT1 (l item))) (projT2 (l item))
              (existing_equivocator_label_forget_proper
                 (IM (projT1 (l item))) Hs)) in
       let sx :=
         equivocators_state_project IM equivocators
           (destination item) in
       proper_equivocator_descriptors IM equivocators'
         s
       ∧ equivocators_transition_item_project
           equivocators item =
         Some
           (Some
              {|
                l := lx;
                input := input item;
                destination := sx;
                output := output item
              |}, equivocators'))
  spec Hproject; [by rewrite (sigT_eta (l item)) in Hv |].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
item: composite_transition_item (equivocator_IM IM)
s: composite_state (equivocator_IM IM)
Hs: proper_existing_equivocator_label
  (IM (projT1 (l item))) 
  (projT2 (l item)) (s (projT1 (l item)))
Hv: composite_valid (equivocator_IM IM) 
  (l item) (s, input item)
Ht: composite_transition (equivocator_IM IM) 
  (l item) (s, input item) =
(destination item, output item)
Hproject: transition
  (l
     (composite_transition_item_projection
        (equivocator_IM IM) item))
  (s (projT1 (l item)),
   input
     (composite_transition_item_projection
        (equivocator_IM IM) item)) =
(destination
   (composite_transition_item_projection
      (equivocator_IM IM) item),
 output
   (composite_transition_item_projection
      (equivocator_IM IM) item))
→ proper_equivocator_label
    (IM (projT1 (l item)))
    (l
       (composite_transition_item_projection
          (equivocator_IM IM) item))
    (s (projT1 (l item)))
  ∧ (∃ dest_eqv : MachineDescriptor
                    (IM (projT1 (l item))),
       existing_descriptor
         (IM (projT1 (l item))) dest_eqv
         (destination
            (composite_transition_item_projection
               (equivocator_IM IM) item))
       ∧ equivocator_vlsm_transition_item_project
           (IM (projT1 (l item)))
           (composite_transition_item_projection
              (equivocator_IM IM) item)
           dest_eqv =
         Some
           (Some
              {|
                l :=
                  existing_equivocator_label_extract
                    (IM (projT1 (l item)))
                    (l
                      (composite_transition_item_projection
                      (equivocator_IM IM)
                      item))
                    (existing_equivocator_label_forget_proper
                      (IM (projT1 (l item)))
                      Hs);
                input :=
                  input
                    (composite_transition_item_projection
                      (equivocator_IM IM)
                      item);
                destination :=
                  equivocator_state_descriptor_project
                    (destination
                      (composite_transition_item_projection
                      (equivocator_IM IM)
                      item)) dest_eqv;
                output :=
                  output
                    (composite_transition_item_projection
                      (equivocator_IM IM)
                      item)
              |},
            equivocator_label_descriptor
              (l
                 (composite_transition_item_projection
                    (equivocator_IM IM)
                    item))))
∃ equivocators : equivocator_descriptors IM,
  not_equivocating_equivocator_descriptors IM
    equivocators (destination item)
  ∧ (∃ equivocators' : equivocator_descriptors IM,
       let lx :=
         existT (projT1 (l item))
           (existing_equivocator_label_extract
              (IM (projT1 (l item))) (projT2 (l item))
              (existing_equivocator_label_forget_proper
                 (IM (projT1 (l item))) Hs)) in
       let sx :=
         equivocators_state_project IM equivocators
           (destination item) in
       proper_equivocator_descriptors IM equivocators'
         s
       ∧ equivocators_transition_item_project
           equivocators item =
         Some
           (Some
              {|
                l := lx;
                input := input item;
                destination := sx;
                output := output item
              |}, equivocators'))
  spec Hproject; [by apply composite_transition_project_active in Ht |].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
item: composite_transition_item (equivocator_IM IM)
s: composite_state (equivocator_IM IM)
Hs: proper_existing_equivocator_label
  (IM (projT1 (l item))) 
  (projT2 (l item)) (s (projT1 (l item)))
Hv: composite_valid (equivocator_IM IM) 
  (l item) (s, input item)
Ht: composite_transition (equivocator_IM IM) 
  (l item) (s, input item) =
(destination item, output item)
Hproject: proper_equivocator_label
  (IM (projT1 (l item)))
  (l
     (composite_transition_item_projection
        (equivocator_IM IM) item))
  (s (projT1 (l item)))
∧ (∃ dest_eqv : MachineDescriptor
                  (IM (projT1 (l item))),
     existing_descriptor
       (IM (projT1 (l item))) dest_eqv
       (destination
          (composite_transition_item_projection
             (equivocator_IM IM) item))
     ∧ equivocator_vlsm_transition_item_project
         (IM (projT1 (l item)))
         (composite_transition_item_projection
            (equivocator_IM IM) item)
         dest_eqv =
       Some
         (Some
            {|
              l :=
                existing_equivocator_label_extract
                  (IM (projT1 (l item)))
                  (l
                     (composite_transition_item_projection
                      (equivocator_IM IM)
                      item))
                  (existing_equivocator_label_forget_proper
                     (IM (projT1 (l item)))
                     Hs);
              input :=
                input
                  (composite_transition_item_projection
                     (equivocator_IM IM)
                     item);
              destination :=
                equivocator_state_descriptor_project
                  (destination
                     (composite_transition_item_projection
                      (equivocator_IM IM)
                      item)) dest_eqv;
              output :=
                output
                  (composite_transition_item_projection
                     (equivocator_IM IM)
                     item)
            |},
          equivocator_label_descriptor
            (l
               (composite_transition_item_projection
                  (equivocator_IM IM) item))))
∃ equivocators : equivocator_descriptors IM,
  not_equivocating_equivocator_descriptors IM
    equivocators (destination item)
  ∧ (∃ equivocators' : equivocator_descriptors IM,
       let lx :=
         existT (projT1 (l item))
           (existing_equivocator_label_extract
              (IM (projT1 (l item))) (projT2 (l item))
              (existing_equivocator_label_forget_proper
                 (IM (projT1 (l item))) Hs)) in
       let sx :=
         equivocators_state_project IM equivocators
           (destination item) in
       proper_equivocator_descriptors IM equivocators'
         s
       ∧ equivocators_transition_item_project
           equivocators item =
         Some
           (Some
              {|
                l := lx;
                input := input item;
                destination := sx;
                output := output item
              |}, equivocators'))
  destruct Hproject as [Heqv' [eqv [Heqv Hproject]]].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
item: composite_transition_item (equivocator_IM IM)
s: composite_state (equivocator_IM IM)
Hs: proper_existing_equivocator_label
  (IM (projT1 (l item))) 
  (projT2 (l item)) (s (projT1 (l item)))
Hv: composite_valid (equivocator_IM IM) 
  (l item) (s, input item)
Ht: composite_transition (equivocator_IM IM) 
  (l item) (s, input item) =
(destination item, output item)
Heqv': proper_equivocator_label
  (IM (projT1 (l item)))
  (l
     (composite_transition_item_projection
        (equivocator_IM IM) item))
  (s (projT1 (l item)))
eqv: MachineDescriptor (IM (projT1 (l item)))
Heqv: existing_descriptor (IM (projT1 (l item))) eqv
  (destination
     (composite_transition_item_projection
        (equivocator_IM IM) item))
Hproject: equivocator_vlsm_transition_item_project
  (IM (projT1 (l item)))
  (composite_transition_item_projection
     (equivocator_IM IM) item) eqv =
Some
  (Some
     {|
       l :=
         existing_equivocator_label_extract
           (IM (projT1 (l item)))
           (l
              (composite_transition_item_projection
                 (equivocator_IM IM) item))
           (existing_equivocator_label_forget_proper
              (IM (projT1 (l item))) Hs);
       input :=
         input
           (composite_transition_item_projection
              (equivocator_IM IM) item);
       destination :=
         equivocator_state_descriptor_project
           (destination
              (composite_transition_item_projection
                 (equivocator_IM IM) item))
           eqv;
       output :=
         output
           (composite_transition_item_projection
              (equivocator_IM IM) item)
     |},
   equivocator_label_descriptor
     (l
        (composite_transition_item_projection
           (equivocator_IM IM) item)))
∃ equivocators : equivocator_descriptors IM,
  not_equivocating_equivocator_descriptors IM
    equivocators (destination item)
  ∧ (∃ equivocators' : equivocator_descriptors IM,
       let lx :=
         existT (projT1 (l item))
           (existing_equivocator_label_extract
              (IM (projT1 (l item))) (projT2 (l item))
              (existing_equivocator_label_forget_proper
                 (IM (projT1 (l item))) Hs)) in
       let sx :=
         equivocators_state_project IM equivocators
           (destination item) in
       proper_equivocator_descriptors IM equivocators'
         s
       ∧ equivocators_transition_item_project
           equivocators item =
         Some
           (Some
              {|
                l := lx;
                input := input item;
                destination := sx;
                output := output item
              |}, equivocators'))
  exists (equivocator_descriptors_update IM (zero_descriptor IM) (projT1 (l item)) eqv).message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
item: composite_transition_item (equivocator_IM IM)
s: composite_state (equivocator_IM IM)
Hs: proper_existing_equivocator_label
  (IM (projT1 (l item))) 
  (projT2 (l item)) (s (projT1 (l item)))
Hv: composite_valid (equivocator_IM IM) 
  (l item) (s, input item)
Ht: composite_transition (equivocator_IM IM) 
  (l item) (s, input item) =
(destination item, output item)
Heqv': proper_equivocator_label
  (IM (projT1 (l item)))
  (l
     (composite_transition_item_projection
        (equivocator_IM IM) item))
  (s (projT1 (l item)))
eqv: MachineDescriptor (IM (projT1 (l item)))
Heqv: existing_descriptor (IM (projT1 (l item))) eqv
  (destination
     (composite_transition_item_projection
        (equivocator_IM IM) item))
Hproject: equivocator_vlsm_transition_item_project
  (IM (projT1 (l item)))
  (composite_transition_item_projection
     (equivocator_IM IM) item) eqv =
Some
  (Some
     {|
       l :=
         existing_equivocator_label_extract
           (IM (projT1 (l item)))
           (l
              (composite_transition_item_projection
                 (equivocator_IM IM) item))
           (existing_equivocator_label_forget_proper
              (IM (projT1 (l item))) Hs);
       input :=
         input
           (composite_transition_item_projection
              (equivocator_IM IM) item);
       destination :=
         equivocator_state_descriptor_project
           (destination
              (composite_transition_item_projection
                 (equivocator_IM IM) item))
           eqv;
       output :=
         output
           (composite_transition_item_projection
              (equivocator_IM IM) item)
     |},
   equivocator_label_descriptor
     (l
        (composite_transition_item_projection
           (equivocator_IM IM) item)))
not_equivocating_equivocator_descriptors IM
  (equivocator_descriptors_update IM
     (zero_descriptor IM) (projT1 (l item)) eqv)
  (destination item)
∧ (∃ equivocators' : equivocator_descriptors IM,
     let lx :=
       existT (projT1 (l item))
         (existing_equivocator_label_extract
            (IM (projT1 (l item))) (projT2 (l item))
            (existing_equivocator_label_forget_proper
               (IM (projT1 (l item))) Hs)) in
     let sx :=
       equivocators_state_project IM
         (equivocator_descriptors_update IM
            (zero_descriptor IM) (projT1 (l item)) eqv)
         (destination item) in
     proper_equivocator_descriptors IM equivocators' s
     ∧ equivocators_transition_item_project
         (equivocator_descriptors_update IM
            (zero_descriptor IM) (projT1 (l item)) eqv)
         item =
       Some
         (Some
            {|
              l := lx;
              input := input item;
              destination := sx;
              output := output item
            |}, equivocators'))
  split.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
item: composite_transition_item (equivocator_IM IM)
s: composite_state (equivocator_IM IM)
Hs: proper_existing_equivocator_label
  (IM (projT1 (l item))) 
  (projT2 (l item)) (s (projT1 (l item)))
Hv: composite_valid (equivocator_IM IM) 
  (l item) (s, input item)
Ht: composite_transition (equivocator_IM IM) 
  (l item) (s, input item) =
(destination item, output item)
Heqv': proper_equivocator_label
  (IM (projT1 (l item)))
  (l
     (composite_transition_item_projection
        (equivocator_IM IM) item))
  (s (projT1 (l item)))
eqv: MachineDescriptor (IM (projT1 (l item)))
Heqv: existing_descriptor (IM (projT1 (l item))) eqv
  (destination
     (composite_transition_item_projection
        (equivocator_IM IM) item))
Hproject: equivocator_vlsm_transition_item_project
  (IM (projT1 (l item)))
  (composite_transition_item_projection
     (equivocator_IM IM) item) eqv =
Some
  (Some
     {|
       l :=
         existing_equivocator_label_extract
           (IM (projT1 (l item)))
           (l
              (composite_transition_item_projection
                 (equivocator_IM IM) item))
           (existing_equivocator_label_forget_proper
              (IM (projT1 (l item))) Hs);
       input :=
         input
           (composite_transition_item_projection
              (equivocator_IM IM) item);
       destination :=
         equivocator_state_descriptor_project
           (destination
              (composite_transition_item_projection
                 (equivocator_IM IM) item))
           eqv;
       output :=
         output
           (composite_transition_item_projection
              (equivocator_IM IM) item)
     |},
   equivocator_label_descriptor
     (l
        (composite_transition_item_projection
           (equivocator_IM IM) item)))
not_equivocating_equivocator_descriptors IM
  (equivocator_descriptors_update IM
     (zero_descriptor IM) (projT1 (l item)) eqv)
  (destination item)
message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
item: composite_transition_item (equivocator_IM IM)
s: composite_state (equivocator_IM IM)
Hs: proper_existing_equivocator_label
  (IM (projT1 (l item))) 
  (projT2 (l item)) (s (projT1 (l item)))
Hv: composite_valid (equivocator_IM IM) 
  (l item) (s, input item)
Ht: composite_transition (equivocator_IM IM) 
  (l item) (s, input item) =
(destination item, output item)
Heqv': proper_equivocator_label
  (IM (projT1 (l item)))
  (l
     (composite_transition_item_projection
        (equivocator_IM IM) item))
  (s (projT1 (l item)))
eqv: MachineDescriptor (IM (projT1 (l item)))
Heqv: existing_descriptor (IM (projT1 (l item))) eqv
  (destination
     (composite_transition_item_projection
        (equivocator_IM IM) item))
Hproject: equivocator_vlsm_transition_item_project
  (IM (projT1 (l item)))
  (composite_transition_item_projection
     (equivocator_IM IM) item) eqv =
Some
  (Some
     {|
       l :=
         existing_equivocator_label_extract
           (IM (projT1 (l item)))
           (l
              (composite_transition_item_projection
                 (equivocator_IM IM) item))
           (existing_equivocator_label_forget_proper
              (IM (projT1 (l item))) Hs);
       input :=
         input
           (composite_transition_item_projection
              (equivocator_IM IM) item);
       destination :=
         equivocator_state_descriptor_project
           (destination
              (composite_transition_item_projection
                 (equivocator_IM IM) item))
           eqv;
       output :=
         output
           (composite_transition_item_projection
              (equivocator_IM IM) item)
     |},
   equivocator_label_descriptor
     (l
        (composite_transition_item_projection
           (equivocator_IM IM) item)))
∃ equivocators' : equivocator_descriptors IM,
  let lx :=
    existT (projT1 (l item))
      (existing_equivocator_label_extract
         (IM (projT1 (l item))) 
         (projT2 (l item))
         (existing_equivocator_label_forget_proper
            (IM (projT1 (l item))) Hs)) in
  let sx :=
    equivocators_state_project IM
      (equivocator_descriptors_update IM
         (zero_descriptor IM) 
         (projT1 (l item)) eqv) 
      (destination item) in
  proper_equivocator_descriptors IM equivocators' s
  ∧ equivocators_transition_item_project
      (equivocator_descriptors_update IM
         (zero_descriptor IM) 
         (projT1 (l item)) eqv) item =
    Some
      (Some
         {|
           l := lx;
           input := input item;
           destination := sx;
           output := output item
         |}, equivocators')
  {message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
item: composite_transition_item (equivocator_IM IM)
s: composite_state (equivocator_IM IM)
Hs: proper_existing_equivocator_label
  (IM (projT1 (l item))) 
  (projT2 (l item)) (s (projT1 (l item)))
Hv: composite_valid (equivocator_IM IM) 
  (l item) (s, input item)
Ht: composite_transition (equivocator_IM IM) 
  (l item) (s, input item) =
(destination item, output item)
Heqv': proper_equivocator_label
  (IM (projT1 (l item)))
  (l
     (composite_transition_item_projection
        (equivocator_IM IM) item))
  (s (projT1 (l item)))
eqv: MachineDescriptor (IM (projT1 (l item)))
Heqv: existing_descriptor (IM (projT1 (l item))) eqv
  (destination
     (composite_transition_item_projection
        (equivocator_IM IM) item))
Hproject: equivocator_vlsm_transition_item_project
  (IM (projT1 (l item)))
  (composite_transition_item_projection
     (equivocator_IM IM) item) eqv =
Some
  (Some
     {|
       l :=
         existing_equivocator_label_extract
           (IM (projT1 (l item)))
           (l
              (composite_transition_item_projection
                 (equivocator_IM IM) item))
           (existing_equivocator_label_forget_proper
              (IM (projT1 (l item))) Hs);
       input :=
         input
           (composite_transition_item_projection
              (equivocator_IM IM) item);
       destination :=
         equivocator_state_descriptor_project
           (destination
              (composite_transition_item_projection
                 (equivocator_IM IM) item))
           eqv;
       output :=
         output
           (composite_transition_item_projection
              (equivocator_IM IM) item)
     |},
   equivocator_label_descriptor
     (l
        (composite_transition_item_projection
           (equivocator_IM IM) item)))
not_equivocating_equivocator_descriptors IM
  (equivocator_descriptors_update IM
     (zero_descriptor IM) (projT1 (l item)) eqv)
  (destination item)
    intro i.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
item: composite_transition_item (equivocator_IM IM)
s: composite_state (equivocator_IM IM)
Hs: proper_existing_equivocator_label
  (IM (projT1 (l item))) 
  (projT2 (l item)) (s (projT1 (l item)))
Hv: composite_valid (equivocator_IM IM) 
  (l item) (s, input item)
Ht: composite_transition (equivocator_IM IM) 
  (l item) (s, input item) =
(destination item, output item)
Heqv': proper_equivocator_label
  (IM (projT1 (l item)))
  (l
     (composite_transition_item_projection
        (equivocator_IM IM) item))
  (s (projT1 (l item)))
eqv: MachineDescriptor (IM (projT1 (l item)))
Heqv: existing_descriptor (IM (projT1 (l item))) eqv
  (destination
     (composite_transition_item_projection
        (equivocator_IM IM) item))
Hproject: equivocator_vlsm_transition_item_project
  (IM (projT1 (l item)))
  (composite_transition_item_projection
     (equivocator_IM IM) item) eqv =
Some
  (Some
     {|
       l :=
         existing_equivocator_label_extract
           (IM (projT1 (l item)))
           (l
              (composite_transition_item_projection
                 (equivocator_IM IM) item))
           (existing_equivocator_label_forget_proper
              (IM (projT1 (l item))) Hs);
       input :=
         input
           (composite_transition_item_projection
              (equivocator_IM IM) item);
       destination :=
         equivocator_state_descriptor_project
           (destination
              (composite_transition_item_projection
                 (equivocator_IM IM) item))
           eqv;
       output :=
         output
           (composite_transition_item_projection
              (equivocator_IM IM) item)
     |},
   equivocator_label_descriptor
     (l
        (composite_transition_item_projection
           (equivocator_IM IM) item)))
i: index
existing_descriptor (IM i)
  (equivocator_descriptors_update IM
     (zero_descriptor IM) (projT1 (l item)) eqv i)
  (destination item i) unfold equivocator_descriptors_update.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
item: composite_transition_item (equivocator_IM IM)
s: composite_state (equivocator_IM IM)
Hs: proper_existing_equivocator_label
  (IM (projT1 (l item))) 
  (projT2 (l item)) (s (projT1 (l item)))
Hv: composite_valid (equivocator_IM IM) 
  (l item) (s, input item)
Ht: composite_transition (equivocator_IM IM) 
  (l item) (s, input item) =
(destination item, output item)
Heqv': proper_equivocator_label
  (IM (projT1 (l item)))
  (l
     (composite_transition_item_projection
        (equivocator_IM IM) item))
  (s (projT1 (l item)))
eqv: MachineDescriptor (IM (projT1 (l item)))
Heqv: existing_descriptor (IM (projT1 (l item))) eqv
  (destination
     (composite_transition_item_projection
        (equivocator_IM IM) item))
Hproject: equivocator_vlsm_transition_item_project
  (IM (projT1 (l item)))
  (composite_transition_item_projection
     (equivocator_IM IM) item) eqv =
Some
  (Some
     {|
       l :=
         existing_equivocator_label_extract
           (IM (projT1 (l item)))
           (l
              (composite_transition_item_projection
                 (equivocator_IM IM) item))
           (existing_equivocator_label_forget_proper
              (IM (projT1 (l item))) Hs);
       input :=
         input
           (composite_transition_item_projection
              (equivocator_IM IM) item);
       destination :=
         equivocator_state_descriptor_project
           (destination
              (composite_transition_item_projection
                 (equivocator_IM IM) item))
           eqv;
       output :=
         output
           (composite_transition_item_projection
              (equivocator_IM IM) item)
     |},
   equivocator_label_descriptor
     (l
        (composite_transition_item_projection
           (equivocator_IM IM) item)))
i: index
existing_descriptor (IM i)
  match decide (i = projT1 (l item)) with
  | left e =>
      eq_rect_r
        (λ i : index, MachineDescriptor (IM i)) eqv e
  | right _ => zero_descriptor IM i
  end (destination item i) destruct (decide (i = projT1 (l item))).message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
item: composite_transition_item (equivocator_IM IM)
s: composite_state (equivocator_IM IM)
Hs: proper_existing_equivocator_label
  (IM (projT1 (l item))) 
  (projT2 (l item)) (s (projT1 (l item)))
Hv: composite_valid (equivocator_IM IM) 
  (l item) (s, input item)
Ht: composite_transition (equivocator_IM IM) 
  (l item) (s, input item) =
(destination item, output item)
Heqv': proper_equivocator_label
  (IM (projT1 (l item)))
  (l
     (composite_transition_item_projection
        (equivocator_IM IM) item))
  (s (projT1 (l item)))
eqv: MachineDescriptor (IM (projT1 (l item)))
Heqv: existing_descriptor (IM (projT1 (l item))) eqv
  (destination
     (composite_transition_item_projection
        (equivocator_IM IM) item))
Hproject: equivocator_vlsm_transition_item_project
  (IM (projT1 (l item)))
  (composite_transition_item_projection
     (equivocator_IM IM) item) eqv =
Some
  (Some
     {|
       l :=
         existing_equivocator_label_extract
           (IM (projT1 (l item)))
           (l
              (composite_transition_item_projection
                 (equivocator_IM IM) item))
           (existing_equivocator_label_forget_proper
              (IM (projT1 (l item))) Hs);
       input :=
         input
           (composite_transition_item_projection
              (equivocator_IM IM) item);
       destination :=
         equivocator_state_descriptor_project
           (destination
              (composite_transition_item_projection
                 (equivocator_IM IM) item))
           eqv;
       output :=
         output
           (composite_transition_item_projection
              (equivocator_IM IM) item)
     |},
   equivocator_label_descriptor
     (l
        (composite_transition_item_projection
           (equivocator_IM IM) item)))
i: index
e: i = projT1 (l item)
existing_descriptor (IM i)
  (eq_rect_r (λ i : index, MachineDescriptor (IM i))
     eqv e) (destination item i)
message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
item: composite_transition_item (equivocator_IM IM)
s: composite_state (equivocator_IM IM)
Hs: proper_existing_equivocator_label
  (IM (projT1 (l item))) 
  (projT2 (l item)) (s (projT1 (l item)))
Hv: composite_valid (equivocator_IM IM) 
  (l item) (s, input item)
Ht: composite_transition (equivocator_IM IM) 
  (l item) (s, input item) =
(destination item, output item)
Heqv': proper_equivocator_label
  (IM (projT1 (l item)))
  (l
     (composite_transition_item_projection
        (equivocator_IM IM) item))
  (s (projT1 (l item)))
eqv: MachineDescriptor (IM (projT1 (l item)))
Heqv: existing_descriptor (IM (projT1 (l item))) eqv
  (destination
     (composite_transition_item_projection
        (equivocator_IM IM) item))
Hproject: equivocator_vlsm_transition_item_project
  (IM (projT1 (l item)))
  (composite_transition_item_projection
     (equivocator_IM IM) item) eqv =
Some
  (Some
     {|
       l :=
         existing_equivocator_label_extract
           (IM (projT1 (l item)))
           (l
              (composite_transition_item_projection
                 (equivocator_IM IM) item))
           (existing_equivocator_label_forget_proper
              (IM (projT1 (l item))) Hs);
       input :=
         input
           (composite_transition_item_projection
              (equivocator_IM IM) item);
       destination :=
         equivocator_state_descriptor_project
           (destination
              (composite_transition_item_projection
                 (equivocator_IM IM) item))
           eqv;
       output :=
         output
           (composite_transition_item_projection
              (equivocator_IM IM) item)
     |},
   equivocator_label_descriptor
     (l
        (composite_transition_item_projection
           (equivocator_IM IM) item)))
i: index
n: i ≠ projT1 (l item)
existing_descriptor (IM i) 
  (zero_descriptor IM i) 
  (destination item i)
    -message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
item: composite_transition_item (equivocator_IM IM)
s: composite_state (equivocator_IM IM)
Hs: proper_existing_equivocator_label
  (IM (projT1 (l item))) 
  (projT2 (l item)) (s (projT1 (l item)))
Hv: composite_valid (equivocator_IM IM) 
  (l item) (s, input item)
Ht: composite_transition (equivocator_IM IM) 
  (l item) (s, input item) =
(destination item, output item)
Heqv': proper_equivocator_label
  (IM (projT1 (l item)))
  (l
     (composite_transition_item_projection
        (equivocator_IM IM) item))
  (s (projT1 (l item)))
eqv: MachineDescriptor (IM (projT1 (l item)))
Heqv: existing_descriptor (IM (projT1 (l item))) eqv
  (destination
     (composite_transition_item_projection
        (equivocator_IM IM) item))
Hproject: equivocator_vlsm_transition_item_project
  (IM (projT1 (l item)))
  (composite_transition_item_projection
     (equivocator_IM IM) item) eqv =
Some
  (Some
     {|
       l :=
         existing_equivocator_label_extract
           (IM (projT1 (l item)))
           (l
              (composite_transition_item_projection
                 (equivocator_IM IM) item))
           (existing_equivocator_label_forget_proper
              (IM (projT1 (l item))) Hs);
       input :=
         input
           (composite_transition_item_projection
              (equivocator_IM IM) item);
       destination :=
         equivocator_state_descriptor_project
           (destination
              (composite_transition_item_projection
                 (equivocator_IM IM) item))
           eqv;
       output :=
         output
           (composite_transition_item_projection
              (equivocator_IM IM) item)
     |},
   equivocator_label_descriptor
     (l
        (composite_transition_item_projection
           (equivocator_IM IM) item)))
i: index
e: i = projT1 (l item)
existing_descriptor (IM i)
  (eq_rect_r (λ i : index, MachineDescriptor (IM i))
     eqv e) (destination item i) by subst; state_update_simpl.
    -message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
item: composite_transition_item (equivocator_IM IM)
s: composite_state (equivocator_IM IM)
Hs: proper_existing_equivocator_label
  (IM (projT1 (l item))) 
  (projT2 (l item)) (s (projT1 (l item)))
Hv: composite_valid (equivocator_IM IM) 
  (l item) (s, input item)
Ht: composite_transition (equivocator_IM IM) 
  (l item) (s, input item) =
(destination item, output item)
Heqv': proper_equivocator_label
  (IM (projT1 (l item)))
  (l
     (composite_transition_item_projection
        (equivocator_IM IM) item))
  (s (projT1 (l item)))
eqv: MachineDescriptor (IM (projT1 (l item)))
Heqv: existing_descriptor (IM (projT1 (l item))) eqv
  (destination
     (composite_transition_item_projection
        (equivocator_IM IM) item))
Hproject: equivocator_vlsm_transition_item_project
  (IM (projT1 (l item)))
  (composite_transition_item_projection
     (equivocator_IM IM) item) eqv =
Some
  (Some
     {|
       l :=
         existing_equivocator_label_extract
           (IM (projT1 (l item)))
           (l
              (composite_transition_item_projection
                 (equivocator_IM IM) item))
           (existing_equivocator_label_forget_proper
              (IM (projT1 (l item))) Hs);
       input :=
         input
           (composite_transition_item_projection
              (equivocator_IM IM) item);
       destination :=
         equivocator_state_descriptor_project
           (destination
              (composite_transition_item_projection
                 (equivocator_IM IM) item))
           eqv;
       output :=
         output
           (composite_transition_item_projection
              (equivocator_IM IM) item)
     |},
   equivocator_label_descriptor
     (l
        (composite_transition_item_projection
           (equivocator_IM IM) item)))
i: index
n: i ≠ projT1 (l item)
existing_descriptor (IM i) (zero_descriptor IM i)
  (destination item i) by cbn; rewrite equivocator_state_project_zero.
  }message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
item: composite_transition_item (equivocator_IM IM)
s: composite_state (equivocator_IM IM)
Hs: proper_existing_equivocator_label
  (IM (projT1 (l item))) 
  (projT2 (l item)) (s (projT1 (l item)))
Hv: composite_valid (equivocator_IM IM) 
  (l item) (s, input item)
Ht: composite_transition (equivocator_IM IM) 
  (l item) (s, input item) =
(destination item, output item)
Heqv': proper_equivocator_label
  (IM (projT1 (l item)))
  (l
     (composite_transition_item_projection
        (equivocator_IM IM) item))
  (s (projT1 (l item)))
eqv: MachineDescriptor (IM (projT1 (l item)))
Heqv: existing_descriptor (IM (projT1 (l item))) eqv
  (destination
     (composite_transition_item_projection
        (equivocator_IM IM) item))
Hproject: equivocator_vlsm_transition_item_project
  (IM (projT1 (l item)))
  (composite_transition_item_projection
     (equivocator_IM IM) item) eqv =
Some
  (Some
     {|
       l :=
         existing_equivocator_label_extract
           (IM (projT1 (l item)))
           (l
              (composite_transition_item_projection
                 (equivocator_IM IM) item))
           (existing_equivocator_label_forget_proper
              (IM (projT1 (l item))) Hs);
       input :=
         input
           (composite_transition_item_projection
              (equivocator_IM IM) item);
       destination :=
         equivocator_state_descriptor_project
           (destination
              (composite_transition_item_projection
                 (equivocator_IM IM) item))
           eqv;
       output :=
         output
           (composite_transition_item_projection
              (equivocator_IM IM) item)
     |},
   equivocator_label_descriptor
     (l
        (composite_transition_item_projection
           (equivocator_IM IM) item)))
∃ equivocators' : equivocator_descriptors IM,
  let lx :=
    existT (projT1 (l item))
      (existing_equivocator_label_extract
         (IM (projT1 (l item))) (projT2 (l item))
         (existing_equivocator_label_forget_proper
            (IM (projT1 (l item))) Hs)) in
  let sx :=
    equivocators_state_project IM
      (equivocator_descriptors_update IM
         (zero_descriptor IM) (projT1 (l item)) eqv)
      (destination item) in
  proper_equivocator_descriptors IM equivocators' s
  ∧ equivocators_transition_item_project
      (equivocator_descriptors_update IM
         (zero_descriptor IM) (projT1 (l item)) eqv)
      item =
    Some
      (Some
         {|
           l := lx;
           input := input item;
           destination := sx;
           output := output item
         |}, equivocators')
  exists (equivocator_descriptors_update IM (zero_descriptor IM) (projT1 (l item))
    (equivocator_label_descriptor (l (composite_transition_item_projection (equivocator_IM IM) item)))).message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
item: composite_transition_item (equivocator_IM IM)
s: composite_state (equivocator_IM IM)
Hs: proper_existing_equivocator_label
  (IM (projT1 (l item))) 
  (projT2 (l item)) (s (projT1 (l item)))
Hv: composite_valid (equivocator_IM IM) 
  (l item) (s, input item)
Ht: composite_transition (equivocator_IM IM) 
  (l item) (s, input item) =
(destination item, output item)
Heqv': proper_equivocator_label
  (IM (projT1 (l item)))
  (l
     (composite_transition_item_projection
        (equivocator_IM IM) item))
  (s (projT1 (l item)))
eqv: MachineDescriptor (IM (projT1 (l item)))
Heqv: existing_descriptor (IM (projT1 (l item))) eqv
  (destination
     (composite_transition_item_projection
        (equivocator_IM IM) item))
Hproject: equivocator_vlsm_transition_item_project
  (IM (projT1 (l item)))
  (composite_transition_item_projection
     (equivocator_IM IM) item) eqv =
Some
  (Some
     {|
       l :=
         existing_equivocator_label_extract
           (IM (projT1 (l item)))
           (l
              (composite_transition_item_projection
                 (equivocator_IM IM) item))
           (existing_equivocator_label_forget_proper
              (IM (projT1 (l item))) Hs);
       input :=
         input
           (composite_transition_item_projection
              (equivocator_IM IM) item);
       destination :=
         equivocator_state_descriptor_project
           (destination
              (composite_transition_item_projection
                 (equivocator_IM IM) item))
           eqv;
       output :=
         output
           (composite_transition_item_projection
              (equivocator_IM IM) item)
     |},
   equivocator_label_descriptor
     (l
        (composite_transition_item_projection
           (equivocator_IM IM) item)))
let lx :=
  existT (projT1 (l item))
    (existing_equivocator_label_extract
       (IM (projT1 (l item))) (projT2 (l item))
       (existing_equivocator_label_forget_proper
          (IM (projT1 (l item))) Hs)) in
let sx :=
  equivocators_state_project IM
    (equivocator_descriptors_update IM
       (zero_descriptor IM) (projT1 (l item)) eqv)
    (destination item) in
proper_equivocator_descriptors IM
  (equivocator_descriptors_update IM
     (zero_descriptor IM) (projT1 (l item))
     (equivocator_label_descriptor
        (l
           (composite_transition_item_projection
              (equivocator_IM IM) item)))) s
∧ equivocators_transition_item_project
    (equivocator_descriptors_update IM
       (zero_descriptor IM) (projT1 (l item)) eqv)
    item =
  Some
    (Some
       {|
         l := lx;
         input := input item;
         destination := sx;
         output := output item
       |},
     equivocator_descriptors_update IM
       (zero_descriptor IM) (projT1 (l item))
       (equivocator_label_descriptor
          (l
             (composite_transition_item_projection
                (equivocator_IM IM) item))))
  split.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
item: composite_transition_item (equivocator_IM IM)
s: composite_state (equivocator_IM IM)
Hs: proper_existing_equivocator_label
  (IM (projT1 (l item))) 
  (projT2 (l item)) (s (projT1 (l item)))
Hv: composite_valid (equivocator_IM IM) 
  (l item) (s, input item)
Ht: composite_transition (equivocator_IM IM) 
  (l item) (s, input item) =
(destination item, output item)
Heqv': proper_equivocator_label
  (IM (projT1 (l item)))
  (l
     (composite_transition_item_projection
        (equivocator_IM IM) item))
  (s (projT1 (l item)))
eqv: MachineDescriptor (IM (projT1 (l item)))
Heqv: existing_descriptor (IM (projT1 (l item))) eqv
  (destination
     (composite_transition_item_projection
        (equivocator_IM IM) item))
Hproject: equivocator_vlsm_transition_item_project
  (IM (projT1 (l item)))
  (composite_transition_item_projection
     (equivocator_IM IM) item) eqv =
Some
  (Some
     {|
       l :=
         existing_equivocator_label_extract
           (IM (projT1 (l item)))
           (l
              (composite_transition_item_projection
                 (equivocator_IM IM) item))
           (existing_equivocator_label_forget_proper
              (IM (projT1 (l item))) Hs);
       input :=
         input
           (composite_transition_item_projection
              (equivocator_IM IM) item);
       destination :=
         equivocator_state_descriptor_project
           (destination
              (composite_transition_item_projection
                 (equivocator_IM IM) item))
           eqv;
       output :=
         output
           (composite_transition_item_projection
              (equivocator_IM IM) item)
     |},
   equivocator_label_descriptor
     (l
        (composite_transition_item_projection
           (equivocator_IM IM) item)))
proper_equivocator_descriptors IM
  (equivocator_descriptors_update IM
     (zero_descriptor IM) (projT1 (l item))
     (equivocator_label_descriptor
        (l
           (composite_transition_item_projection
              (equivocator_IM IM) item)))) s
message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
item: composite_transition_item (equivocator_IM IM)
s: composite_state (equivocator_IM IM)
Hs: proper_existing_equivocator_label
  (IM (projT1 (l item))) 
  (projT2 (l item)) (s (projT1 (l item)))
Hv: composite_valid (equivocator_IM IM) 
  (l item) (s, input item)
Ht: composite_transition (equivocator_IM IM) 
  (l item) (s, input item) =
(destination item, output item)
Heqv': proper_equivocator_label
  (IM (projT1 (l item)))
  (l
     (composite_transition_item_projection
        (equivocator_IM IM) item))
  (s (projT1 (l item)))
eqv: MachineDescriptor (IM (projT1 (l item)))
Heqv: existing_descriptor (IM (projT1 (l item))) eqv
  (destination
     (composite_transition_item_projection
        (equivocator_IM IM) item))
Hproject: equivocator_vlsm_transition_item_project
  (IM (projT1 (l item)))
  (composite_transition_item_projection
     (equivocator_IM IM) item) eqv =
Some
  (Some
     {|
       l :=
         existing_equivocator_label_extract
           (IM (projT1 (l item)))
           (l
              (composite_transition_item_projection
                 (equivocator_IM IM) item))
           (existing_equivocator_label_forget_proper
              (IM (projT1 (l item))) Hs);
       input :=
         input
           (composite_transition_item_projection
              (equivocator_IM IM) item);
       destination :=
         equivocator_state_descriptor_project
           (destination
              (composite_transition_item_projection
                 (equivocator_IM IM) item))
           eqv;
       output :=
         output
           (composite_transition_item_projection
              (equivocator_IM IM) item)
     |},
   equivocator_label_descriptor
     (l
        (composite_transition_item_projection
           (equivocator_IM IM) item)))
equivocators_transition_item_project
  (equivocator_descriptors_update IM
     (zero_descriptor IM) 
     (projT1 (l item)) eqv) item =
Some
  (Some
     {|
       l :=
         existT (projT1 (l item))
           (existing_equivocator_label_extract
              (IM (projT1 (l item))) 
              (projT2 (l item))
              (existing_equivocator_label_forget_proper
                 (IM (projT1 (l item))) Hs));
       input := input item;
       destination :=
         equivocators_state_project IM
           (equivocator_descriptors_update IM
              (zero_descriptor IM) 
              (projT1 (l item)) eqv)
           (destination item);
       output := output item
     |},
   equivocator_descriptors_update IM
     (zero_descriptor IM) 
     (projT1 (l item))
     (equivocator_label_descriptor
        (l
           (composite_transition_item_projection
              (equivocator_IM IM) item))))
  {message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
item: composite_transition_item (equivocator_IM IM)
s: composite_state (equivocator_IM IM)
Hs: proper_existing_equivocator_label
  (IM (projT1 (l item))) 
  (projT2 (l item)) (s (projT1 (l item)))
Hv: composite_valid (equivocator_IM IM) 
  (l item) (s, input item)
Ht: composite_transition (equivocator_IM IM) 
  (l item) (s, input item) =
(destination item, output item)
Heqv': proper_equivocator_label
  (IM (projT1 (l item)))
  (l
     (composite_transition_item_projection
        (equivocator_IM IM) item))
  (s (projT1 (l item)))
eqv: MachineDescriptor (IM (projT1 (l item)))
Heqv: existing_descriptor (IM (projT1 (l item))) eqv
  (destination
     (composite_transition_item_projection
        (equivocator_IM IM) item))
Hproject: equivocator_vlsm_transition_item_project
  (IM (projT1 (l item)))
  (composite_transition_item_projection
     (equivocator_IM IM) item) eqv =
Some
  (Some
     {|
       l :=
         existing_equivocator_label_extract
           (IM (projT1 (l item)))
           (l
              (composite_transition_item_projection
                 (equivocator_IM IM) item))
           (existing_equivocator_label_forget_proper
              (IM (projT1 (l item))) Hs);
       input :=
         input
           (composite_transition_item_projection
              (equivocator_IM IM) item);
       destination :=
         equivocator_state_descriptor_project
           (destination
              (composite_transition_item_projection
                 (equivocator_IM IM) item))
           eqv;
       output :=
         output
           (composite_transition_item_projection
              (equivocator_IM IM) item)
     |},
   equivocator_label_descriptor
     (l
        (composite_transition_item_projection
           (equivocator_IM IM) item)))
proper_equivocator_descriptors IM
  (equivocator_descriptors_update IM
     (zero_descriptor IM) (projT1 (l item))
     (equivocator_label_descriptor
        (l
           (composite_transition_item_projection
              (equivocator_IM IM) item)))) s intro i.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
item: composite_transition_item (equivocator_IM IM)
s: composite_state (equivocator_IM IM)
Hs: proper_existing_equivocator_label
  (IM (projT1 (l item))) 
  (projT2 (l item)) (s (projT1 (l item)))
Hv: composite_valid (equivocator_IM IM) 
  (l item) (s, input item)
Ht: composite_transition (equivocator_IM IM) 
  (l item) (s, input item) =
(destination item, output item)
Heqv': proper_equivocator_label
  (IM (projT1 (l item)))
  (l
     (composite_transition_item_projection
        (equivocator_IM IM) item))
  (s (projT1 (l item)))
eqv: MachineDescriptor (IM (projT1 (l item)))
Heqv: existing_descriptor (IM (projT1 (l item))) eqv
  (destination
     (composite_transition_item_projection
        (equivocator_IM IM) item))
Hproject: equivocator_vlsm_transition_item_project
  (IM (projT1 (l item)))
  (composite_transition_item_projection
     (equivocator_IM IM) item) eqv =
Some
  (Some
     {|
       l :=
         existing_equivocator_label_extract
           (IM (projT1 (l item)))
           (l
              (composite_transition_item_projection
                 (equivocator_IM IM) item))
           (existing_equivocator_label_forget_proper
              (IM (projT1 (l item))) Hs);
       input :=
         input
           (composite_transition_item_projection
              (equivocator_IM IM) item);
       destination :=
         equivocator_state_descriptor_project
           (destination
              (composite_transition_item_projection
                 (equivocator_IM IM) item))
           eqv;
       output :=
         output
           (composite_transition_item_projection
              (equivocator_IM IM) item)
     |},
   equivocator_label_descriptor
     (l
        (composite_transition_item_projection
           (equivocator_IM IM) item)))
i: index
proper_descriptor (IM i)
  (equivocator_descriptors_update IM
     (zero_descriptor IM) (projT1 (l item))
     (equivocator_label_descriptor
        (l
           (composite_transition_item_projection
              (equivocator_IM IM) item))) i) (s i) unfold equivocator_descriptors_update.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
item: composite_transition_item (equivocator_IM IM)
s: composite_state (equivocator_IM IM)
Hs: proper_existing_equivocator_label
  (IM (projT1 (l item))) 
  (projT2 (l item)) (s (projT1 (l item)))
Hv: composite_valid (equivocator_IM IM) 
  (l item) (s, input item)
Ht: composite_transition (equivocator_IM IM) 
  (l item) (s, input item) =
(destination item, output item)
Heqv': proper_equivocator_label
  (IM (projT1 (l item)))
  (l
     (composite_transition_item_projection
        (equivocator_IM IM) item))
  (s (projT1 (l item)))
eqv: MachineDescriptor (IM (projT1 (l item)))
Heqv: existing_descriptor (IM (projT1 (l item))) eqv
  (destination
     (composite_transition_item_projection
        (equivocator_IM IM) item))
Hproject: equivocator_vlsm_transition_item_project
  (IM (projT1 (l item)))
  (composite_transition_item_projection
     (equivocator_IM IM) item) eqv =
Some
  (Some
     {|
       l :=
         existing_equivocator_label_extract
           (IM (projT1 (l item)))
           (l
              (composite_transition_item_projection
                 (equivocator_IM IM) item))
           (existing_equivocator_label_forget_proper
              (IM (projT1 (l item))) Hs);
       input :=
         input
           (composite_transition_item_projection
              (equivocator_IM IM) item);
       destination :=
         equivocator_state_descriptor_project
           (destination
              (composite_transition_item_projection
                 (equivocator_IM IM) item))
           eqv;
       output :=
         output
           (composite_transition_item_projection
              (equivocator_IM IM) item)
     |},
   equivocator_label_descriptor
     (l
        (composite_transition_item_projection
           (equivocator_IM IM) item)))
i: index
proper_descriptor (IM i)
  match decide (i = projT1 (l item)) with
  | left e =>
      eq_rect_r
        (λ i : index, MachineDescriptor (IM i))
        (equivocator_label_descriptor
           (l
              (composite_transition_item_projection
                 (equivocator_IM IM) item))) e
  | right _ => zero_descriptor IM i
  end (s i) destruct (decide (i = projT1 (l item))).message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
item: composite_transition_item (equivocator_IM IM)
s: composite_state (equivocator_IM IM)
Hs: proper_existing_equivocator_label
  (IM (projT1 (l item))) 
  (projT2 (l item)) (s (projT1 (l item)))
Hv: composite_valid (equivocator_IM IM) 
  (l item) (s, input item)
Ht: composite_transition (equivocator_IM IM) 
  (l item) (s, input item) =
(destination item, output item)
Heqv': proper_equivocator_label
  (IM (projT1 (l item)))
  (l
     (composite_transition_item_projection
        (equivocator_IM IM) item))
  (s (projT1 (l item)))
eqv: MachineDescriptor (IM (projT1 (l item)))
Heqv: existing_descriptor (IM (projT1 (l item))) eqv
  (destination
     (composite_transition_item_projection
        (equivocator_IM IM) item))
Hproject: equivocator_vlsm_transition_item_project
  (IM (projT1 (l item)))
  (composite_transition_item_projection
     (equivocator_IM IM) item) eqv =
Some
  (Some
     {|
       l :=
         existing_equivocator_label_extract
           (IM (projT1 (l item)))
           (l
              (composite_transition_item_projection
                 (equivocator_IM IM) item))
           (existing_equivocator_label_forget_proper
              (IM (projT1 (l item))) Hs);
       input :=
         input
           (composite_transition_item_projection
              (equivocator_IM IM) item);
       destination :=
         equivocator_state_descriptor_project
           (destination
              (composite_transition_item_projection
                 (equivocator_IM IM) item))
           eqv;
       output :=
         output
           (composite_transition_item_projection
              (equivocator_IM IM) item)
     |},
   equivocator_label_descriptor
     (l
        (composite_transition_item_projection
           (equivocator_IM IM) item)))
i: index
e: i = projT1 (l item)
proper_descriptor (IM i)
  (eq_rect_r (λ i : index, MachineDescriptor (IM i))
     (equivocator_label_descriptor
        (l
           (composite_transition_item_projection
              (equivocator_IM IM) item))) e) (s i)
message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
item: composite_transition_item (equivocator_IM IM)
s: composite_state (equivocator_IM IM)
Hs: proper_existing_equivocator_label
  (IM (projT1 (l item))) 
  (projT2 (l item)) (s (projT1 (l item)))
Hv: composite_valid (equivocator_IM IM) 
  (l item) (s, input item)
Ht: composite_transition (equivocator_IM IM) 
  (l item) (s, input item) =
(destination item, output item)
Heqv': proper_equivocator_label
  (IM (projT1 (l item)))
  (l
     (composite_transition_item_projection
        (equivocator_IM IM) item))
  (s (projT1 (l item)))
eqv: MachineDescriptor (IM (projT1 (l item)))
Heqv: existing_descriptor (IM (projT1 (l item))) eqv
  (destination
     (composite_transition_item_projection
        (equivocator_IM IM) item))
Hproject: equivocator_vlsm_transition_item_project
  (IM (projT1 (l item)))
  (composite_transition_item_projection
     (equivocator_IM IM) item) eqv =
Some
  (Some
     {|
       l :=
         existing_equivocator_label_extract
           (IM (projT1 (l item)))
           (l
              (composite_transition_item_projection
                 (equivocator_IM IM) item))
           (existing_equivocator_label_forget_proper
              (IM (projT1 (l item))) Hs);
       input :=
         input
           (composite_transition_item_projection
              (equivocator_IM IM) item);
       destination :=
         equivocator_state_descriptor_project
           (destination
              (composite_transition_item_projection
                 (equivocator_IM IM) item))
           eqv;
       output :=
         output
           (composite_transition_item_projection
              (equivocator_IM IM) item)
     |},
   equivocator_label_descriptor
     (l
        (composite_transition_item_projection
           (equivocator_IM IM) item)))
i: index
n: i ≠ projT1 (l item)
proper_descriptor (IM i) (zero_descriptor IM i) (s i)
    -message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
item: composite_transition_item (equivocator_IM IM)
s: composite_state (equivocator_IM IM)
Hs: proper_existing_equivocator_label
  (IM (projT1 (l item))) 
  (projT2 (l item)) (s (projT1 (l item)))
Hv: composite_valid (equivocator_IM IM) 
  (l item) (s, input item)
Ht: composite_transition (equivocator_IM IM) 
  (l item) (s, input item) =
(destination item, output item)
Heqv': proper_equivocator_label
  (IM (projT1 (l item)))
  (l
     (composite_transition_item_projection
        (equivocator_IM IM) item))
  (s (projT1 (l item)))
eqv: MachineDescriptor (IM (projT1 (l item)))
Heqv: existing_descriptor (IM (projT1 (l item))) eqv
  (destination
     (composite_transition_item_projection
        (equivocator_IM IM) item))
Hproject: equivocator_vlsm_transition_item_project
  (IM (projT1 (l item)))
  (composite_transition_item_projection
     (equivocator_IM IM) item) eqv =
Some
  (Some
     {|
       l :=
         existing_equivocator_label_extract
           (IM (projT1 (l item)))
           (l
              (composite_transition_item_projection
                 (equivocator_IM IM) item))
           (existing_equivocator_label_forget_proper
              (IM (projT1 (l item))) Hs);
       input :=
         input
           (composite_transition_item_projection
              (equivocator_IM IM) item);
       destination :=
         equivocator_state_descriptor_project
           (destination
              (composite_transition_item_projection
                 (equivocator_IM IM) item))
           eqv;
       output :=
         output
           (composite_transition_item_projection
              (equivocator_IM IM) item)
     |},
   equivocator_label_descriptor
     (l
        (composite_transition_item_projection
           (equivocator_IM IM) item)))
i: index
e: i = projT1 (l item)
proper_descriptor (IM i)
  (eq_rect_r (λ i : index, MachineDescriptor (IM i))
     (equivocator_label_descriptor
        (l
           (composite_transition_item_projection
              (equivocator_IM IM) item))) e) (s i) by subst; state_update_simpl.
    -message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
item: composite_transition_item (equivocator_IM IM)
s: composite_state (equivocator_IM IM)
Hs: proper_existing_equivocator_label
  (IM (projT1 (l item))) 
  (projT2 (l item)) (s (projT1 (l item)))
Hv: composite_valid (equivocator_IM IM) 
  (l item) (s, input item)
Ht: composite_transition (equivocator_IM IM) 
  (l item) (s, input item) =
(destination item, output item)
Heqv': proper_equivocator_label
  (IM (projT1 (l item)))
  (l
     (composite_transition_item_projection
        (equivocator_IM IM) item))
  (s (projT1 (l item)))
eqv: MachineDescriptor (IM (projT1 (l item)))
Heqv: existing_descriptor (IM (projT1 (l item))) eqv
  (destination
     (composite_transition_item_projection
        (equivocator_IM IM) item))
Hproject: equivocator_vlsm_transition_item_project
  (IM (projT1 (l item)))
  (composite_transition_item_projection
     (equivocator_IM IM) item) eqv =
Some
  (Some
     {|
       l :=
         existing_equivocator_label_extract
           (IM (projT1 (l item)))
           (l
              (composite_transition_item_projection
                 (equivocator_IM IM) item))
           (existing_equivocator_label_forget_proper
              (IM (projT1 (l item))) Hs);
       input :=
         input
           (composite_transition_item_projection
              (equivocator_IM IM) item);
       destination :=
         equivocator_state_descriptor_project
           (destination
              (composite_transition_item_projection
                 (equivocator_IM IM) item))
           eqv;
       output :=
         output
           (composite_transition_item_projection
              (equivocator_IM IM) item)
     |},
   equivocator_label_descriptor
     (l
        (composite_transition_item_projection
           (equivocator_IM IM) item)))
i: index
n: i ≠ projT1 (l item)
proper_descriptor (IM i) (zero_descriptor IM i) (s i) by cbn; rewrite equivocator_state_project_zero.
  }message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
item: composite_transition_item (equivocator_IM IM)
s: composite_state (equivocator_IM IM)
Hs: proper_existing_equivocator_label
  (IM (projT1 (l item))) 
  (projT2 (l item)) (s (projT1 (l item)))
Hv: composite_valid (equivocator_IM IM) 
  (l item) (s, input item)
Ht: composite_transition (equivocator_IM IM) 
  (l item) (s, input item) =
(destination item, output item)
Heqv': proper_equivocator_label
  (IM (projT1 (l item)))
  (l
     (composite_transition_item_projection
        (equivocator_IM IM) item))
  (s (projT1 (l item)))
eqv: MachineDescriptor (IM (projT1 (l item)))
Heqv: existing_descriptor (IM (projT1 (l item))) eqv
  (destination
     (composite_transition_item_projection
        (equivocator_IM IM) item))
Hproject: equivocator_vlsm_transition_item_project
  (IM (projT1 (l item)))
  (composite_transition_item_projection
     (equivocator_IM IM) item) eqv =
Some
  (Some
     {|
       l :=
         existing_equivocator_label_extract
           (IM (projT1 (l item)))
           (l
              (composite_transition_item_projection
                 (equivocator_IM IM) item))
           (existing_equivocator_label_forget_proper
              (IM (projT1 (l item))) Hs);
       input :=
         input
           (composite_transition_item_projection
              (equivocator_IM IM) item);
       destination :=
         equivocator_state_descriptor_project
           (destination
              (composite_transition_item_projection
                 (equivocator_IM IM) item))
           eqv;
       output :=
         output
           (composite_transition_item_projection
              (equivocator_IM IM) item)
     |},
   equivocator_label_descriptor
     (l
        (composite_transition_item_projection
           (equivocator_IM IM) item)))
equivocators_transition_item_project
  (equivocator_descriptors_update IM
     (zero_descriptor IM) (projT1 (l item)) eqv) item =
Some
  (Some
     {|
       l :=
         existT (projT1 (l item))
           (existing_equivocator_label_extract
              (IM (projT1 (l item))) (projT2 (l item))
              (existing_equivocator_label_forget_proper
                 (IM (projT1 (l item))) Hs));
       input := input item;
       destination :=
         equivocators_state_project IM
           (equivocator_descriptors_update IM
              (zero_descriptor IM) (projT1 (l item))
              eqv) (destination item);
       output := output item
     |},
   equivocator_descriptors_update IM
     (zero_descriptor IM) (projT1 (l item))
     (equivocator_label_descriptor
        (l
           (composite_transition_item_projection
              (equivocator_IM IM) item))))
  unfold equivocators_transition_item_project.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
item: composite_transition_item (equivocator_IM IM)
s: composite_state (equivocator_IM IM)
Hs: proper_existing_equivocator_label
  (IM (projT1 (l item))) 
  (projT2 (l item)) (s (projT1 (l item)))
Hv: composite_valid (equivocator_IM IM) 
  (l item) (s, input item)
Ht: composite_transition (equivocator_IM IM) 
  (l item) (s, input item) =
(destination item, output item)
Heqv': proper_equivocator_label
  (IM (projT1 (l item)))
  (l
     (composite_transition_item_projection
        (equivocator_IM IM) item))
  (s (projT1 (l item)))
eqv: MachineDescriptor (IM (projT1 (l item)))
Heqv: existing_descriptor (IM (projT1 (l item))) eqv
  (destination
     (composite_transition_item_projection
        (equivocator_IM IM) item))
Hproject: equivocator_vlsm_transition_item_project
  (IM (projT1 (l item)))
  (composite_transition_item_projection
     (equivocator_IM IM) item) eqv =
Some
  (Some
     {|
       l :=
         existing_equivocator_label_extract
           (IM (projT1 (l item)))
           (l
              (composite_transition_item_projection
                 (equivocator_IM IM) item))
           (existing_equivocator_label_forget_proper
              (IM (projT1 (l item))) Hs);
       input :=
         input
           (composite_transition_item_projection
              (equivocator_IM IM) item);
       destination :=
         equivocator_state_descriptor_project
           (destination
              (composite_transition_item_projection
                 (equivocator_IM IM) item))
           eqv;
       output :=
         output
           (composite_transition_item_projection
              (equivocator_IM IM) item)
     |},
   equivocator_label_descriptor
     (l
        (composite_transition_item_projection
           (equivocator_IM IM) item)))
match
  equivocator_vlsm_transition_item_project
    (IM (projT1 (l item)))
    (composite_transition_item_projection
       (equivocator_IM IM) item)
    (equivocator_descriptors_update IM
       (zero_descriptor IM) (projT1 (l item)) eqv
       (projT1 (l item)))
with
| Some (Some item', deqv') =>
    Some
      (Some
         {|
           l := existT (projT1 (l item)) (l item');
           input := input item;
           destination :=
             equivocators_state_project IM
               (equivocator_descriptors_update IM
                  (zero_descriptor IM)
                  (projT1 (l item)) eqv)
               (destination item);
           output := output item
         |},
       equivocator_descriptors_update IM
         (equivocator_descriptors_update IM
            (zero_descriptor IM) (projT1 (l item)) eqv)
         (projT1 (l item)) deqv')
| Some (None, deqv') =>
    Some
      (None,
       equivocator_descriptors_update IM
         (equivocator_descriptors_update IM
            (zero_descriptor IM) (projT1 (l item)) eqv)
         (projT1 (l item)) deqv')
| None => None
end =
Some
  (Some
     {|
       l :=
         existT (projT1 (l item))
           (existing_equivocator_label_extract
              (IM (projT1 (l item))) (projT2 (l item))
              (existing_equivocator_label_forget_proper
                 (IM (projT1 (l item))) Hs));
       input := input item;
       destination :=
         equivocators_state_project IM
           (equivocator_descriptors_update IM
              (zero_descriptor IM) (projT1 (l item))
              eqv) (destination item);
       output := output item
     |},
   equivocator_descriptors_update IM
     (zero_descriptor IM) (projT1 (l item))
     (equivocator_label_descriptor
        (l
           (composite_transition_item_projection
              (equivocator_IM IM) item))))
  state_update_simpl.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
item: composite_transition_item (equivocator_IM IM)
s: composite_state (equivocator_IM IM)
Hs: proper_existing_equivocator_label
  (IM (projT1 (l item))) 
  (projT2 (l item)) (s (projT1 (l item)))
Hv: composite_valid (equivocator_IM IM) 
  (l item) (s, input item)
Ht: composite_transition (equivocator_IM IM) 
  (l item) (s, input item) =
(destination item, output item)
Heqv': proper_equivocator_label
  (IM (projT1 (l item)))
  (l
     (composite_transition_item_projection
        (equivocator_IM IM) item))
  (s (projT1 (l item)))
eqv: MachineDescriptor (IM (projT1 (l item)))
Heqv: existing_descriptor (IM (projT1 (l item))) eqv
  (destination
     (composite_transition_item_projection
        (equivocator_IM IM) item))
Hproject: equivocator_vlsm_transition_item_project
  (IM (projT1 (l item)))
  (composite_transition_item_projection
     (equivocator_IM IM) item) eqv =
Some
  (Some
     {|
       l :=
         existing_equivocator_label_extract
           (IM (projT1 (l item)))
           (l
              (composite_transition_item_projection
                 (equivocator_IM IM) item))
           (existing_equivocator_label_forget_proper
              (IM (projT1 (l item))) Hs);
       input :=
         input
           (composite_transition_item_projection
              (equivocator_IM IM) item);
       destination :=
         equivocator_state_descriptor_project
           (destination
              (composite_transition_item_projection
                 (equivocator_IM IM) item))
           eqv;
       output :=
         output
           (composite_transition_item_projection
              (equivocator_IM IM) item)
     |},
   equivocator_label_descriptor
     (l
        (composite_transition_item_projection
           (equivocator_IM IM) item)))
match
  equivocator_vlsm_transition_item_project
    (IM (projT1 (l item)))
    (composite_transition_item_projection
       (equivocator_IM IM) item) eqv
with
| Some (Some item', deqv') =>
    Some
      (Some
         {|
           l := existT (projT1 (l item)) (l item');
           input := input item;
           destination :=
             equivocators_state_project IM
               (equivocator_descriptors_update IM
                  (zero_descriptor IM)
                  (projT1 (l item)) eqv)
               (destination item);
           output := output item
         |},
       equivocator_descriptors_update IM
         (equivocator_descriptors_update IM
            (zero_descriptor IM) (projT1 (l item)) eqv)
         (projT1 (l item)) deqv')
| Some (None, deqv') =>
    Some
      (None,
       equivocator_descriptors_update IM
         (equivocator_descriptors_update IM
            (zero_descriptor IM) (projT1 (l item)) eqv)
         (projT1 (l item)) deqv')
| None => None
end =
Some
  (Some
     {|
       l :=
         existT (projT1 (l item))
           (existing_equivocator_label_extract
              (IM (projT1 (l item))) (projT2 (l item))
              (existing_equivocator_label_forget_proper
                 (IM (projT1 (l item))) Hs));
       input := input item;
       destination :=
         equivocators_state_project IM
           (equivocator_descriptors_update IM
              (zero_descriptor IM) (projT1 (l item))
              eqv) (destination item);
       output := output item
     |},
   equivocator_descriptors_update IM
     (zero_descriptor IM) (projT1 (l item))
     (equivocator_label_descriptor
        (l
           (composite_transition_item_projection
              (equivocator_IM IM) item))))
  by rewrite Hproject, equivocator_descriptors_update_twice.
Qed.

Lemma equivocators_transition_item_project_proper_descriptor_characterization
  (eqv_descriptors : equivocator_descriptors IM)
  (item : composite_transition_item (equivocator_IM IM))
  (i := projT1 (l item))
  (Hproper : proper_descriptor (IM i) (eqv_descriptors i) (destination item i))
  : exists oitem eqv_descriptors',
    equivocators_transition_item_project eqv_descriptors item = Some (oitem, eqv_descriptors')
    /\ match oitem with
      | Some itemx =>
        (exists (Hex : existing_equivocator_label _ (projT2 (l item))),
          existT i (existing_equivocator_label_extract _ _ Hex) = l itemx) /\
        input item = input itemx /\ output item = output itemx /\
        (equivocators_state_project IM eqv_descriptors (destination item) = destination itemx)
        /\ eqv_descriptors' i = (equivocator_label_descriptor (projT2 (l item)))
      | None => True
      end
    /\ forall
      (s : composite_state (equivocator_IM IM))
      (Hv : composite_valid (equivocator_IM IM) (l item) (s, input item))
      (Ht : composite_transition (equivocator_IM IM) (l item) (s, input item) =
              (destination item, output item)),
      proper_descriptor (IM i) (eqv_descriptors' i) (s i) /\
      eqv_descriptors' = equivocator_descriptors_update IM eqv_descriptors i (eqv_descriptors' i) /\
      s = state_update (equivocator_IM IM) (destination item) i (s i) /\
      previous_state_descriptor_prop (IM i) (eqv_descriptors i) (s i) (eqv_descriptors' i) /\
      match oitem with
      | Some itemx =>
        forall (sx : composite_state IM)
          (Hsx : sx = equivocators_state_project IM eqv_descriptors' s),
          composite_valid IM (l itemx) (sx, input itemx) /\
          composite_transition IM (l itemx) (sx, input itemx) = (destination itemx, output itemx)
      | None =>
        equivocators_state_project IM eqv_descriptors (destination item) =
        equivocators_state_project IM eqv_descriptors' s
      end.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
eqv_descriptors: equivocator_descriptors IM
item: composite_transition_item (equivocator_IM IM)
i:= projT1 (l item): index
Hproper: proper_descriptor (IM i) 
  (eqv_descriptors i) (destination item i)
∃ (oitem : option (composite_transition_item IM)) 
  (eqv_descriptors' : equivocator_descriptors IM),
  equivocators_transition_item_project eqv_descriptors
    item = Some (oitem, eqv_descriptors')
  ∧ match oitem with
    | Some itemx =>
        (∃ Hex : existing_equivocator_label
                   (IM (projT1 (l item)))
                   (projT2 (l item)),
           existT i
             (existing_equivocator_label_extract
                (IM (projT1 (l item)))
                (projT2 (l item)) Hex) = l itemx)
        ∧ input item = input itemx
          ∧ output item = output itemx
            ∧ equivocators_state_project IM
                eqv_descriptors (destination item) =
              destination itemx
              ∧ eqv_descriptors' i =
                equivocator_label_descriptor
                  (projT2 (l item))
    | None => True
    end
    ∧ (∀ s : composite_state (equivocator_IM IM),
         composite_valid (equivocator_IM IM) (l item)
           (s, input item)
         → composite_transition (equivocator_IM IM)
             (l item) (s, input item) =
           (destination item, output item)
           → proper_descriptor (IM i)
               (eqv_descriptors' i) (s i)
             ∧ eqv_descriptors' =
               equivocator_descriptors_update IM
                 eqv_descriptors i
                 (eqv_descriptors' i)
               ∧ s =
                 state_update (equivocator_IM IM)
                   (destination item) i (s i)
                 ∧ previous_state_descriptor_prop
                     (IM i) (eqv_descriptors i) (s i)
                     (eqv_descriptors' i)
                   ∧ match oitem with
                     | Some itemx =>
                         ∀ sx : composite_state IM,
                           sx =
                           equivocators_state_project
                             IM eqv_descriptors' s
                           → composite_valid IM
                               (l itemx)
                               (sx, input itemx)
                             ∧ composite_transition IM
                                 (l itemx)
                                 (sx, input itemx) =
                               (destination itemx,
                                output itemx)
                     | None =>
                         equivocators_state_project IM
                           eqv_descriptors
                           (destination item) =
                         equivocators_state_project IM
                           eqv_descriptors' s
                     end)
Proof.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
eqv_descriptors: equivocator_descriptors IM
item: composite_transition_item (equivocator_IM IM)
i:= projT1 (l item): index
Hproper: proper_descriptor (IM i) 
  (eqv_descriptors i) (destination item i)
∃ (oitem : option (composite_transition_item IM)) 
  (eqv_descriptors' : equivocator_descriptors IM),
  equivocators_transition_item_project eqv_descriptors
    item = Some (oitem, eqv_descriptors')
  ∧ match oitem with
    | Some itemx =>
        (∃ Hex : existing_equivocator_label
                   (IM (projT1 (l item)))
                   (projT2 (l item)),
           existT i
             (existing_equivocator_label_extract
                (IM (projT1 (l item)))
                (projT2 (l item)) Hex) = l itemx)
        ∧ input item = input itemx
          ∧ output item = output itemx
            ∧ equivocators_state_project IM
                eqv_descriptors (destination item) =
              destination itemx
              ∧ eqv_descriptors' i =
                equivocator_label_descriptor
                  (projT2 (l item))
    | None => True
    end
    ∧ (∀ s : composite_state (equivocator_IM IM),
         composite_valid (equivocator_IM IM) (l item)
           (s, input item)
         → composite_transition (equivocator_IM IM)
             (l item) (s, input item) =
           (destination item, output item)
           → proper_descriptor (IM i)
               (eqv_descriptors' i) (s i)
             ∧ eqv_descriptors' =
               equivocator_descriptors_update IM
                 eqv_descriptors i
                 (eqv_descriptors' i)
               ∧ s =
                 state_update (equivocator_IM IM)
                   (destination item) i (s i)
                 ∧ previous_state_descriptor_prop
                     (IM i) (eqv_descriptors i) (s i)
                     (eqv_descriptors' i)
                   ∧ match oitem with
                     | Some itemx =>
                         ∀ sx : composite_state IM,
                           sx =
                           equivocators_state_project
                             IM eqv_descriptors' s
                           → composite_valid IM
                               (l itemx)
                               (sx, input itemx)
                             ∧ composite_transition IM
                                 (l itemx)
                                 (sx, input itemx) =
                               (destination itemx,
                                output itemx)
                     | None =>
                         equivocators_state_project IM
                           eqv_descriptors
                           (destination item) =
                         equivocators_state_project IM
                           eqv_descriptors' s
                     end)
  destruct
    (equivocator_transition_item_project_proper_characterization (IM i)
      (composite_transition_item_projection (equivocator_IM IM) item)
      (eqv_descriptors i) Hproper)
    as (oitemi & eqv_descriptorsi' & Hoitemi & Hitemx & Hchar).message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
eqv_descriptors: equivocator_descriptors IM
item: composite_transition_item (equivocator_IM IM)
i:= projT1 (l item): index
Hproper: proper_descriptor (IM i) 
  (eqv_descriptors i) (destination item i)
oitemi: option transition_item
eqv_descriptorsi': MachineDescriptor (IM i)
Hoitemi: equivocator_vlsm_transition_item_project
  (IM i)
  (composite_transition_item_projection
     (equivocator_IM IM) item)
  (eqv_descriptors i) =
Some (oitemi, eqv_descriptorsi')
Hitemx: match oitemi with
| Some itemx =>
    (∃ Hex : existing_equivocator_label
               (IM i)
               (l
                  (composite_transition_item_projection
                     (equivocator_IM IM) item)),
       l itemx =
       existing_equivocator_label_extract
         (IM i)
         (l
            (composite_transition_item_projection
               (equivocator_IM IM) item)) Hex)
    ∧ input
        (composite_transition_item_projection
           (equivocator_IM IM) item) =
      input itemx
      ∧ output
          (composite_transition_item_projection
             (equivocator_IM IM) item) =
        output itemx
        ∧ equivocator_state_descriptor_project
            (destination
               (composite_transition_item_projection
                  (equivocator_IM IM) item))
            (eqv_descriptors i) =
          destination itemx
          ∧ eqv_descriptorsi' =
            equivocator_label_descriptor
              (l
                 (composite_transition_item_projection
                    (equivocator_IM IM) item))
| None => True
end
Hchar: ∀ s : state (equivocator_vlsm (IM i)),
  valid
    (l
       (composite_transition_item_projection
          (equivocator_IM IM) item))
    (s,
     input
       (composite_transition_item_projection
          (equivocator_IM IM) item))
  → transition
      (l
         (composite_transition_item_projection
            (equivocator_IM IM) item))
      (s,
       input
         (composite_transition_item_projection
            (equivocator_IM IM) item)) =
    (destination
       (composite_transition_item_projection
          (equivocator_IM IM) item),
     output
       (composite_transition_item_projection
          (equivocator_IM IM) item))
    → proper_descriptor 
        (IM i) eqv_descriptorsi' s
      ∧ previous_state_descriptor_prop 
          (IM i) (eqv_descriptors i) s
          eqv_descriptorsi'
        ∧ match oitemi with
          | Some itemx =>
              ∀ sx : state (IM i),
                sx =
                equivocator_state_descriptor_project
                  s eqv_descriptorsi'
                → valid 
                    (l itemx)
                    (sx, input itemx)
                  ∧ transition 
                      (l itemx)
                      (
                      sx, input itemx) =
                    (destination itemx,
                     output itemx)
          | None =>
              equivocator_state_descriptor_project
                (destination
                   (composite_transition_item_projection
                      (equivocator_IM IM) item))
                (eqv_descriptors i) =
              equivocator_state_descriptor_project
                s eqv_descriptorsi'
          end
∃ (oitem : option (composite_transition_item IM)) 
  (eqv_descriptors' : equivocator_descriptors IM),
  equivocators_transition_item_project eqv_descriptors
    item = Some (oitem, eqv_descriptors')
  ∧ match oitem with
    | Some itemx =>
        (∃ Hex : existing_equivocator_label
                   (IM (projT1 (l item)))
                   (projT2 (l item)),
           existT i
             (existing_equivocator_label_extract
                (IM (projT1 (l item)))
                (projT2 (l item)) Hex) = l itemx)
        ∧ input item = input itemx
          ∧ output item = output itemx
            ∧ equivocators_state_project IM
                eqv_descriptors (destination item) =
              destination itemx
              ∧ eqv_descriptors' i =
                equivocator_label_descriptor
                  (projT2 (l item))
    | None => True
    end
    ∧ (∀ s : composite_state (equivocator_IM IM),
         composite_valid (equivocator_IM IM) (l item)
           (s, input item)
         → composite_transition (equivocator_IM IM)
             (l item) (s, input item) =
           (destination item, output item)
           → proper_descriptor (IM i)
               (eqv_descriptors' i) (s i)
             ∧ eqv_descriptors' =
               equivocator_descriptors_update IM
                 eqv_descriptors i
                 (eqv_descriptors' i)
               ∧ s =
                 state_update (equivocator_IM IM)
                   (destination item) i (s i)
                 ∧ previous_state_descriptor_prop
                     (IM i) (eqv_descriptors i) (s i)
                     (eqv_descriptors' i)
                   ∧ match oitem with
                     | Some itemx =>
                         ∀ sx : composite_state IM,
                           sx =
                           equivocators_state_project
                             IM eqv_descriptors' s
                           → composite_valid IM
                               (l itemx)
                               (sx, input itemx)
                             ∧ composite_transition IM
                                 (l itemx)
                                 (sx, input itemx) =
                               (destination itemx,
                                output itemx)
                     | None =>
                         equivocators_state_project IM
                           eqv_descriptors
                           (destination item) =
                         equivocators_state_project IM
                           eqv_descriptors' s
                     end)
  subst i.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
eqv_descriptors: equivocator_descriptors IM
item: composite_transition_item (equivocator_IM IM)
Hproper: proper_descriptor (IM (projT1 (l item)))
  (eqv_descriptors (projT1 (l item)))
  (destination item (projT1 (l item)))
oitemi: option transition_item
eqv_descriptorsi': MachineDescriptor
  (IM (projT1 (l item)))
Hoitemi: equivocator_vlsm_transition_item_project
  (IM (projT1 (l item)))
  (composite_transition_item_projection
     (equivocator_IM IM) item)
  (eqv_descriptors (projT1 (l item))) =
Some (oitemi, eqv_descriptorsi')
Hitemx: match oitemi with
| Some itemx =>
    (∃ Hex : existing_equivocator_label
               (IM (projT1 (l item)))
               (l
                  (composite_transition_item_projection
                     (equivocator_IM IM) item)),
       l itemx =
       existing_equivocator_label_extract
         (IM (projT1 (l item)))
         (l
            (composite_transition_item_projection
               (equivocator_IM IM) item)) Hex)
    ∧ input
        (composite_transition_item_projection
           (equivocator_IM IM) item) =
      input itemx
      ∧ output
          (composite_transition_item_projection
             (equivocator_IM IM) item) =
        output itemx
        ∧ equivocator_state_descriptor_project
            (destination
               (composite_transition_item_projection
                  (equivocator_IM IM) item))
            (eqv_descriptors
               (projT1 (l item))) =
          destination itemx
          ∧ eqv_descriptorsi' =
            equivocator_label_descriptor
              (l
                 (composite_transition_item_projection
                    (equivocator_IM IM) item))
| None => True
end
Hchar: ∀ s : state
        (equivocator_vlsm
           (IM (projT1 (l item)))),
  valid
    (l
       (composite_transition_item_projection
          (equivocator_IM IM) item))
    (s,
     input
       (composite_transition_item_projection
          (equivocator_IM IM) item))
  → transition
      (l
         (composite_transition_item_projection
            (equivocator_IM IM) item))
      (s,
       input
         (composite_transition_item_projection
            (equivocator_IM IM) item)) =
    (destination
       (composite_transition_item_projection
          (equivocator_IM IM) item),
     output
       (composite_transition_item_projection
          (equivocator_IM IM) item))
    → proper_descriptor 
        (IM (projT1 (l item)))
        eqv_descriptorsi' s
      ∧ previous_state_descriptor_prop
          (IM (projT1 (l item)))
          (eqv_descriptors (projT1 (l item)))
          s eqv_descriptorsi'
        ∧ match oitemi with
          | Some itemx =>
              ∀ sx : state
                      (IM (projT1 (l item))),
                sx =
                equivocator_state_descriptor_project
                  s eqv_descriptorsi'
                → valid 
                    (l itemx)
                    (sx, input itemx)
                  ∧ transition 
                      (l itemx)
                      (
                      sx, input itemx) =
                    (destination itemx,
                     output itemx)
          | None =>
              equivocator_state_descriptor_project
                (destination
                   (composite_transition_item_projection
                      (equivocator_IM IM) item))
                (eqv_descriptors
                   (projT1 (l item))) =
              equivocator_state_descriptor_project
                s eqv_descriptorsi'
          end
∃ (oitem : option (composite_transition_item IM)) 
  (eqv_descriptors' : equivocator_descriptors IM),
  equivocators_transition_item_project eqv_descriptors
    item = Some (oitem, eqv_descriptors')
  ∧ match oitem with
    | Some itemx =>
        (∃ Hex : existing_equivocator_label
                   (IM (projT1 (l item)))
                   (projT2 (l item)),
           existT (projT1 (l item))
             (existing_equivocator_label_extract
                (IM (projT1 (l item)))
                (projT2 (l item)) Hex) = l itemx)
        ∧ input item = input itemx
          ∧ output item = output itemx
            ∧ equivocators_state_project IM
                eqv_descriptors (destination item) =
              destination itemx
              ∧ eqv_descriptors' (projT1 (l item)) =
                equivocator_label_descriptor
                  (projT2 (l item))
    | None => True
    end
    ∧ (∀ s : composite_state (equivocator_IM IM),
         composite_valid (equivocator_IM IM) (l item)
           (s, input item)
         → composite_transition (equivocator_IM IM)
             (l item) (s, input item) =
           (destination item, output item)
           → proper_descriptor (IM (projT1 (l item)))
               (eqv_descriptors' (projT1 (l item)))
               (s (projT1 (l item)))
             ∧ eqv_descriptors' =
               equivocator_descriptors_update IM
                 eqv_descriptors (projT1 (l item))
                 (eqv_descriptors' (projT1 (l item)))
               ∧ s =
                 state_update (equivocator_IM IM)
                   (destination item)
                   (projT1 (l item))
                   (s (projT1 (l item)))
                 ∧ previous_state_descriptor_prop
                     (IM (projT1 (l item)))
                     (eqv_descriptors
                        (projT1 (l item)))
                     (s (projT1 (l item)))
                     (eqv_descriptors'
                        (projT1 (l item)))
                   ∧ match oitem with
                     | Some itemx =>
                         ∀ sx : composite_state IM,
                           sx =
                           equivocators_state_project
                             IM eqv_descriptors' s
                           → composite_valid IM
                               (l itemx)
                               (sx, input itemx)
                             ∧ composite_transition IM
                                 (l itemx)
                                 (sx, input itemx) =
                               (destination itemx,
                                output itemx)
                     | None =>
                         equivocators_state_project IM
                           eqv_descriptors
                           (destination item) =
                         equivocators_state_project IM
                           eqv_descriptors' s
                     end)
  unfold equivocators_transition_item_project.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
eqv_descriptors: equivocator_descriptors IM
item: composite_transition_item (equivocator_IM IM)
Hproper: proper_descriptor (IM (projT1 (l item)))
  (eqv_descriptors (projT1 (l item)))
  (destination item (projT1 (l item)))
oitemi: option transition_item
eqv_descriptorsi': MachineDescriptor
  (IM (projT1 (l item)))
Hoitemi: equivocator_vlsm_transition_item_project
  (IM (projT1 (l item)))
  (composite_transition_item_projection
     (equivocator_IM IM) item)
  (eqv_descriptors (projT1 (l item))) =
Some (oitemi, eqv_descriptorsi')
Hitemx: match oitemi with
| Some itemx =>
    (∃ Hex : existing_equivocator_label
               (IM (projT1 (l item)))
               (l
                  (composite_transition_item_projection
                     (equivocator_IM IM) item)),
       l itemx =
       existing_equivocator_label_extract
         (IM (projT1 (l item)))
         (l
            (composite_transition_item_projection
               (equivocator_IM IM) item)) Hex)
    ∧ input
        (composite_transition_item_projection
           (equivocator_IM IM) item) =
      input itemx
      ∧ output
          (composite_transition_item_projection
             (equivocator_IM IM) item) =
        output itemx
        ∧ equivocator_state_descriptor_project
            (destination
               (composite_transition_item_projection
                  (equivocator_IM IM) item))
            (eqv_descriptors
               (projT1 (l item))) =
          destination itemx
          ∧ eqv_descriptorsi' =
            equivocator_label_descriptor
              (l
                 (composite_transition_item_projection
                    (equivocator_IM IM) item))
| None => True
end
Hchar: ∀ s : state
        (equivocator_vlsm
           (IM (projT1 (l item)))),
  valid
    (l
       (composite_transition_item_projection
          (equivocator_IM IM) item))
    (s,
     input
       (composite_transition_item_projection
          (equivocator_IM IM) item))
  → transition
      (l
         (composite_transition_item_projection
            (equivocator_IM IM) item))
      (s,
       input
         (composite_transition_item_projection
            (equivocator_IM IM) item)) =
    (destination
       (composite_transition_item_projection
          (equivocator_IM IM) item),
     output
       (composite_transition_item_projection
          (equivocator_IM IM) item))
    → proper_descriptor 
        (IM (projT1 (l item)))
        eqv_descriptorsi' s
      ∧ previous_state_descriptor_prop
          (IM (projT1 (l item)))
          (eqv_descriptors (projT1 (l item)))
          s eqv_descriptorsi'
        ∧ match oitemi with
          | Some itemx =>
              ∀ sx : state
                      (IM (projT1 (l item))),
                sx =
                equivocator_state_descriptor_project
                  s eqv_descriptorsi'
                → valid 
                    (l itemx)
                    (sx, input itemx)
                  ∧ transition 
                      (l itemx)
                      (
                      sx, input itemx) =
                    (destination itemx,
                     output itemx)
          | None =>
              equivocator_state_descriptor_project
                (destination
                   (composite_transition_item_projection
                      (equivocator_IM IM) item))
                (eqv_descriptors
                   (projT1 (l item))) =
              equivocator_state_descriptor_project
                s eqv_descriptorsi'
          end
∃ (oitem : option (composite_transition_item IM)) 
  (eqv_descriptors' : equivocator_descriptors IM),
  match
    equivocator_vlsm_transition_item_project
      (IM (projT1 (l item)))
      (composite_transition_item_projection
         (equivocator_IM IM) item)
      (eqv_descriptors (projT1 (l item)))
  with
  | Some (Some item', deqv') =>
      Some
        (Some
           {|
             l := existT (projT1 (l item)) (l item');
             input := input item;
             destination :=
               equivocators_state_project IM
                 eqv_descriptors (destination item);
             output := output item
           |},
         equivocator_descriptors_update IM
           eqv_descriptors (projT1 (l item)) deqv')
  | Some (None, deqv') =>
      Some
        (None,
         equivocator_descriptors_update IM
           eqv_descriptors (projT1 (l item)) deqv')
  | None => None
  end = Some (oitem, eqv_descriptors')
  ∧ match oitem with
    | Some itemx =>
        (∃ Hex : existing_equivocator_label
                   (IM (projT1 (l item)))
                   (projT2 (l item)),
           existT (projT1 (l item))
             (existing_equivocator_label_extract
                (IM (projT1 (l item)))
                (projT2 (l item)) Hex) = l itemx)
        ∧ input item = input itemx
          ∧ output item = output itemx
            ∧ equivocators_state_project IM
                eqv_descriptors (destination item) =
              destination itemx
              ∧ eqv_descriptors' (projT1 (l item)) =
                equivocator_label_descriptor
                  (projT2 (l item))
    | None => True
    end
    ∧ (∀ s : composite_state (equivocator_IM IM),
         composite_valid (equivocator_IM IM) (l item)
           (s, input item)
         → composite_transition (equivocator_IM IM)
             (l item) (s, input item) =
           (destination item, output item)
           → proper_descriptor (IM (projT1 (l item)))
               (eqv_descriptors' (projT1 (l item)))
               (s (projT1 (l item)))
             ∧ eqv_descriptors' =
               equivocator_descriptors_update IM
                 eqv_descriptors (projT1 (l item))
                 (eqv_descriptors' (projT1 (l item)))
               ∧ s =
                 state_update (equivocator_IM IM)
                   (destination item)
                   (projT1 (l item))
                   (s (projT1 (l item)))
                 ∧ previous_state_descriptor_prop
                     (IM (projT1 (l item)))
                     (eqv_descriptors
                        (projT1 (l item)))
                     (s (projT1 (l item)))
                     (eqv_descriptors'
                        (projT1 (l item)))
                   ∧ match oitem with
                     | Some itemx =>
                         ∀ sx : composite_state IM,
                           sx =
                           equivocators_state_project
                             IM eqv_descriptors' s
                           → composite_valid IM
                               (l itemx)
                               (sx, input itemx)
                             ∧ composite_transition IM
                                 (l itemx)
                                 (sx, input itemx) =
                               (destination itemx,
                                output itemx)
                     | None =>
                         equivocators_state_project IM
                           eqv_descriptors
                           (destination item) =
                         equivocators_state_project IM
                           eqv_descriptors' s
                     end)
  rewrite Hoitemi.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
eqv_descriptors: equivocator_descriptors IM
item: composite_transition_item (equivocator_IM IM)
Hproper: proper_descriptor (IM (projT1 (l item)))
  (eqv_descriptors (projT1 (l item)))
  (destination item (projT1 (l item)))
oitemi: option transition_item
eqv_descriptorsi': MachineDescriptor
  (IM (projT1 (l item)))
Hoitemi: equivocator_vlsm_transition_item_project
  (IM (projT1 (l item)))
  (composite_transition_item_projection
     (equivocator_IM IM) item)
  (eqv_descriptors (projT1 (l item))) =
Some (oitemi, eqv_descriptorsi')
Hitemx: match oitemi with
| Some itemx =>
    (∃ Hex : existing_equivocator_label
               (IM (projT1 (l item)))
               (l
                  (composite_transition_item_projection
                     (equivocator_IM IM) item)),
       l itemx =
       existing_equivocator_label_extract
         (IM (projT1 (l item)))
         (l
            (composite_transition_item_projection
               (equivocator_IM IM) item)) Hex)
    ∧ input
        (composite_transition_item_projection
           (equivocator_IM IM) item) =
      input itemx
      ∧ output
          (composite_transition_item_projection
             (equivocator_IM IM) item) =
        output itemx
        ∧ equivocator_state_descriptor_project
            (destination
               (composite_transition_item_projection
                  (equivocator_IM IM) item))
            (eqv_descriptors
               (projT1 (l item))) =
          destination itemx
          ∧ eqv_descriptorsi' =
            equivocator_label_descriptor
              (l
                 (composite_transition_item_projection
                    (equivocator_IM IM) item))
| None => True
end
Hchar: ∀ s : state
        (equivocator_vlsm
           (IM (projT1 (l item)))),
  valid
    (l
       (composite_transition_item_projection
          (equivocator_IM IM) item))
    (s,
     input
       (composite_transition_item_projection
          (equivocator_IM IM) item))
  → transition
      (l
         (composite_transition_item_projection
            (equivocator_IM IM) item))
      (s,
       input
         (composite_transition_item_projection
            (equivocator_IM IM) item)) =
    (destination
       (composite_transition_item_projection
          (equivocator_IM IM) item),
     output
       (composite_transition_item_projection
          (equivocator_IM IM) item))
    → proper_descriptor 
        (IM (projT1 (l item)))
        eqv_descriptorsi' s
      ∧ previous_state_descriptor_prop
          (IM (projT1 (l item)))
          (eqv_descriptors (projT1 (l item)))
          s eqv_descriptorsi'
        ∧ match oitemi with
          | Some itemx =>
              ∀ sx : state
                      (IM (projT1 (l item))),
                sx =
                equivocator_state_descriptor_project
                  s eqv_descriptorsi'
                → valid 
                    (l itemx)
                    (sx, input itemx)
                  ∧ transition 
                      (l itemx)
                      (
                      sx, input itemx) =
                    (destination itemx,
                     output itemx)
          | None =>
              equivocator_state_descriptor_project
                (destination
                   (composite_transition_item_projection
                      (equivocator_IM IM) item))
                (eqv_descriptors
                   (projT1 (l item))) =
              equivocator_state_descriptor_project
                s eqv_descriptorsi'
          end
∃ (oitem : option (composite_transition_item IM)) 
  (eqv_descriptors' : equivocator_descriptors IM),
  match oitemi with
  | Some item' =>
      Some
        (Some
           {|
             l := existT (projT1 (l item)) (l item');
             input := input item;
             destination :=
               equivocators_state_project IM
                 eqv_descriptors (destination item);
             output := output item
           |},
         equivocator_descriptors_update IM
           eqv_descriptors (projT1 (l item))
           eqv_descriptorsi')
  | None =>
      Some
        (None,
         equivocator_descriptors_update IM
           eqv_descriptors (projT1 (l item))
           eqv_descriptorsi')
  end = Some (oitem, eqv_descriptors')
  ∧ match oitem with
    | Some itemx =>
        (∃ Hex : existing_equivocator_label
                   (IM (projT1 (l item)))
                   (projT2 (l item)),
           existT (projT1 (l item))
             (existing_equivocator_label_extract
                (IM (projT1 (l item)))
                (projT2 (l item)) Hex) = l itemx)
        ∧ input item = input itemx
          ∧ output item = output itemx
            ∧ equivocators_state_project IM
                eqv_descriptors (destination item) =
              destination itemx
              ∧ eqv_descriptors' (projT1 (l item)) =
                equivocator_label_descriptor
                  (projT2 (l item))
    | None => True
    end
    ∧ (∀ s : composite_state (equivocator_IM IM),
         composite_valid (equivocator_IM IM) (l item)
           (s, input item)
         → composite_transition (equivocator_IM IM)
             (l item) (s, input item) =
           (destination item, output item)
           → proper_descriptor (IM (projT1 (l item)))
               (eqv_descriptors' (projT1 (l item)))
               (s (projT1 (l item)))
             ∧ eqv_descriptors' =
               equivocator_descriptors_update IM
                 eqv_descriptors (projT1 (l item))
                 (eqv_descriptors' (projT1 (l item)))
               ∧ s =
                 state_update (equivocator_IM IM)
                   (destination item)
                   (projT1 (l item))
                   (s (projT1 (l item)))
                 ∧ previous_state_descriptor_prop
                     (IM (projT1 (l item)))
                     (eqv_descriptors
                        (projT1 (l item)))
                     (s (projT1 (l item)))
                     (eqv_descriptors'
                        (projT1 (l item)))
                   ∧ match oitem with
                     | Some itemx =>
                         ∀ sx : composite_state IM,
                           sx =
                           equivocators_state_project
                             IM eqv_descriptors' s
                           → composite_valid IM
                               (l itemx)
                               (sx, input itemx)
                             ∧ composite_transition IM
                                 (l itemx)
                                 (sx, input itemx) =
                               (destination itemx,
                                output itemx)
                     | None =>
                         equivocators_state_project IM
                           eqv_descriptors
                           (destination item) =
                         equivocators_state_project IM
                           eqv_descriptors' s
                     end) clear Hoitemi.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
eqv_descriptors: equivocator_descriptors IM
item: composite_transition_item (equivocator_IM IM)
Hproper: proper_descriptor (IM (projT1 (l item)))
  (eqv_descriptors (projT1 (l item)))
  (destination item (projT1 (l item)))
oitemi: option transition_item
eqv_descriptorsi': MachineDescriptor
  (IM (projT1 (l item)))
Hitemx: match oitemi with
| Some itemx =>
    (∃ Hex : existing_equivocator_label
               (IM (projT1 (l item)))
               (l
                  (composite_transition_item_projection
                     (equivocator_IM IM) item)),
       l itemx =
       existing_equivocator_label_extract
         (IM (projT1 (l item)))
         (l
            (composite_transition_item_projection
               (equivocator_IM IM) item)) Hex)
    ∧ input
        (composite_transition_item_projection
           (equivocator_IM IM) item) =
      input itemx
      ∧ output
          (composite_transition_item_projection
             (equivocator_IM IM) item) =
        output itemx
        ∧ equivocator_state_descriptor_project
            (destination
               (composite_transition_item_projection
                  (equivocator_IM IM) item))
            (eqv_descriptors
               (projT1 (l item))) =
          destination itemx
          ∧ eqv_descriptorsi' =
            equivocator_label_descriptor
              (l
                 (composite_transition_item_projection
                    (equivocator_IM IM) item))
| None => True
end
Hchar: ∀ s : state
        (equivocator_vlsm
           (IM (projT1 (l item)))),
  valid
    (l
       (composite_transition_item_projection
          (equivocator_IM IM) item))
    (s,
     input
       (composite_transition_item_projection
          (equivocator_IM IM) item))
  → transition
      (l
         (composite_transition_item_projection
            (equivocator_IM IM) item))
      (s,
       input
         (composite_transition_item_projection
            (equivocator_IM IM) item)) =
    (destination
       (composite_transition_item_projection
          (equivocator_IM IM) item),
     output
       (composite_transition_item_projection
          (equivocator_IM IM) item))
    → proper_descriptor 
        (IM (projT1 (l item)))
        eqv_descriptorsi' s
      ∧ previous_state_descriptor_prop
          (IM (projT1 (l item)))
          (eqv_descriptors (projT1 (l item)))
          s eqv_descriptorsi'
        ∧ match oitemi with
          | Some itemx =>
              ∀ sx : state
                      (IM (projT1 (l item))),
                sx =
                equivocator_state_descriptor_project
                  s eqv_descriptorsi'
                → valid 
                    (l itemx)
                    (sx, input itemx)
                  ∧ transition 
                      (l itemx)
                      (
                      sx, input itemx) =
                    (destination itemx,
                     output itemx)
          | None =>
              equivocator_state_descriptor_project
                (destination
                   (composite_transition_item_projection
                      (equivocator_IM IM) item))
                (eqv_descriptors
                   (projT1 (l item))) =
              equivocator_state_descriptor_project
                s eqv_descriptorsi'
          end
∃ (oitem : option (composite_transition_item IM)) 
  (eqv_descriptors' : equivocator_descriptors IM),
  match oitemi with
  | Some item' =>
      Some
        (Some
           {|
             l := existT (projT1 (l item)) (l item');
             input := input item;
             destination :=
               equivocators_state_project IM
                 eqv_descriptors (destination item);
             output := output item
           |},
         equivocator_descriptors_update IM
           eqv_descriptors (projT1 (l item))
           eqv_descriptorsi')
  | None =>
      Some
        (None,
         equivocator_descriptors_update IM
           eqv_descriptors (projT1 (l item))
           eqv_descriptorsi')
  end = Some (oitem, eqv_descriptors')
  ∧ match oitem with
    | Some itemx =>
        (∃ Hex : existing_equivocator_label
                   (IM (projT1 (l item)))
                   (projT2 (l item)),
           existT (projT1 (l item))
             (existing_equivocator_label_extract
                (IM (projT1 (l item)))
                (projT2 (l item)) Hex) = l itemx)
        ∧ input item = input itemx
          ∧ output item = output itemx
            ∧ equivocators_state_project IM
                eqv_descriptors (destination item) =
              destination itemx
              ∧ eqv_descriptors' (projT1 (l item)) =
                equivocator_label_descriptor
                  (projT2 (l item))
    | None => True
    end
    ∧ (∀ s : composite_state (equivocator_IM IM),
         composite_valid (equivocator_IM IM) (l item)
           (s, input item)
         → composite_transition (equivocator_IM IM)
             (l item) (s, input item) =
           (destination item, output item)
           → proper_descriptor (IM (projT1 (l item)))
               (eqv_descriptors' (projT1 (l item)))
               (s (projT1 (l item)))
             ∧ eqv_descriptors' =
               equivocator_descriptors_update IM
                 eqv_descriptors (projT1 (l item))
                 (eqv_descriptors' (projT1 (l item)))
               ∧ s =
                 state_update (equivocator_IM IM)
                   (destination item)
                   (projT1 (l item))
                   (s (projT1 (l item)))
                 ∧ previous_state_descriptor_prop
                     (IM (projT1 (l item)))
                     (eqv_descriptors
                        (projT1 (l item)))
                     (s (projT1 (l item)))
                     (eqv_descriptors'
                        (projT1 (l item)))
                   ∧ match oitem with
                     | Some itemx =>
                         ∀ sx : composite_state IM,
                           sx =
                           equivocators_state_project
                             IM eqv_descriptors' s
                           → composite_valid IM
                               (l itemx)
                               (sx, input itemx)
                             ∧ composite_transition IM
                                 (l itemx)
                                 (sx, input itemx) =
                               (destination itemx,
                                output itemx)
                     | None =>
                         equivocators_state_project IM
                           eqv_descriptors
                           (destination item) =
                         equivocators_state_project IM
                           eqv_descriptors' s
                     end)
  destruct item.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
eqv_descriptors: equivocator_descriptors IM
l: label (composite_type (equivocator_IM IM))
input: option message
destination: state
  (composite_type (equivocator_IM IM))
output: option message
Hproper: proper_descriptor
  (IM
     (projT1
        (VLSM.l
           {|
             l := l;
             input := input;
             destination := destination;
             output := output
           |})))
  (eqv_descriptors
     (projT1
        (VLSM.l
           {|
             l := l;
             input := input;
             destination := destination;
             output := output
           |})))
  (VLSM.destination
     {|
       l := l;
       input := input;
       destination := destination;
       output := output
     |}
     (projT1
        (VLSM.l
           {|
             l := l;
             input := input;
             destination := destination;
             output := output
           |})))
oitemi: option transition_item
eqv_descriptorsi': MachineDescriptor
  (IM
     (projT1
        (VLSM.l
           {|
             l := l;
             input := input;
             destination :=
               destination;
             output := output
           |})))
Hitemx: match oitemi with
| Some itemx =>
    (∃ Hex : existing_equivocator_label
               (IM
                  (projT1
                     (VLSM.l
                      {|
                      l := l;
                      input := input;
                      destination :=
                      destination;
                      output := output
                      |})))
               (VLSM.l
                  (composite_transition_item_projection
                     (equivocator_IM IM)
                     {|
                      l := l;
                      input := input;
                      destination :=
                      destination;
                      output := output
                     |})),
       VLSM.l itemx =
       existing_equivocator_label_extract
         (IM
            (projT1
               (VLSM.l
                  {|
                    l := l;
                    input := input;
                    destination :=
                      destination;
                    output := output
                  |})))
         (VLSM.l
            (composite_transition_item_projection
               (equivocator_IM IM)
               {|
                 l := l;
                 input := input;
                 destination := destination;
                 output := output
               |})) Hex)
    ∧ VLSM.input
        (composite_transition_item_projection
           (equivocator_IM IM)
           {|
             l := l;
             input := input;
             destination := destination;
             output := output
           |}) = VLSM.input itemx
      ∧ VLSM.output
          (composite_transition_item_projection
             (equivocator_IM IM)
             {|
               l := l;
               input := input;
               destination := destination;
               output := output
             |}) = VLSM.output itemx
        ∧ equivocator_state_descriptor_project
            (VLSM.destination
               (composite_transition_item_projection
                  (equivocator_IM IM)
                  {|
                    l := l;
                    input := input;
                    destination :=
                      destination;
                    output := output
                  |}))
            (eqv_descriptors
               (projT1
                  (VLSM.l
                     {|
                      l := l;
                      input := input;
                      destination :=
                      destination;
                      output := output
                     |}))) =
          VLSM.destination itemx
          ∧ eqv_descriptorsi' =
            equivocator_label_descriptor
              (VLSM.l
                 (composite_transition_item_projection
                    (equivocator_IM IM)
                    {|
                      l := l;
                      input := input;
                      destination :=
                      destination;
                      output := output
                    |}))
| None => True
end
Hchar: ∀ s : state
        (equivocator_vlsm
           (IM
              (projT1
                 (VLSM.l
                    {|
                      l := l;
                      input := input;
                      destination :=
                      destination;
                      output := output
                    |})))),
  valid
    (VLSM.l
       (composite_transition_item_projection
          (equivocator_IM IM)
          {|
            l := l;
            input := input;
            destination := destination;
            output := output
          |}))
    (s,
     VLSM.input
       (composite_transition_item_projection
          (equivocator_IM IM)
          {|
            l := l;
            input := input;
            destination := destination;
            output := output
          |}))
  → transition
      (VLSM.l
         (composite_transition_item_projection
            (equivocator_IM IM)
            {|
              l := l;
              input := input;
              destination := destination;
              output := output
            |}))
      (s,
       VLSM.input
         (composite_transition_item_projection
            (equivocator_IM IM)
            {|
              l := l;
              input := input;
              destination := destination;
              output := output
            |})) =
    (VLSM.destination
       (composite_transition_item_projection
          (equivocator_IM IM)
          {|
            l := l;
            input := input;
            destination := destination;
            output := output
          |}),
     VLSM.output
       (composite_transition_item_projection
          (equivocator_IM IM)
          {|
            l := l;
            input := input;
            destination := destination;
            output := output
          |}))
    → proper_descriptor
        (IM
           (projT1
              (VLSM.l
                 {|
                   l := l;
                   input := input;
                   destination := destination;
                   output := output
                 |}))) eqv_descriptorsi' s
      ∧ previous_state_descriptor_prop
          (IM
             (projT1
                (VLSM.l
                   {|
                     l := l;
                     input := input;
                     destination :=
                      destination;
                     output := output
                   |})))
          (eqv_descriptors
             (projT1
                (VLSM.l
                   {|
                     l := l;
                     input := input;
                     destination :=
                      destination;
                     output := output
                   |}))) s eqv_descriptorsi'
        ∧ match oitemi with
          | Some itemx =>
              ∀ sx : state
                      (IM
                      (projT1
                      (VLSM.l
                      {|
                      l := l;
                      input := input;
                      destination :=
                      destination;
                      output := output
                      |}))),
                sx =
                equivocator_state_descriptor_project
                  s eqv_descriptorsi'
                → valid 
                    (VLSM.l itemx)
                    (sx, VLSM.input itemx)
                  ∧ transition 
                      (VLSM.l itemx)
                      (
                      sx, VLSM.input itemx) =
                    (VLSM.destination itemx,
                     VLSM.output itemx)
          | None =>
              equivocator_state_descriptor_project
                (VLSM.destination
                   (composite_transition_item_projection
                      (equivocator_IM IM)
                      {|
                      l := l;
                      input := input;
                      destination :=
                      destination;
                      output := output
                      |}))
                (eqv_descriptors
                   (projT1
                      (VLSM.l
                      {|
                      l := l;
                      input := input;
                      destination :=
                      destination;
                      output := output
                      |}))) =
              equivocator_state_descriptor_project
                s eqv_descriptorsi'
          end
∃ (oitem : option (composite_transition_item IM)) 
  (eqv_descriptors' : equivocator_descriptors IM),
  match oitemi with
  | Some item' =>
      Some
        (Some
           {|
             l :=
               existT
                 (projT1
                    (VLSM.l
                       {|
                         l := l;
                         input := input;
                         destination := destination;
                         output := output
                       |})) (VLSM.l item');
             input :=
               VLSM.input
                 {|
                   l := l;
                   input := input;
                   destination := destination;
                   output := output
                 |};
             destination :=
               equivocators_state_project IM
                 eqv_descriptors
                 (VLSM.destination
                    {|
                      l := l;
                      input := input;
                      destination := destination;
                      output := output
                    |});
             output :=
               VLSM.output
                 {|
                   l := l;
                   input := input;
                   destination := destination;
                   output := output
                 |}
           |},
         equivocator_descriptors_update IM
           eqv_descriptors
           (projT1
              (VLSM.l
                 {|
                   l := l;
                   input := input;
                   destination := destination;
                   output := output
                 |})) eqv_descriptorsi')
  | None =>
      Some
        (None,
         equivocator_descriptors_update IM
           eqv_descriptors
           (projT1
              (VLSM.l
                 {|
                   l := l;
                   input := input;
                   destination := destination;
                   output := output
                 |})) eqv_descriptorsi')
  end = Some (oitem, eqv_descriptors')
  ∧ match oitem with
    | Some itemx =>
        (∃ Hex : existing_equivocator_label
                   (IM
                      (projT1
                         (VLSM.l
                            {|
                              l := l;
                              input := input;
                              destination :=
                                destination;
                              output := output
                            |})))
                   (projT2
                      (VLSM.l
                         {|
                           l := l;
                           input := input;
                           destination := destination;
                           output := output
                         |})),
           existT
             (projT1
                (VLSM.l
                   {|
                     l := l;
                     input := input;
                     destination := destination;
                     output := output
                   |}))
             (existing_equivocator_label_extract
                (IM
                   (projT1
                      (VLSM.l
                         {|
                           l := l;
                           input := input;
                           destination := destination;
                           output := output
                         |})))
                (projT2
                   (VLSM.l
                      {|
                        l := l;
                        input := input;
                        destination := destination;
                        output := output
                      |})) Hex) = VLSM.l itemx)
        ∧ VLSM.input
            {|
              l := l;
              input := input;
              destination := destination;
              output := output
            |} = VLSM.input itemx
          ∧ VLSM.output
              {|
                l := l;
                input := input;
                destination := destination;
                output := output
              |} = VLSM.output itemx
            ∧ equivocators_state_project IM
                eqv_descriptors
                (VLSM.destination
                   {|
                     l := l;
                     input := input;
                     destination := destination;
                     output := output
                   |}) = VLSM.destination itemx
              ∧ eqv_descriptors'
                  (projT1
                     (VLSM.l
                        {|
                          l := l;
                          input := input;
                          destination := destination;
                          output := output
                        |})) =
                equivocator_label_descriptor
                  (projT2
                     (VLSM.l
                        {|
                          l := l;
                          input := input;
                          destination := destination;
                          output := output
                        |}))
    | None => True
    end
    ∧ (∀ s : composite_state (equivocator_IM IM),
         composite_valid (equivocator_IM IM)
           (VLSM.l
              {|
                l := l;
                input := input;
                destination := destination;
                output := output
              |})
           (s,
            VLSM.input
              {|
                l := l;
                input := input;
                destination := destination;
                output := output
              |})
         → composite_transition (equivocator_IM IM)
             (VLSM.l
                {|
                  l := l;
                  input := input;
                  destination := destination;
                  output := output
                |})
             (s,
              VLSM.input
                {|
                  l := l;
                  input := input;
                  destination := destination;
                  output := output
                |}) =
           (VLSM.destination
              {|
                l := l;
                input := input;
                destination := destination;
                output := output
              |},
            VLSM.output
              {|
                l := l;
                input := input;
                destination := destination;
                output := output
              |})
           → proper_descriptor
               (IM
                  (projT1
                     (VLSM.l
                        {|
                          l := l;
                          input := input;
                          destination := destination;
                          output := output
                        |})))
               (eqv_descriptors'
                  (projT1
                     (VLSM.l
                        {|
                          l := l;
                          input := input;
                          destination := destination;
                          output := output
                        |})))
               (s
                  (projT1
                     (VLSM.l
                        {|
                          l := l;
                          input := input;
                          destination := destination;
                          output := output
                        |})))
             ∧ eqv_descriptors' =
               equivocator_descriptors_update IM
                 eqv_descriptors
                 (projT1
                    (VLSM.l
                       {|
                         l := l;
                         input := input;
                         destination := destination;
                         output := output
                       |}))
                 (eqv_descriptors'
                    (projT1
                       (VLSM.l
                          {|
                            l := l;
                            input := input;
                            destination := destination;
                            output := output
                          |})))
               ∧ s =
                 state_update (equivocator_IM IM)
                   (VLSM.destination
                      {|
                        l := l;
                        input := input;
                        destination := destination;
                        output := output
                      |})
                   (projT1
                      (VLSM.l
                         {|
                           l := l;
                           input := input;
                           destination := destination;
                           output := output
                         |}))
                   (s
                      (projT1
                         (VLSM.l
                            {|
                              l := l;
                              input := input;
                              destination :=
                                destination;
                              output := output
                            |})))
                 ∧ previous_state_descriptor_prop
                     (IM
                        (projT1
                           (VLSM.l
                              {|
                                l := l;
                                input := input;
                                destination :=
                                  destination;
                                output := output
                              |})))
                     (eqv_descriptors
                        (projT1
                           (VLSM.l
                              {|
                                l := l;
                                input := input;
                                destination :=
                                  destination;
                                output := output
                              |})))
                     (s
                        (projT1
                           (VLSM.l
                              {|
                                l := l;
                                input := input;
                                destination :=
                                  destination;
                                output := output
                              |})))
                     (eqv_descriptors'
                        (projT1
                           (VLSM.l
                              {|
                                l := l;
                                input := input;
                                destination :=
                                  destination;
                                output := output
                              |})))
                   ∧ match oitem with
                     | Some itemx =>
                         ∀ sx : composite_state IM,
                           sx =
                           equivocators_state_project
                             IM eqv_descriptors' s
                           → composite_valid IM
                               (VLSM.l itemx)
                               (sx, VLSM.input itemx)
                             ∧ composite_transition IM
                                 (VLSM.l itemx)
                                 (sx, VLSM.input itemx) =
                               (VLSM.destination itemx,
                                VLSM.output itemx)
                     | None =>
                         equivocators_state_project IM
                           eqv_descriptors
                           (VLSM.destination
                              {|
                                l := l;
                                input := input;
                                destination :=
                                  destination;
                                output := output
                              |}) =
                         equivocators_state_project IM
                           eqv_descriptors' s
                     end) simpl in *.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
eqv_descriptors: equivocator_descriptors IM
l: composite_label (equivocator_IM IM)
input: option message
destination: composite_state (equivocator_IM IM)
output: option message
Hproper: proper_descriptor (IM (projT1 l))
  (eqv_descriptors (projT1 l))
  (destination (projT1 l))
oitemi: option transition_item
eqv_descriptorsi': MachineDescriptor (IM (projT1 l))
Hitemx: match oitemi with
| Some itemx =>
    (∃ Hex : existing_equivocator_label
               (IM (projT1 l))
               (eq_rect_r
                  (λ n : index,
                     EquivocatorLabel (IM n))
                  (projT2 l) eq_refl),
       VLSM.l itemx =
       existing_equivocator_label_extract
         (IM (projT1 l))
         (eq_rect_r
            (λ n : index,
               EquivocatorLabel (IM n))
            (projT2 l) eq_refl) Hex)
    ∧ input = VLSM.input itemx
      ∧ output = VLSM.output itemx
        ∧ equivocator_state_descriptor_project
            (destination (projT1 l))
            (eqv_descriptors (projT1 l)) =
          VLSM.destination itemx
          ∧ eqv_descriptorsi' =
            equivocator_label_descriptor
              (eq_rect_r
                 (λ n : index,
                    EquivocatorLabel (IM n))
                 (projT2 l) eq_refl)
| None => True
end
Hchar: ∀ s : Equivocators.bounded_state_copies
        (IM (projT1 l)),
  equivocator_valid (IM (projT1 l))
    (eq_rect_r
       (λ n : index, EquivocatorLabel (IM n))
       (projT2 l) eq_refl) (
    s, input)
  → equivocator_transition 
      (IM (projT1 l))
      (eq_rect_r
         (λ n : index, EquivocatorLabel (IM n))
         (projT2 l) eq_refl) (
      s, input) =
    (destination (projT1 l), output)
    → proper_descriptor 
        (IM (projT1 l)) eqv_descriptorsi' s
      ∧ previous_state_descriptor_prop
          (IM (projT1 l))
          (eqv_descriptors (projT1 l)) s
          eqv_descriptorsi'
        ∧ match oitemi with
          | Some itemx =>
              ∀ sx : state (IM (projT1 l)),
                sx =
                equivocator_state_descriptor_project
                  s eqv_descriptorsi'
                → valid 
                    (VLSM.l itemx)
                    (sx, VLSM.input itemx)
                  ∧ transition 
                      (VLSM.l itemx)
                      (
                      sx, VLSM.input itemx) =
                    (VLSM.destination itemx,
                     VLSM.output itemx)
          | None =>
              equivocator_state_descriptor_project
                (destination (projT1 l))
                (eqv_descriptors (projT1 l)) =
              equivocator_state_descriptor_project
                s eqv_descriptorsi'
          end
∃ (oitem : option (composite_transition_item IM)) 
  (eqv_descriptors' : equivocator_descriptors IM),
  match oitemi with
  | Some item' =>
      Some
        (Some
           {|
             l := existT (projT1 l) (VLSM.l item');
             input := input;
             destination :=
               equivocators_state_project IM
                 eqv_descriptors destination;
             output := output
           |},
         equivocator_descriptors_update IM
           eqv_descriptors (projT1 l)
           eqv_descriptorsi')
  | None =>
      Some
        (None,
         equivocator_descriptors_update IM
           eqv_descriptors (projT1 l)
           eqv_descriptorsi')
  end = Some (oitem, eqv_descriptors')
  ∧ match oitem with
    | Some itemx =>
        (∃ Hex : existing_equivocator_label
                   (IM (projT1 l)) (projT2 l),
           existT (projT1 l)
             (existing_equivocator_label_extract
                (IM (projT1 l)) (projT2 l) Hex) =
           VLSM.l itemx)
        ∧ input = VLSM.input itemx
          ∧ output = VLSM.output itemx
            ∧ equivocators_state_project IM
                eqv_descriptors destination =
              VLSM.destination itemx
              ∧ eqv_descriptors' (projT1 l) =
                equivocator_label_descriptor
                  (projT2 l)
    | None => True
    end
    ∧ (∀ s : composite_state (equivocator_IM IM),
         (let (i, li) := l in
          equivocator_valid (IM i) li (s i, input))
         → (let (i, li) := l in
            let (si', om') :=
              equivocator_transition (IM i) li
                (s i, input) in
            (state_update (equivocator_IM IM) s i si',
             om')) = (destination, output)
           → proper_descriptor (IM (projT1 l))
               (eqv_descriptors' (projT1 l))
               (s (projT1 l))
             ∧ eqv_descriptors' =
               equivocator_descriptors_update IM
                 eqv_descriptors (projT1 l)
                 (eqv_descriptors' (projT1 l))
               ∧ s =
                 state_update (equivocator_IM IM)
                   destination (projT1 l)
                   (s (projT1 l))
                 ∧ previous_state_descriptor_prop
                     (IM (projT1 l))
                     (eqv_descriptors (projT1 l))
                     (s (projT1 l))
                     (eqv_descriptors' (projT1 l))
                   ∧ match oitem with
                     | Some itemx =>
                         ∀ sx : composite_state IM,
                           sx =
                           equivocators_state_project
                             IM eqv_descriptors' s
                           → (let (i, li) :=
                                VLSM.l itemx in
                              valid li
                                (sx i,
                                 VLSM.input itemx))
                             ∧ (let (i, li) :=
                                  VLSM.l itemx in
                                let (si', om') :=
                                  transition li
                                    (sx i,
                                     VLSM.input itemx) in
                                (state_update IM sx i
                                   si', om')) =
                               (VLSM.destination itemx,
                                VLSM.output itemx)
                     | None =>
                         equivocators_state_project IM
                           eqv_descriptors destination =
                         equivocators_state_project IM
                           eqv_descriptors' s
                     end) destruct l as (i, li).message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
eqv_descriptors: equivocator_descriptors IM
i: index
li: label (equivocator_IM IM i)
input: option message
destination: composite_state (equivocator_IM IM)
output: option message
Hproper: proper_descriptor
  (IM (projT1 (existT i li)))
  (eqv_descriptors (projT1 (existT i li)))
  (destination (projT1 (existT i li)))
oitemi: option transition_item
eqv_descriptorsi': MachineDescriptor
  (IM (projT1 (existT i li)))
Hitemx: match oitemi with
| Some itemx =>
    (∃ Hex : existing_equivocator_label
               (IM (projT1 (existT i li)))
               (eq_rect_r
                  (λ n : index,
                     EquivocatorLabel (IM n))
                  (projT2 (existT i li))
                  eq_refl),
       l itemx =
       existing_equivocator_label_extract
         (IM (projT1 (existT i li)))
         (eq_rect_r
            (λ n : index,
               EquivocatorLabel (IM n))
            (projT2 (existT i li)) eq_refl)
         Hex)
    ∧ input = VLSM.input itemx
      ∧ output = VLSM.output itemx
        ∧ equivocator_state_descriptor_project
            (destination
               (projT1 (existT i li)))
            (eqv_descriptors
               (projT1 (existT i li))) =
          VLSM.destination itemx
          ∧ eqv_descriptorsi' =
            equivocator_label_descriptor
              (eq_rect_r
                 (λ n : index,
                    EquivocatorLabel (IM n))
                 (projT2 (existT i li))
                 eq_refl)
| None => True
end
Hchar: ∀ s : Equivocators.bounded_state_copies
        (IM (projT1 (existT i li))),
  equivocator_valid
    (IM (projT1 (existT i li)))
    (eq_rect_r
       (λ n : index, EquivocatorLabel (IM n))
       (projT2 (existT i li)) eq_refl)
    (s, input)
  → equivocator_transition
      (IM (projT1 (existT i li)))
      (eq_rect_r
         (λ n : index, EquivocatorLabel (IM n))
         (projT2 (existT i li)) eq_refl)
      (s, input) =
    (destination (projT1 (existT i li)),
     output)
    → proper_descriptor
        (IM (projT1 (existT i li)))
        eqv_descriptorsi' s
      ∧ previous_state_descriptor_prop
          (IM (projT1 (existT i li)))
          (eqv_descriptors
             (projT1 (existT i li))) s
          eqv_descriptorsi'
        ∧ match oitemi with
          | Some itemx =>
              ∀ sx : state
                      (IM
                      (projT1 (existT i li))),
                sx =
                equivocator_state_descriptor_project
                  s eqv_descriptorsi'
                → valid 
                    (l itemx)
                    (sx, VLSM.input itemx)
                  ∧ transition 
                      (l itemx)
                      (
                      sx, VLSM.input itemx) =
                    (VLSM.destination itemx,
                     VLSM.output itemx)
          | None =>
              equivocator_state_descriptor_project
                (destination
                   (projT1 (existT i li)))
                (eqv_descriptors
                   (projT1 (existT i li))) =
              equivocator_state_descriptor_project
                s eqv_descriptorsi'
          end
∃ (oitem : option (composite_transition_item IM)) 
  (eqv_descriptors' : equivocator_descriptors IM),
  match oitemi with
  | Some item' =>
      Some
        (Some
           {|
             l :=
               existT (projT1 (existT i li)) (l item');
             input := input;
             destination :=
               equivocators_state_project IM
                 eqv_descriptors destination;
             output := output
           |},
         equivocator_descriptors_update IM
           eqv_descriptors (projT1 (existT i li))
           eqv_descriptorsi')
  | None =>
      Some
        (None,
         equivocator_descriptors_update IM
           eqv_descriptors (projT1 (existT i li))
           eqv_descriptorsi')
  end = Some (oitem, eqv_descriptors')
  ∧ match oitem with
    | Some itemx =>
        (∃ Hex : existing_equivocator_label
                   (IM (projT1 (existT i li)))
                   (projT2 (existT i li)),
           existT (projT1 (existT i li))
             (existing_equivocator_label_extract
                (IM (projT1 (existT i li)))
                (projT2 (existT i li)) Hex) = l itemx)
        ∧ input = VLSM.input itemx
          ∧ output = VLSM.output itemx
            ∧ equivocators_state_project IM
                eqv_descriptors destination =
              VLSM.destination itemx
              ∧ eqv_descriptors'
                  (projT1 (existT i li)) =
                equivocator_label_descriptor
                  (projT2 (existT i li))
    | None => True
    end
    ∧ (∀ s : composite_state (equivocator_IM IM),
         equivocator_valid (IM i) li (s i, input)
         → (let (si', om') :=
              equivocator_transition (IM i) li
                (s i, input) in
            (state_update (equivocator_IM IM) s i si',
             om')) = (destination, output)
           → proper_descriptor
               (IM (projT1 (existT i li)))
               (eqv_descriptors'
                  (projT1 (existT i li)))
               (s (projT1 (existT i li)))
             ∧ eqv_descriptors' =
               equivocator_descriptors_update IM
                 eqv_descriptors
                 (projT1 (existT i li))
                 (eqv_descriptors'
                    (projT1 (existT i li)))
               ∧ s =
                 state_update (equivocator_IM IM)
                   destination (projT1 (existT i li))
                   (s (projT1 (existT i li)))
                 ∧ previous_state_descriptor_prop
                     (IM (projT1 (existT i li)))
                     (eqv_descriptors
                        (projT1 (existT i li)))
                     (s (projT1 (existT i li)))
                     (eqv_descriptors'
                        (projT1 (existT i li)))
                   ∧ match oitem with
                     | Some itemx =>
                         ∀ sx : composite_state IM,
                           sx =
                           equivocators_state_project
                             IM eqv_descriptors' s
                           → (let (i, li) :=
                                l itemx in
                              valid li
                                (sx i,
                                 VLSM.input itemx))
                             ∧ (let (i, li) :=
                                  l itemx in
                                let (si', om') :=
                                  transition li
                                    (sx i,
                                     VLSM.input itemx) in
                                (state_update IM sx i
                                   si', om')) =
                               (VLSM.destination itemx,
                                VLSM.output itemx)
                     | None =>
                         equivocators_state_project IM
                           eqv_descriptors destination =
                         equivocators_state_project IM
                           eqv_descriptors' s
                     end) simpl in *.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
eqv_descriptors: equivocator_descriptors IM
i: index
li: EquivocatorLabel (IM i)
input: option message
destination: composite_state (equivocator_IM IM)
output: option message
Hproper: proper_descriptor (IM i) 
  (eqv_descriptors i) (destination i)
oitemi: option transition_item
eqv_descriptorsi': MachineDescriptor (IM i)
Hitemx: match oitemi with
| Some itemx =>
    (∃ Hex : existing_equivocator_label
               (IM i)
               (eq_rect_r
                  (λ n : index,
                     EquivocatorLabel (IM n))
                  li eq_refl),
       l itemx =
       existing_equivocator_label_extract
         (IM i)
         (eq_rect_r
            (λ n : index,
               EquivocatorLabel (IM n)) li
            eq_refl) Hex)
    ∧ input = VLSM.input itemx
      ∧ output = VLSM.output itemx
        ∧ equivocator_state_descriptor_project
            (destination i)
            (eqv_descriptors i) =
          VLSM.destination itemx
          ∧ eqv_descriptorsi' =
            equivocator_label_descriptor
              (eq_rect_r
                 (λ n : index,
                    EquivocatorLabel (IM n))
                 li eq_refl)
| None => True
end
Hchar: ∀ s : Equivocators.bounded_state_copies (IM i),
  equivocator_valid (IM i)
    (eq_rect_r
       (λ n : index, EquivocatorLabel (IM n))
       li eq_refl) (s, input)
  → equivocator_transition 
      (IM i)
      (eq_rect_r
         (λ n : index, EquivocatorLabel (IM n))
         li eq_refl) (s, input) =
    (destination i, output)
    → proper_descriptor 
        (IM i) eqv_descriptorsi' s
      ∧ previous_state_descriptor_prop 
          (IM i) (eqv_descriptors i) s
          eqv_descriptorsi'
        ∧ match oitemi with
          | Some itemx =>
              ∀ sx : state (IM i),
                sx =
                equivocator_state_descriptor_project
                  s eqv_descriptorsi'
                → valid 
                    (l itemx)
                    (sx, VLSM.input itemx)
                  ∧ transition 
                      (l itemx)
                      (
                      sx, VLSM.input itemx) =
                    (VLSM.destination itemx,
                     VLSM.output itemx)
          | None =>
              equivocator_state_descriptor_project
                (destination i)
                (eqv_descriptors i) =
              equivocator_state_descriptor_project
                s eqv_descriptorsi'
          end
∃ (oitem : option (composite_transition_item IM)) 
  (eqv_descriptors' : equivocator_descriptors IM),
  match oitemi with
  | Some item' =>
      Some
        (Some
           {|
             l := existT i (l item');
             input := input;
             destination :=
               equivocators_state_project IM
                 eqv_descriptors destination;
             output := output
           |},
         equivocator_descriptors_update IM
           eqv_descriptors i eqv_descriptorsi')
  | None =>
      Some
        (None,
         equivocator_descriptors_update IM
           eqv_descriptors i eqv_descriptorsi')
  end = Some (oitem, eqv_descriptors')
  ∧ match oitem with
    | Some itemx =>
        (∃ Hex : existing_equivocator_label (IM i) li,
           existT i
             (existing_equivocator_label_extract
                (IM i) li Hex) = l itemx)
        ∧ input = VLSM.input itemx
          ∧ output = VLSM.output itemx
            ∧ equivocators_state_project IM
                eqv_descriptors destination =
              VLSM.destination itemx
              ∧ eqv_descriptors' i =
                equivocator_label_descriptor li
    | None => True
    end
    ∧ (∀ s : composite_state (equivocator_IM IM),
         equivocator_valid (IM i) li (s i, input)
         → (let (si', om') :=
              equivocator_transition (IM i) li
                (s i, input) in
            (state_update (equivocator_IM IM) s i si',
             om')) = (destination, output)
           → proper_descriptor (IM i)
               (eqv_descriptors' i) (s i)
             ∧ eqv_descriptors' =
               equivocator_descriptors_update IM
                 eqv_descriptors i
                 (eqv_descriptors' i)
               ∧ s =
                 state_update (equivocator_IM IM)
                   destination i (s i)
                 ∧ previous_state_descriptor_prop
                     (IM i) (eqv_descriptors i) (s i)
                     (eqv_descriptors' i)
                   ∧ match oitem with
                     | Some itemx =>
                         ∀ sx : composite_state IM,
                           sx =
                           equivocators_state_project
                             IM eqv_descriptors' s
                           → (let (i, li) :=
                                l itemx in
                              valid li
                                (sx i,
                                 VLSM.input itemx))
                             ∧ (let (i, li) :=
                                  l itemx in
                                let (si', om') :=
                                  transition li
                                    (sx i,
                                     VLSM.input itemx) in
                                (state_update IM sx i
                                   si', om')) =
                               (VLSM.destination itemx,
                                VLSM.output itemx)
                     | None =>
                         equivocators_state_project IM
                           eqv_descriptors destination =
                         equivocators_state_project IM
                           eqv_descriptors' s
                     end)
  destruct oitemi as [itemi' |]; eexists _; eexists _; (split; [done |])
  ; [| split; [done |]]
  ; [destruct Hitemx as [[Hex Hli] [Hinputi [Houtputi [Hdestinationi Hdescriptori]]]]
  ; rewrite Hli; subst; split; [repeat split |]
    |]
  ; [by exists Hex | apply equivocator_descriptors_update_eq | ..]
  ; intros
  ; case_match eqn: Ht'
  ; inversion Ht; subst; clear Ht
  ; rewrite state_update_eq in Hchar
  ; specialize (Hchar _ Hv Ht')
  ; simpl in *
  ; destruct Hchar as (Hproper' & Hex_new & Hchar)
  .message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
eqv_descriptors: equivocator_descriptors IM
i: index
li: EquivocatorLabel (IM i)
s: composite_state (equivocator_IM IM)
e: equivocator_state (IM i)
Hproper: proper_descriptor (IM i) 
  (eqv_descriptors i)
  (state_update (equivocator_IM IM) s i e i)
itemi': transition_item
Hex: existing_equivocator_label 
  (IM i)
  (eq_rect_r
     (λ n : index, EquivocatorLabel (IM n)) li
     eq_refl)
Hli: l itemi' =
existing_equivocator_label_extract 
  (IM i)
  (eq_rect_r
     (λ n : index, EquivocatorLabel (IM n)) li
     eq_refl) Hex
Hproper': proper_descriptor (IM i)
  (equivocator_label_descriptor
     (eq_rect_r
        (λ n : index,
           EquivocatorLabel (IM n)) li
        eq_refl)) (s i)
Hex_new: previous_state_descriptor_prop 
  (IM i) (eqv_descriptors i) 
  (s i)
  (equivocator_label_descriptor
     (eq_rect_r
        (λ n : index,
           EquivocatorLabel (IM n)) li
        eq_refl))
Hchar: ∀ sx : state (IM i),
  sx =
  equivocator_state_descriptor_project 
    (s i)
    (equivocator_label_descriptor
       (eq_rect_r
          (λ n : index,
             EquivocatorLabel (IM n)) li
          eq_refl))
  → valid (l itemi') (sx, input itemi')
    ∧ transition (l itemi') (sx, input itemi') =
      (destination itemi', output itemi')
Hdestinationi: equivocator_state_descriptor_project
  (state_update (equivocator_IM IM) s
     i e i) (eqv_descriptors i) =
destination itemi'
Hv: equivocator_valid (IM i) li (s i, input itemi')
Ht': equivocator_transition 
  (IM i) li (s i, input itemi') =
(e, output itemi')
proper_descriptor (IM i)
  (equivocator_descriptors_update IM eqv_descriptors i
     (equivocator_label_descriptor
        (eq_rect_r
           (λ n : index, EquivocatorLabel (IM n)) li
           eq_refl)) i) (s i)
∧ equivocator_descriptors_update IM eqv_descriptors i
    (equivocator_label_descriptor
       (eq_rect_r
          (λ n : index, EquivocatorLabel (IM n)) li
          eq_refl)) =
  equivocator_descriptors_update IM eqv_descriptors i
    (equivocator_descriptors_update IM eqv_descriptors
       i
       (equivocator_label_descriptor
          (eq_rect_r
             (λ n : index, EquivocatorLabel (IM n)) li
             eq_refl)) i)
  ∧ s =
    state_update (equivocator_IM IM)
      (state_update (equivocator_IM IM) s i e) i (s i)
    ∧ previous_state_descriptor_prop (IM i)
        (eqv_descriptors i) (s i)
        (equivocator_descriptors_update IM
           eqv_descriptors i
           (equivocator_label_descriptor
              (eq_rect_r
                 (λ n : index, EquivocatorLabel (IM n))
                 li eq_refl)) i)
      ∧ (∀ sx : composite_state IM,
           sx =
           equivocators_state_project IM
             (equivocator_descriptors_update IM
                eqv_descriptors i
                (equivocator_label_descriptor
                   (eq_rect_r
                      (λ n : index,
                         EquivocatorLabel (IM n)) li
                      eq_refl))) s
           → valid
               (existing_equivocator_label_extract
                  (IM i)
                  (eq_rect_r
                     (λ n : index,
                        EquivocatorLabel (IM n)) li
                     eq_refl) Hex)
               (sx i, input itemi')
             ∧ (let (si', om') :=
                  transition
                    (existing_equivocator_label_extract
                       (IM i)
                       (eq_rect_r
                          (λ n : index,
                             EquivocatorLabel (IM n))
                          li eq_refl) Hex)
                    (sx i, input itemi') in
                (state_update IM sx i si', om')) =
               (equivocators_state_project IM
                  eqv_descriptors
                  (state_update (equivocator_IM IM) s
                     i e), output itemi'))
message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
eqv_descriptors: equivocator_descriptors IM
i: index
li: EquivocatorLabel (IM i)
input, output: option message
s: composite_state (equivocator_IM IM)
e: equivocator_state (IM i)
Hproper: proper_descriptor (IM i) 
  (eqv_descriptors i)
  (state_update (equivocator_IM IM) s i e i)
eqv_descriptorsi': MachineDescriptor (IM i)
Hitemx: True
Hproper': proper_descriptor (IM i) eqv_descriptorsi'
  (s i)
Hex_new: previous_state_descriptor_prop 
  (IM i) (eqv_descriptors i) 
  (s i) eqv_descriptorsi'
Hchar: equivocator_state_descriptor_project e
  (eqv_descriptors i) =
equivocator_state_descriptor_project 
  (s i) eqv_descriptorsi'
Hv: equivocator_valid (IM i) li (s i, input)
Ht': equivocator_transition (IM i) li (s i, input) =
(e, output)
proper_descriptor (IM i)
  (equivocator_descriptors_update IM eqv_descriptors i
     eqv_descriptorsi' i) 
  (s i)
∧ equivocator_descriptors_update IM eqv_descriptors i
    eqv_descriptorsi' =
  equivocator_descriptors_update IM eqv_descriptors i
    (equivocator_descriptors_update IM eqv_descriptors
       i eqv_descriptorsi' i)
  ∧ s =
    state_update (equivocator_IM IM)
      (state_update (equivocator_IM IM) s i e) i 
      (s i)
    ∧ previous_state_descriptor_prop 
        (IM i) (eqv_descriptors i) 
        (s i)
        (equivocator_descriptors_update IM
           eqv_descriptors i eqv_descriptorsi' i)
      ∧ equivocators_state_project IM eqv_descriptors
          (state_update (equivocator_IM IM) s i e) =
        equivocators_state_project IM
          (equivocator_descriptors_update IM
             eqv_descriptors i eqv_descriptorsi') s
  -message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
eqv_descriptors: equivocator_descriptors IM
i: index
li: EquivocatorLabel (IM i)
s: composite_state (equivocator_IM IM)
e: equivocator_state (IM i)
Hproper: proper_descriptor (IM i) 
  (eqv_descriptors i)
  (state_update (equivocator_IM IM) s i e i)
itemi': transition_item
Hex: existing_equivocator_label 
  (IM i)
  (eq_rect_r
     (λ n : index, EquivocatorLabel (IM n)) li
     eq_refl)
Hli: l itemi' =
existing_equivocator_label_extract 
  (IM i)
  (eq_rect_r
     (λ n : index, EquivocatorLabel (IM n)) li
     eq_refl) Hex
Hproper': proper_descriptor (IM i)
  (equivocator_label_descriptor
     (eq_rect_r
        (λ n : index,
           EquivocatorLabel (IM n)) li
        eq_refl)) (s i)
Hex_new: previous_state_descriptor_prop 
  (IM i) (eqv_descriptors i) 
  (s i)
  (equivocator_label_descriptor
     (eq_rect_r
        (λ n : index,
           EquivocatorLabel (IM n)) li
        eq_refl))
Hchar: ∀ sx : state (IM i),
  sx =
  equivocator_state_descriptor_project 
    (s i)
    (equivocator_label_descriptor
       (eq_rect_r
          (λ n : index,
             EquivocatorLabel (IM n)) li
          eq_refl))
  → valid (l itemi') (sx, input itemi')
    ∧ transition (l itemi') (sx, input itemi') =
      (destination itemi', output itemi')
Hdestinationi: equivocator_state_descriptor_project
  (state_update (equivocator_IM IM) s
     i e i) (eqv_descriptors i) =
destination itemi'
Hv: equivocator_valid (IM i) li (s i, input itemi')
Ht': equivocator_transition 
  (IM i) li (s i, input itemi') =
(e, output itemi')
proper_descriptor (IM i)
  (equivocator_descriptors_update IM eqv_descriptors i
     (equivocator_label_descriptor
        (eq_rect_r
           (λ n : index, EquivocatorLabel (IM n)) li
           eq_refl)) i) (s i)
∧ equivocator_descriptors_update IM eqv_descriptors i
    (equivocator_label_descriptor
       (eq_rect_r
          (λ n : index, EquivocatorLabel (IM n)) li
          eq_refl)) =
  equivocator_descriptors_update IM eqv_descriptors i
    (equivocator_descriptors_update IM eqv_descriptors
       i
       (equivocator_label_descriptor
          (eq_rect_r
             (λ n : index, EquivocatorLabel (IM n)) li
             eq_refl)) i)
  ∧ s =
    state_update (equivocator_IM IM)
      (state_update (equivocator_IM IM) s i e) i (s i)
    ∧ previous_state_descriptor_prop (IM i)
        (eqv_descriptors i) (s i)
        (equivocator_descriptors_update IM
           eqv_descriptors i
           (equivocator_label_descriptor
              (eq_rect_r
                 (λ n : index, EquivocatorLabel (IM n))
                 li eq_refl)) i)
      ∧ (∀ sx : composite_state IM,
           sx =
           equivocators_state_project IM
             (equivocator_descriptors_update IM
                eqv_descriptors i
                (equivocator_label_descriptor
                   (eq_rect_r
                      (λ n : index,
                         EquivocatorLabel (IM n)) li
                      eq_refl))) s
           → valid
               (existing_equivocator_label_extract
                  (IM i)
                  (eq_rect_r
                     (λ n : index,
                        EquivocatorLabel (IM n)) li
                     eq_refl) Hex)
               (sx i, input itemi')
             ∧ (let (si', om') :=
                  transition
                    (existing_equivocator_label_extract
                       (IM i)
                       (eq_rect_r
                          (λ n : index,
                             EquivocatorLabel (IM n))
                          li eq_refl) Hex)
                    (sx i, input itemi') in
                (state_update IM sx i si', om')) =
               (equivocators_state_project IM
                  eqv_descriptors
                  (state_update (equivocator_IM IM) s
                     i e), output itemi')) repeat split.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
eqv_descriptors: equivocator_descriptors IM
i: index
li: EquivocatorLabel (IM i)
s: composite_state (equivocator_IM IM)
e: equivocator_state (IM i)
Hproper: proper_descriptor (IM i) 
  (eqv_descriptors i)
  (state_update (equivocator_IM IM) s i e i)
itemi': transition_item
Hex: existing_equivocator_label 
  (IM i)
  (eq_rect_r
     (λ n : index, EquivocatorLabel (IM n)) li
     eq_refl)
Hli: l itemi' =
existing_equivocator_label_extract 
  (IM i)
  (eq_rect_r
     (λ n : index, EquivocatorLabel (IM n)) li
     eq_refl) Hex
Hproper': proper_descriptor (IM i)
  (equivocator_label_descriptor
     (eq_rect_r
        (λ n : index,
           EquivocatorLabel (IM n)) li
        eq_refl)) (s i)
Hex_new: previous_state_descriptor_prop 
  (IM i) (eqv_descriptors i) 
  (s i)
  (equivocator_label_descriptor
     (eq_rect_r
        (λ n : index,
           EquivocatorLabel (IM n)) li
        eq_refl))
Hchar: ∀ sx : state (IM i),
  sx =
  equivocator_state_descriptor_project 
    (s i)
    (equivocator_label_descriptor
       (eq_rect_r
          (λ n : index,
             EquivocatorLabel (IM n)) li
          eq_refl))
  → valid (l itemi') (sx, input itemi')
    ∧ transition (l itemi') (sx, input itemi') =
      (destination itemi', output itemi')
Hdestinationi: equivocator_state_descriptor_project
  (state_update (equivocator_IM IM) s
     i e i) (eqv_descriptors i) =
destination itemi'
Hv: equivocator_valid (IM i) li (s i, input itemi')
Ht': equivocator_transition 
  (IM i) li (s i, input itemi') =
(e, output itemi')
proper_descriptor (IM i)
  (equivocator_descriptors_update IM eqv_descriptors i
     (equivocator_label_descriptor
        (eq_rect_r
           (λ n : index, EquivocatorLabel (IM n)) li
           eq_refl)) i) (s i)
message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
eqv_descriptors: equivocator_descriptors IM
i: index
li: EquivocatorLabel (IM i)
s: composite_state (equivocator_IM IM)
e: equivocator_state (IM i)
Hproper: proper_descriptor (IM i) 
  (eqv_descriptors i)
  (state_update (equivocator_IM IM) s i e i)
itemi': transition_item
Hex: existing_equivocator_label 
  (IM i)
  (eq_rect_r
     (λ n : index, EquivocatorLabel (IM n)) li
     eq_refl)
Hli: l itemi' =
existing_equivocator_label_extract 
  (IM i)
  (eq_rect_r
     (λ n : index, EquivocatorLabel (IM n)) li
     eq_refl) Hex
Hproper': proper_descriptor (IM i)
  (equivocator_label_descriptor
     (eq_rect_r
        (λ n : index,
           EquivocatorLabel (IM n)) li
        eq_refl)) (s i)
Hex_new: previous_state_descriptor_prop 
  (IM i) (eqv_descriptors i) 
  (s i)
  (equivocator_label_descriptor
     (eq_rect_r
        (λ n : index,
           EquivocatorLabel (IM n)) li
        eq_refl))
Hchar: ∀ sx : state (IM i),
  sx =
  equivocator_state_descriptor_project 
    (s i)
    (equivocator_label_descriptor
       (eq_rect_r
          (λ n : index,
             EquivocatorLabel (IM n)) li
          eq_refl))
  → valid (l itemi') (sx, input itemi')
    ∧ transition (l itemi') (sx, input itemi') =
      (destination itemi', output itemi')
Hdestinationi: equivocator_state_descriptor_project
  (state_update (equivocator_IM IM) s
     i e i) (eqv_descriptors i) =
destination itemi'
Hv: equivocator_valid (IM i) li (s i, input itemi')
Ht': equivocator_transition 
  (IM i) li (s i, input itemi') =
(e, output itemi')
equivocator_descriptors_update IM eqv_descriptors i
  (equivocator_label_descriptor
     (eq_rect_r (λ n : index, EquivocatorLabel (IM n))
        li eq_refl)) =
equivocator_descriptors_update IM eqv_descriptors i
  (equivocator_descriptors_update IM eqv_descriptors i
     (equivocator_label_descriptor
        (eq_rect_r
           (λ n : index, EquivocatorLabel (IM n)) li
           eq_refl)) i)
message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
eqv_descriptors: equivocator_descriptors IM
i: index
li: EquivocatorLabel (IM i)
s: composite_state (equivocator_IM IM)
e: equivocator_state (IM i)
Hproper: proper_descriptor (IM i) 
  (eqv_descriptors i)
  (state_update (equivocator_IM IM) s i e i)
itemi': transition_item
Hex: existing_equivocator_label 
  (IM i)
  (eq_rect_r
     (λ n : index, EquivocatorLabel (IM n)) li
     eq_refl)
Hli: l itemi' =
existing_equivocator_label_extract 
  (IM i)
  (eq_rect_r
     (λ n : index, EquivocatorLabel (IM n)) li
     eq_refl) Hex
Hproper': proper_descriptor (IM i)
  (equivocator_label_descriptor
     (eq_rect_r
        (λ n : index,
           EquivocatorLabel (IM n)) li
        eq_refl)) (s i)
Hex_new: previous_state_descriptor_prop 
  (IM i) (eqv_descriptors i) 
  (s i)
  (equivocator_label_descriptor
     (eq_rect_r
        (λ n : index,
           EquivocatorLabel (IM n)) li
        eq_refl))
Hchar: ∀ sx : state (IM i),
  sx =
  equivocator_state_descriptor_project 
    (s i)
    (equivocator_label_descriptor
       (eq_rect_r
          (λ n : index,
             EquivocatorLabel (IM n)) li
          eq_refl))
  → valid (l itemi') (sx, input itemi')
    ∧ transition (l itemi') (sx, input itemi') =
      (destination itemi', output itemi')
Hdestinationi: equivocator_state_descriptor_project
  (state_update (equivocator_IM IM) s
     i e i) (eqv_descriptors i) =
destination itemi'
Hv: equivocator_valid (IM i) li (s i, input itemi')
Ht': equivocator_transition 
  (IM i) li (s i, input itemi') =
(e, output itemi')
s =
state_update (equivocator_IM IM)
  (state_update (equivocator_IM IM) s i e) i 
  (s i)
message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
eqv_descriptors: equivocator_descriptors IM
i: index
li: EquivocatorLabel (IM i)
s: composite_state (equivocator_IM IM)
e: equivocator_state (IM i)
Hproper: proper_descriptor (IM i) 
  (eqv_descriptors i)
  (state_update (equivocator_IM IM) s i e i)
itemi': transition_item
Hex: existing_equivocator_label 
  (IM i)
  (eq_rect_r
     (λ n : index, EquivocatorLabel (IM n)) li
     eq_refl)
Hli: l itemi' =
existing_equivocator_label_extract 
  (IM i)
  (eq_rect_r
     (λ n : index, EquivocatorLabel (IM n)) li
     eq_refl) Hex
Hproper': proper_descriptor (IM i)
  (equivocator_label_descriptor
     (eq_rect_r
        (λ n : index,
           EquivocatorLabel (IM n)) li
        eq_refl)) (s i)
Hex_new: previous_state_descriptor_prop 
  (IM i) (eqv_descriptors i) 
  (s i)
  (equivocator_label_descriptor
     (eq_rect_r
        (λ n : index,
           EquivocatorLabel (IM n)) li
        eq_refl))
Hchar: ∀ sx : state (IM i),
  sx =
  equivocator_state_descriptor_project 
    (s i)
    (equivocator_label_descriptor
       (eq_rect_r
          (λ n : index,
             EquivocatorLabel (IM n)) li
          eq_refl))
  → valid (l itemi') (sx, input itemi')
    ∧ transition (l itemi') (sx, input itemi') =
      (destination itemi', output itemi')
Hdestinationi: equivocator_state_descriptor_project
  (state_update (equivocator_IM IM) s
     i e i) (eqv_descriptors i) =
destination itemi'
Hv: equivocator_valid (IM i) li (s i, input itemi')
Ht': equivocator_transition 
  (IM i) li (s i, input itemi') =
(e, output itemi')
previous_state_descriptor_prop 
  (IM i) (eqv_descriptors i) 
  (s i)
  (equivocator_descriptors_update IM eqv_descriptors i
     (equivocator_label_descriptor
        (eq_rect_r
           (λ n : index, EquivocatorLabel (IM n)) li
           eq_refl)) i)
message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
eqv_descriptors: equivocator_descriptors IM
i: index
li: EquivocatorLabel (IM i)
s: composite_state (equivocator_IM IM)
e: equivocator_state (IM i)
Hproper: proper_descriptor (IM i) 
  (eqv_descriptors i)
  (state_update (equivocator_IM IM) s i e i)
itemi': transition_item
Hex: existing_equivocator_label 
  (IM i)
  (eq_rect_r
     (λ n : index, EquivocatorLabel (IM n)) li
     eq_refl)
Hli: l itemi' =
existing_equivocator_label_extract 
  (IM i)
  (eq_rect_r
     (λ n : index, EquivocatorLabel (IM n)) li
     eq_refl) Hex
Hproper': proper_descriptor (IM i)
  (equivocator_label_descriptor
     (eq_rect_r
        (λ n : index,
           EquivocatorLabel (IM n)) li
        eq_refl)) (s i)
Hex_new: previous_state_descriptor_prop 
  (IM i) (eqv_descriptors i) 
  (s i)
  (equivocator_label_descriptor
     (eq_rect_r
        (λ n : index,
           EquivocatorLabel (IM n)) li
        eq_refl))
Hchar: ∀ sx : state (IM i),
  sx =
  equivocator_state_descriptor_project 
    (s i)
    (equivocator_label_descriptor
       (eq_rect_r
          (λ n : index,
             EquivocatorLabel (IM n)) li
          eq_refl))
  → valid (l itemi') (sx, input itemi')
    ∧ transition (l itemi') (sx, input itemi') =
      (destination itemi', output itemi')
Hdestinationi: equivocator_state_descriptor_project
  (state_update (equivocator_IM IM) s
     i e i) (eqv_descriptors i) =
destination itemi'
Hv: equivocator_valid (IM i) li (s i, input itemi')
Ht': equivocator_transition 
  (IM i) li (s i, input itemi') =
(e, output itemi')
sx: composite_state IM
Hsx: sx =
equivocators_state_project IM
  (equivocator_descriptors_update IM
     eqv_descriptors i
     (equivocator_label_descriptor
        (eq_rect_r
           (λ n : index, EquivocatorLabel (IM n))
           li eq_refl))) s
valid
  (existing_equivocator_label_extract 
     (IM i)
     (eq_rect_r (λ n : index, EquivocatorLabel (IM n))
        li eq_refl) Hex) (
  sx i, input itemi')
message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
eqv_descriptors: equivocator_descriptors IM
i: index
li: EquivocatorLabel (IM i)
s: composite_state (equivocator_IM IM)
e: equivocator_state (IM i)
Hproper: proper_descriptor (IM i) 
  (eqv_descriptors i)
  (state_update (equivocator_IM IM) s i e i)
itemi': transition_item
Hex: existing_equivocator_label 
  (IM i)
  (eq_rect_r
     (λ n : index, EquivocatorLabel (IM n)) li
     eq_refl)
Hli: l itemi' =
existing_equivocator_label_extract 
  (IM i)
  (eq_rect_r
     (λ n : index, EquivocatorLabel (IM n)) li
     eq_refl) Hex
Hproper': proper_descriptor (IM i)
  (equivocator_label_descriptor
     (eq_rect_r
        (λ n : index,
           EquivocatorLabel (IM n)) li
        eq_refl)) (s i)
Hex_new: previous_state_descriptor_prop 
  (IM i) (eqv_descriptors i) 
  (s i)
  (equivocator_label_descriptor
     (eq_rect_r
        (λ n : index,
           EquivocatorLabel (IM n)) li
        eq_refl))
Hchar: ∀ sx : state (IM i),
  sx =
  equivocator_state_descriptor_project 
    (s i)
    (equivocator_label_descriptor
       (eq_rect_r
          (λ n : index,
             EquivocatorLabel (IM n)) li
          eq_refl))
  → valid (l itemi') (sx, input itemi')
    ∧ transition (l itemi') (sx, input itemi') =
      (destination itemi', output itemi')
Hdestinationi: equivocator_state_descriptor_project
  (state_update (equivocator_IM IM) s
     i e i) (eqv_descriptors i) =
destination itemi'
Hv: equivocator_valid (IM i) li (s i, input itemi')
Ht': equivocator_transition 
  (IM i) li (s i, input itemi') =
(e, output itemi')
sx: composite_state IM
Hsx: sx =
equivocators_state_project IM
  (equivocator_descriptors_update IM
     eqv_descriptors i
     (equivocator_label_descriptor
        (eq_rect_r
           (λ n : index, EquivocatorLabel (IM n))
           li eq_refl))) s
(let (si', om') :=
   transition
     (existing_equivocator_label_extract 
        (IM i)
        (eq_rect_r
           (λ n : index, EquivocatorLabel (IM n)) li
           eq_refl) Hex) (
     sx i, input itemi') in
 (state_update IM sx i si', om')) =
(equivocators_state_project IM eqv_descriptors
   (state_update (equivocator_IM IM) s i e),
 output itemi')
    +message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
eqv_descriptors: equivocator_descriptors IM
i: index
li: EquivocatorLabel (IM i)
s: composite_state (equivocator_IM IM)
e: equivocator_state (IM i)
Hproper: proper_descriptor (IM i) 
  (eqv_descriptors i)
  (state_update (equivocator_IM IM) s i e i)
itemi': transition_item
Hex: existing_equivocator_label 
  (IM i)
  (eq_rect_r
     (λ n : index, EquivocatorLabel (IM n)) li
     eq_refl)
Hli: l itemi' =
existing_equivocator_label_extract 
  (IM i)
  (eq_rect_r
     (λ n : index, EquivocatorLabel (IM n)) li
     eq_refl) Hex
Hproper': proper_descriptor (IM i)
  (equivocator_label_descriptor
     (eq_rect_r
        (λ n : index,
           EquivocatorLabel (IM n)) li
        eq_refl)) (s i)
Hex_new: previous_state_descriptor_prop 
  (IM i) (eqv_descriptors i) 
  (s i)
  (equivocator_label_descriptor
     (eq_rect_r
        (λ n : index,
           EquivocatorLabel (IM n)) li
        eq_refl))
Hchar: ∀ sx : state (IM i),
  sx =
  equivocator_state_descriptor_project 
    (s i)
    (equivocator_label_descriptor
       (eq_rect_r
          (λ n : index,
             EquivocatorLabel (IM n)) li
          eq_refl))
  → valid (l itemi') (sx, input itemi')
    ∧ transition (l itemi') (sx, input itemi') =
      (destination itemi', output itemi')
Hdestinationi: equivocator_state_descriptor_project
  (state_update (equivocator_IM IM) s
     i e i) (eqv_descriptors i) =
destination itemi'
Hv: equivocator_valid (IM i) li (s i, input itemi')
Ht': equivocator_transition 
  (IM i) li (s i, input itemi') =
(e, output itemi')
proper_descriptor (IM i)
  (equivocator_descriptors_update IM eqv_descriptors i
     (equivocator_label_descriptor
        (eq_rect_r
           (λ n : index, EquivocatorLabel (IM n)) li
           eq_refl)) i) (s i) by state_update_simpl.
    +message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
eqv_descriptors: equivocator_descriptors IM
i: index
li: EquivocatorLabel (IM i)
s: composite_state (equivocator_IM IM)
e: equivocator_state (IM i)
Hproper: proper_descriptor (IM i) 
  (eqv_descriptors i)
  (state_update (equivocator_IM IM) s i e i)
itemi': transition_item
Hex: existing_equivocator_label 
  (IM i)
  (eq_rect_r
     (λ n : index, EquivocatorLabel (IM n)) li
     eq_refl)
Hli: l itemi' =
existing_equivocator_label_extract 
  (IM i)
  (eq_rect_r
     (λ n : index, EquivocatorLabel (IM n)) li
     eq_refl) Hex
Hproper': proper_descriptor (IM i)
  (equivocator_label_descriptor
     (eq_rect_r
        (λ n : index,
           EquivocatorLabel (IM n)) li
        eq_refl)) (s i)
Hex_new: previous_state_descriptor_prop 
  (IM i) (eqv_descriptors i) 
  (s i)
  (equivocator_label_descriptor
     (eq_rect_r
        (λ n : index,
           EquivocatorLabel (IM n)) li
        eq_refl))
Hchar: ∀ sx : state (IM i),
  sx =
  equivocator_state_descriptor_project 
    (s i)
    (equivocator_label_descriptor
       (eq_rect_r
          (λ n : index,
             EquivocatorLabel (IM n)) li
          eq_refl))
  → valid (l itemi') (sx, input itemi')
    ∧ transition (l itemi') (sx, input itemi') =
      (destination itemi', output itemi')
Hdestinationi: equivocator_state_descriptor_project
  (state_update (equivocator_IM IM) s
     i e i) (eqv_descriptors i) =
destination itemi'
Hv: equivocator_valid (IM i) li (s i, input itemi')
Ht': equivocator_transition 
  (IM i) li (s i, input itemi') =
(e, output itemi')
equivocator_descriptors_update IM eqv_descriptors i
  (equivocator_label_descriptor
     (eq_rect_r (λ n : index, EquivocatorLabel (IM n))
        li eq_refl)) =
equivocator_descriptors_update IM eqv_descriptors i
  (equivocator_descriptors_update IM eqv_descriptors i
     (equivocator_label_descriptor
        (eq_rect_r
           (λ n : index, EquivocatorLabel (IM n)) li
           eq_refl)) i) by state_update_simpl.
    +message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
eqv_descriptors: equivocator_descriptors IM
i: index
li: EquivocatorLabel (IM i)
s: composite_state (equivocator_IM IM)
e: equivocator_state (IM i)
Hproper: proper_descriptor (IM i) 
  (eqv_descriptors i)
  (state_update (equivocator_IM IM) s i e i)
itemi': transition_item
Hex: existing_equivocator_label 
  (IM i)
  (eq_rect_r
     (λ n : index, EquivocatorLabel (IM n)) li
     eq_refl)
Hli: l itemi' =
existing_equivocator_label_extract 
  (IM i)
  (eq_rect_r
     (λ n : index, EquivocatorLabel (IM n)) li
     eq_refl) Hex
Hproper': proper_descriptor (IM i)
  (equivocator_label_descriptor
     (eq_rect_r
        (λ n : index,
           EquivocatorLabel (IM n)) li
        eq_refl)) (s i)
Hex_new: previous_state_descriptor_prop 
  (IM i) (eqv_descriptors i) 
  (s i)
  (equivocator_label_descriptor
     (eq_rect_r
        (λ n : index,
           EquivocatorLabel (IM n)) li
        eq_refl))
Hchar: ∀ sx : state (IM i),
  sx =
  equivocator_state_descriptor_project 
    (s i)
    (equivocator_label_descriptor
       (eq_rect_r
          (λ n : index,
             EquivocatorLabel (IM n)) li
          eq_refl))
  → valid (l itemi') (sx, input itemi')
    ∧ transition (l itemi') (sx, input itemi') =
      (destination itemi', output itemi')
Hdestinationi: equivocator_state_descriptor_project
  (state_update (equivocator_IM IM) s
     i e i) (eqv_descriptors i) =
destination itemi'
Hv: equivocator_valid (IM i) li (s i, input itemi')
Ht': equivocator_transition 
  (IM i) li (s i, input itemi') =
(e, output itemi')
s =
state_update (equivocator_IM IM)
  (state_update (equivocator_IM IM) s i e) i (s i) by extensionality j; destruct (decide (i = j)); subst; state_update_simpl.
    +message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
eqv_descriptors: equivocator_descriptors IM
i: index
li: EquivocatorLabel (IM i)
s: composite_state (equivocator_IM IM)
e: equivocator_state (IM i)
Hproper: proper_descriptor (IM i) 
  (eqv_descriptors i)
  (state_update (equivocator_IM IM) s i e i)
itemi': transition_item
Hex: existing_equivocator_label 
  (IM i)
  (eq_rect_r
     (λ n : index, EquivocatorLabel (IM n)) li
     eq_refl)
Hli: l itemi' =
existing_equivocator_label_extract 
  (IM i)
  (eq_rect_r
     (λ n : index, EquivocatorLabel (IM n)) li
     eq_refl) Hex
Hproper': proper_descriptor (IM i)
  (equivocator_label_descriptor
     (eq_rect_r
        (λ n : index,
           EquivocatorLabel (IM n)) li
        eq_refl)) (s i)
Hex_new: previous_state_descriptor_prop 
  (IM i) (eqv_descriptors i) 
  (s i)
  (equivocator_label_descriptor
     (eq_rect_r
        (λ n : index,
           EquivocatorLabel (IM n)) li
        eq_refl))
Hchar: ∀ sx : state (IM i),
  sx =
  equivocator_state_descriptor_project 
    (s i)
    (equivocator_label_descriptor
       (eq_rect_r
          (λ n : index,
             EquivocatorLabel (IM n)) li
          eq_refl))
  → valid (l itemi') (sx, input itemi')
    ∧ transition (l itemi') (sx, input itemi') =
      (destination itemi', output itemi')
Hdestinationi: equivocator_state_descriptor_project
  (state_update (equivocator_IM IM) s
     i e i) (eqv_descriptors i) =
destination itemi'
Hv: equivocator_valid (IM i) li (s i, input itemi')
Ht': equivocator_transition 
  (IM i) li (s i, input itemi') =
(e, output itemi')
previous_state_descriptor_prop (IM i)
  (eqv_descriptors i) (s i)
  (equivocator_descriptors_update IM eqv_descriptors i
     (equivocator_label_descriptor
        (eq_rect_r
           (λ n : index, EquivocatorLabel (IM n)) li
           eq_refl)) i) by state_update_simpl.
    +message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
eqv_descriptors: equivocator_descriptors IM
i: index
li: EquivocatorLabel (IM i)
s: composite_state (equivocator_IM IM)
e: equivocator_state (IM i)
Hproper: proper_descriptor (IM i) 
  (eqv_descriptors i)
  (state_update (equivocator_IM IM) s i e i)
itemi': transition_item
Hex: existing_equivocator_label 
  (IM i)
  (eq_rect_r
     (λ n : index, EquivocatorLabel (IM n)) li
     eq_refl)
Hli: l itemi' =
existing_equivocator_label_extract 
  (IM i)
  (eq_rect_r
     (λ n : index, EquivocatorLabel (IM n)) li
     eq_refl) Hex
Hproper': proper_descriptor (IM i)
  (equivocator_label_descriptor
     (eq_rect_r
        (λ n : index,
           EquivocatorLabel (IM n)) li
        eq_refl)) (s i)
Hex_new: previous_state_descriptor_prop 
  (IM i) (eqv_descriptors i) 
  (s i)
  (equivocator_label_descriptor
     (eq_rect_r
        (λ n : index,
           EquivocatorLabel (IM n)) li
        eq_refl))
Hchar: ∀ sx : state (IM i),
  sx =
  equivocator_state_descriptor_project 
    (s i)
    (equivocator_label_descriptor
       (eq_rect_r
          (λ n : index,
             EquivocatorLabel (IM n)) li
          eq_refl))
  → valid (l itemi') (sx, input itemi')
    ∧ transition (l itemi') (sx, input itemi') =
      (destination itemi', output itemi')
Hdestinationi: equivocator_state_descriptor_project
  (state_update (equivocator_IM IM) s
     i e i) (eqv_descriptors i) =
destination itemi'
Hv: equivocator_valid (IM i) li (s i, input itemi')
Ht': equivocator_transition 
  (IM i) li (s i, input itemi') =
(e, output itemi')
sx: composite_state IM
Hsx: sx =
equivocators_state_project IM
  (equivocator_descriptors_update IM
     eqv_descriptors i
     (equivocator_label_descriptor
        (eq_rect_r
           (λ n : index, EquivocatorLabel (IM n))
           li eq_refl))) s
valid
  (existing_equivocator_label_extract (IM i)
     (eq_rect_r (λ n : index, EquivocatorLabel (IM n))
        li eq_refl) Hex) (sx i, input itemi') subst.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
eqv_descriptors: equivocator_descriptors IM
i: index
li: EquivocatorLabel (IM i)
s: composite_state (equivocator_IM IM)
e: equivocator_state (IM i)
Hproper: proper_descriptor (IM i) 
  (eqv_descriptors i)
  (state_update (equivocator_IM IM) s i e i)
itemi': transition_item
Hex: existing_equivocator_label 
  (IM i)
  (eq_rect_r
     (λ n : index, EquivocatorLabel (IM n)) li
     eq_refl)
Hli: l itemi' =
existing_equivocator_label_extract 
  (IM i)
  (eq_rect_r
     (λ n : index, EquivocatorLabel (IM n)) li
     eq_refl) Hex
Hproper': proper_descriptor (IM i)
  (equivocator_label_descriptor
     (eq_rect_r
        (λ n : index,
           EquivocatorLabel (IM n)) li
        eq_refl)) (s i)
Hex_new: previous_state_descriptor_prop 
  (IM i) (eqv_descriptors i) 
  (s i)
  (equivocator_label_descriptor
     (eq_rect_r
        (λ n : index,
           EquivocatorLabel (IM n)) li
        eq_refl))
Hchar: ∀ sx : state (IM i),
  sx =
  equivocator_state_descriptor_project 
    (s i)
    (equivocator_label_descriptor
       (eq_rect_r
          (λ n : index,
             EquivocatorLabel (IM n)) li
          eq_refl))
  → valid (l itemi') (sx, input itemi')
    ∧ transition (l itemi') (sx, input itemi') =
      (destination itemi', output itemi')
Hdestinationi: equivocator_state_descriptor_project
  (state_update (equivocator_IM IM) s
     i e i) (eqv_descriptors i) =
destination itemi'
Hv: equivocator_valid (IM i) li (s i, input itemi')
Ht': equivocator_transition 
  (IM i) li (s i, input itemi') =
(e, output itemi')
valid
  (existing_equivocator_label_extract (IM i)
     (eq_rect_r (λ n : index, EquivocatorLabel (IM n))
        li eq_refl) Hex)
  (equivocators_state_project IM
     (equivocator_descriptors_update IM
        eqv_descriptors i
        (equivocator_label_descriptor
           (eq_rect_r
              (λ n : index, EquivocatorLabel (IM n))
              li eq_refl))) s i, input itemi') specialize (Hchar _ eq_refl) as [Hvx Htx].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
eqv_descriptors: equivocator_descriptors IM
i: index
li: EquivocatorLabel (IM i)
s: composite_state (equivocator_IM IM)
e: equivocator_state (IM i)
Hproper: proper_descriptor (IM i) 
  (eqv_descriptors i)
  (state_update (equivocator_IM IM) s i e i)
itemi': transition_item
Hex: existing_equivocator_label 
  (IM i)
  (eq_rect_r
     (λ n : index, EquivocatorLabel (IM n)) li
     eq_refl)
Hli: l itemi' =
existing_equivocator_label_extract 
  (IM i)
  (eq_rect_r
     (λ n : index, EquivocatorLabel (IM n)) li
     eq_refl) Hex
Hproper': proper_descriptor (IM i)
  (equivocator_label_descriptor
     (eq_rect_r
        (λ n : index,
           EquivocatorLabel (IM n)) li
        eq_refl)) (s i)
Hex_new: previous_state_descriptor_prop 
  (IM i) (eqv_descriptors i) 
  (s i)
  (equivocator_label_descriptor
     (eq_rect_r
        (λ n : index,
           EquivocatorLabel (IM n)) li
        eq_refl))
Hdestinationi: equivocator_state_descriptor_project
  (state_update (equivocator_IM IM) s
     i e i) (eqv_descriptors i) =
destination itemi'
Hv: equivocator_valid (IM i) li (s i, input itemi')
Ht': equivocator_transition 
  (IM i) li (s i, input itemi') =
(e, output itemi')
Hvx: valid (l itemi')
  (equivocator_state_descriptor_project 
     (s i)
     (equivocator_label_descriptor
        (eq_rect_r
           (λ n : index, EquivocatorLabel (IM n))
           li eq_refl)), 
   input itemi')
Htx: transition (l itemi')
  (equivocator_state_descriptor_project 
     (s i)
     (equivocator_label_descriptor
        (eq_rect_r
           (λ n : index, EquivocatorLabel (IM n))
           li eq_refl)), 
   input itemi') =
(destination itemi', output itemi')
valid
  (existing_equivocator_label_extract (IM i)
     (eq_rect_r (λ n : index, EquivocatorLabel (IM n))
        li eq_refl) Hex)
  (equivocators_state_project IM
     (equivocator_descriptors_update IM
        eqv_descriptors i
        (equivocator_label_descriptor
           (eq_rect_r
              (λ n : index, EquivocatorLabel (IM n))
              li eq_refl))) s i, input itemi')
      unfold equivocators_state_project, EquivocatorsComposition.equivocators_state_project.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
eqv_descriptors: equivocator_descriptors IM
i: index
li: EquivocatorLabel (IM i)
s: composite_state (equivocator_IM IM)
e: equivocator_state (IM i)
Hproper: proper_descriptor (IM i) 
  (eqv_descriptors i)
  (state_update (equivocator_IM IM) s i e i)
itemi': transition_item
Hex: existing_equivocator_label 
  (IM i)
  (eq_rect_r
     (λ n : index, EquivocatorLabel (IM n)) li
     eq_refl)
Hli: l itemi' =
existing_equivocator_label_extract 
  (IM i)
  (eq_rect_r
     (λ n : index, EquivocatorLabel (IM n)) li
     eq_refl) Hex
Hproper': proper_descriptor (IM i)
  (equivocator_label_descriptor
     (eq_rect_r
        (λ n : index,
           EquivocatorLabel (IM n)) li
        eq_refl)) (s i)
Hex_new: previous_state_descriptor_prop 
  (IM i) (eqv_descriptors i) 
  (s i)
  (equivocator_label_descriptor
     (eq_rect_r
        (λ n : index,
           EquivocatorLabel (IM n)) li
        eq_refl))
Hdestinationi: equivocator_state_descriptor_project
  (state_update (equivocator_IM IM) s
     i e i) (eqv_descriptors i) =
destination itemi'
Hv: equivocator_valid (IM i) li (s i, input itemi')
Ht': equivocator_transition 
  (IM i) li (s i, input itemi') =
(e, output itemi')
Hvx: valid (l itemi')
  (equivocator_state_descriptor_project 
     (s i)
     (equivocator_label_descriptor
        (eq_rect_r
           (λ n : index, EquivocatorLabel (IM n))
           li eq_refl)), 
   input itemi')
Htx: transition (l itemi')
  (equivocator_state_descriptor_project 
     (s i)
     (equivocator_label_descriptor
        (eq_rect_r
           (λ n : index, EquivocatorLabel (IM n))
           li eq_refl)), 
   input itemi') =
(destination itemi', output itemi')
valid
  (existing_equivocator_label_extract (IM i)
     (eq_rect_r (λ n : index, EquivocatorLabel (IM n))
        li eq_refl) Hex)
  (equivocator_state_descriptor_project (s i)
     (equivocator_descriptors_update IM
        eqv_descriptors i
        (equivocator_label_descriptor
           (eq_rect_r
              (λ n : index, EquivocatorLabel (IM n))
              li eq_refl)) i), input itemi')
      rewrite Hli in Hvx.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
eqv_descriptors: equivocator_descriptors IM
i: index
li: EquivocatorLabel (IM i)
s: composite_state (equivocator_IM IM)
e: equivocator_state (IM i)
Hproper: proper_descriptor (IM i) 
  (eqv_descriptors i)
  (state_update (equivocator_IM IM) s i e i)
itemi': transition_item
Hex: existing_equivocator_label 
  (IM i)
  (eq_rect_r
     (λ n : index, EquivocatorLabel (IM n)) li
     eq_refl)
Hli: l itemi' =
existing_equivocator_label_extract 
  (IM i)
  (eq_rect_r
     (λ n : index, EquivocatorLabel (IM n)) li
     eq_refl) Hex
Hproper': proper_descriptor (IM i)
  (equivocator_label_descriptor
     (eq_rect_r
        (λ n : index,
           EquivocatorLabel (IM n)) li
        eq_refl)) (s i)
Hex_new: previous_state_descriptor_prop 
  (IM i) (eqv_descriptors i) 
  (s i)
  (equivocator_label_descriptor
     (eq_rect_r
        (λ n : index,
           EquivocatorLabel (IM n)) li
        eq_refl))
Hdestinationi: equivocator_state_descriptor_project
  (state_update (equivocator_IM IM) s
     i e i) (eqv_descriptors i) =
destination itemi'
Hv: equivocator_valid (IM i) li (s i, input itemi')
Ht': equivocator_transition 
  (IM i) li (s i, input itemi') =
(e, output itemi')
Hvx: valid
  (existing_equivocator_label_extract 
     (IM i)
     (eq_rect_r
        (λ n : index, EquivocatorLabel (IM n))
        li eq_refl) Hex)
  (equivocator_state_descriptor_project 
     (s i)
     (equivocator_label_descriptor
        (eq_rect_r
           (λ n : index, EquivocatorLabel (IM n))
           li eq_refl)), 
   input itemi')
Htx: transition (l itemi')
  (equivocator_state_descriptor_project 
     (s i)
     (equivocator_label_descriptor
        (eq_rect_r
           (λ n : index, EquivocatorLabel (IM n))
           li eq_refl)), 
   input itemi') =
(destination itemi', output itemi')
valid
  (existing_equivocator_label_extract (IM i)
     (eq_rect_r (λ n : index, EquivocatorLabel (IM n))
        li eq_refl) Hex)
  (equivocator_state_descriptor_project (s i)
     (equivocator_descriptors_update IM
        eqv_descriptors i
        (equivocator_label_descriptor
           (eq_rect_r
              (λ n : index, EquivocatorLabel (IM n))
              li eq_refl)) i), input itemi')
      by state_update_simpl.
    +message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
eqv_descriptors: equivocator_descriptors IM
i: index
li: EquivocatorLabel (IM i)
s: composite_state (equivocator_IM IM)
e: equivocator_state (IM i)
Hproper: proper_descriptor (IM i) 
  (eqv_descriptors i)
  (state_update (equivocator_IM IM) s i e i)
itemi': transition_item
Hex: existing_equivocator_label 
  (IM i)
  (eq_rect_r
     (λ n : index, EquivocatorLabel (IM n)) li
     eq_refl)
Hli: l itemi' =
existing_equivocator_label_extract 
  (IM i)
  (eq_rect_r
     (λ n : index, EquivocatorLabel (IM n)) li
     eq_refl) Hex
Hproper': proper_descriptor (IM i)
  (equivocator_label_descriptor
     (eq_rect_r
        (λ n : index,
           EquivocatorLabel (IM n)) li
        eq_refl)) (s i)
Hex_new: previous_state_descriptor_prop 
  (IM i) (eqv_descriptors i) 
  (s i)
  (equivocator_label_descriptor
     (eq_rect_r
        (λ n : index,
           EquivocatorLabel (IM n)) li
        eq_refl))
Hchar: ∀ sx : state (IM i),
  sx =
  equivocator_state_descriptor_project 
    (s i)
    (equivocator_label_descriptor
       (eq_rect_r
          (λ n : index,
             EquivocatorLabel (IM n)) li
          eq_refl))
  → valid (l itemi') (sx, input itemi')
    ∧ transition (l itemi') (sx, input itemi') =
      (destination itemi', output itemi')
Hdestinationi: equivocator_state_descriptor_project
  (state_update (equivocator_IM IM) s
     i e i) (eqv_descriptors i) =
destination itemi'
Hv: equivocator_valid (IM i) li (s i, input itemi')
Ht': equivocator_transition 
  (IM i) li (s i, input itemi') =
(e, output itemi')
sx: composite_state IM
Hsx: sx =
equivocators_state_project IM
  (equivocator_descriptors_update IM
     eqv_descriptors i
     (equivocator_label_descriptor
        (eq_rect_r
           (λ n : index, EquivocatorLabel (IM n))
           li eq_refl))) s
(let (si', om') :=
   transition
     (existing_equivocator_label_extract (IM i)
        (eq_rect_r
           (λ n : index, EquivocatorLabel (IM n)) li
           eq_refl) Hex) (sx i, input itemi') in
 (state_update IM sx i si', om')) =
(equivocators_state_project IM eqv_descriptors
   (state_update (equivocator_IM IM) s i e),
 output itemi') subst.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
eqv_descriptors: equivocator_descriptors IM
i: index
li: EquivocatorLabel (IM i)
s: composite_state (equivocator_IM IM)
e: equivocator_state (IM i)
Hproper: proper_descriptor (IM i) 
  (eqv_descriptors i)
  (state_update (equivocator_IM IM) s i e i)
itemi': transition_item
Hex: existing_equivocator_label 
  (IM i)
  (eq_rect_r
     (λ n : index, EquivocatorLabel (IM n)) li
     eq_refl)
Hli: l itemi' =
existing_equivocator_label_extract 
  (IM i)
  (eq_rect_r
     (λ n : index, EquivocatorLabel (IM n)) li
     eq_refl) Hex
Hproper': proper_descriptor (IM i)
  (equivocator_label_descriptor
     (eq_rect_r
        (λ n : index,
           EquivocatorLabel (IM n)) li
        eq_refl)) (s i)
Hex_new: previous_state_descriptor_prop 
  (IM i) (eqv_descriptors i) 
  (s i)
  (equivocator_label_descriptor
     (eq_rect_r
        (λ n : index,
           EquivocatorLabel (IM n)) li
        eq_refl))
Hchar: ∀ sx : state (IM i),
  sx =
  equivocator_state_descriptor_project 
    (s i)
    (equivocator_label_descriptor
       (eq_rect_r
          (λ n : index,
             EquivocatorLabel (IM n)) li
          eq_refl))
  → valid (l itemi') (sx, input itemi')
    ∧ transition (l itemi') (sx, input itemi') =
      (destination itemi', output itemi')
Hdestinationi: equivocator_state_descriptor_project
  (state_update (equivocator_IM IM) s
     i e i) (eqv_descriptors i) =
destination itemi'
Hv: equivocator_valid (IM i) li (s i, input itemi')
Ht': equivocator_transition 
  (IM i) li (s i, input itemi') =
(e, output itemi')
(let (si', om') :=
   transition
     (existing_equivocator_label_extract (IM i)
        (eq_rect_r
           (λ n : index, EquivocatorLabel (IM n)) li
           eq_refl) Hex)
     (equivocators_state_project IM
        (equivocator_descriptors_update IM
           eqv_descriptors i
           (equivocator_label_descriptor
              (eq_rect_r
                 (λ n : index, EquivocatorLabel (IM n))
                 li eq_refl))) s i, input itemi') in
 (state_update IM
    (equivocators_state_project IM
       (equivocator_descriptors_update IM
          eqv_descriptors i
          (equivocator_label_descriptor
             (eq_rect_r
                (λ n : index, EquivocatorLabel (IM n))
                li eq_refl))) s) i si', om')) =
(equivocators_state_project IM eqv_descriptors
   (state_update (equivocator_IM IM) s i e),
 output itemi') specialize (Hchar _ eq_refl) as [Hvx Htx].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
eqv_descriptors: equivocator_descriptors IM
i: index
li: EquivocatorLabel (IM i)
s: composite_state (equivocator_IM IM)
e: equivocator_state (IM i)
Hproper: proper_descriptor (IM i) 
  (eqv_descriptors i)
  (state_update (equivocator_IM IM) s i e i)
itemi': transition_item
Hex: existing_equivocator_label 
  (IM i)
  (eq_rect_r
     (λ n : index, EquivocatorLabel (IM n)) li
     eq_refl)
Hli: l itemi' =
existing_equivocator_label_extract 
  (IM i)
  (eq_rect_r
     (λ n : index, EquivocatorLabel (IM n)) li
     eq_refl) Hex
Hproper': proper_descriptor (IM i)
  (equivocator_label_descriptor
     (eq_rect_r
        (λ n : index,
           EquivocatorLabel (IM n)) li
        eq_refl)) (s i)
Hex_new: previous_state_descriptor_prop 
  (IM i) (eqv_descriptors i) 
  (s i)
  (equivocator_label_descriptor
     (eq_rect_r
        (λ n : index,
           EquivocatorLabel (IM n)) li
        eq_refl))
Hdestinationi: equivocator_state_descriptor_project
  (state_update (equivocator_IM IM) s
     i e i) (eqv_descriptors i) =
destination itemi'
Hv: equivocator_valid (IM i) li (s i, input itemi')
Ht': equivocator_transition 
  (IM i) li (s i, input itemi') =
(e, output itemi')
Hvx: valid (l itemi')
  (equivocator_state_descriptor_project 
     (s i)
     (equivocator_label_descriptor
        (eq_rect_r
           (λ n : index, EquivocatorLabel (IM n))
           li eq_refl)), 
   input itemi')
Htx: transition (l itemi')
  (equivocator_state_descriptor_project 
     (s i)
     (equivocator_label_descriptor
        (eq_rect_r
           (λ n : index, EquivocatorLabel (IM n))
           li eq_refl)), 
   input itemi') =
(destination itemi', output itemi')
(let (si', om') :=
   transition
     (existing_equivocator_label_extract (IM i)
        (eq_rect_r
           (λ n : index, EquivocatorLabel (IM n)) li
           eq_refl) Hex)
     (equivocators_state_project IM
        (equivocator_descriptors_update IM
           eqv_descriptors i
           (equivocator_label_descriptor
              (eq_rect_r
                 (λ n : index, EquivocatorLabel (IM n))
                 li eq_refl))) s i, input itemi') in
 (state_update IM
    (equivocators_state_project IM
       (equivocator_descriptors_update IM
          eqv_descriptors i
          (equivocator_label_descriptor
             (eq_rect_r
                (λ n : index, EquivocatorLabel (IM n))
                li eq_refl))) s) i si', om')) =
(equivocators_state_project IM eqv_descriptors
   (state_update (equivocator_IM IM) s i e),
 output itemi')
      unfold equivocators_state_project, EquivocatorsComposition.equivocators_state_project.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
eqv_descriptors: equivocator_descriptors IM
i: index
li: EquivocatorLabel (IM i)
s: composite_state (equivocator_IM IM)
e: equivocator_state (IM i)
Hproper: proper_descriptor (IM i) 
  (eqv_descriptors i)
  (state_update (equivocator_IM IM) s i e i)
itemi': transition_item
Hex: existing_equivocator_label 
  (IM i)
  (eq_rect_r
     (λ n : index, EquivocatorLabel (IM n)) li
     eq_refl)
Hli: l itemi' =
existing_equivocator_label_extract 
  (IM i)
  (eq_rect_r
     (λ n : index, EquivocatorLabel (IM n)) li
     eq_refl) Hex
Hproper': proper_descriptor (IM i)
  (equivocator_label_descriptor
     (eq_rect_r
        (λ n : index,
           EquivocatorLabel (IM n)) li
        eq_refl)) (s i)
Hex_new: previous_state_descriptor_prop 
  (IM i) (eqv_descriptors i) 
  (s i)
  (equivocator_label_descriptor
     (eq_rect_r
        (λ n : index,
           EquivocatorLabel (IM n)) li
        eq_refl))
Hdestinationi: equivocator_state_descriptor_project
  (state_update (equivocator_IM IM) s
     i e i) (eqv_descriptors i) =
destination itemi'
Hv: equivocator_valid (IM i) li (s i, input itemi')
Ht': equivocator_transition 
  (IM i) li (s i, input itemi') =
(e, output itemi')
Hvx: valid (l itemi')
  (equivocator_state_descriptor_project 
     (s i)
     (equivocator_label_descriptor
        (eq_rect_r
           (λ n : index, EquivocatorLabel (IM n))
           li eq_refl)), 
   input itemi')
Htx: transition (l itemi')
  (equivocator_state_descriptor_project 
     (s i)
     (equivocator_label_descriptor
        (eq_rect_r
           (λ n : index, EquivocatorLabel (IM n))
           li eq_refl)), 
   input itemi') =
(destination itemi', output itemi')
(let (si', om') :=
   transition
     (existing_equivocator_label_extract (IM i)
        (eq_rect_r
           (λ n : index, EquivocatorLabel (IM n)) li
           eq_refl) Hex)
     (equivocator_state_descriptor_project (s i)
        (equivocator_descriptors_update IM
           eqv_descriptors i
           (equivocator_label_descriptor
              (eq_rect_r
                 (λ n : index, EquivocatorLabel (IM n))
                 li eq_refl)) i), input itemi') in
 (state_update IM
    (λ eqv : index,
       equivocator_state_descriptor_project (s eqv)
         (equivocator_descriptors_update IM
            eqv_descriptors i
            (equivocator_label_descriptor
               (eq_rect_r
                  (λ n : index,
                     EquivocatorLabel (IM n)) li
                  eq_refl)) eqv)) i si', om')) =
(λ eqv : index,
   equivocator_state_descriptor_project
     (state_update (equivocator_IM IM) s i e eqv)
     (eqv_descriptors eqv), output itemi')
      state_update_simpl.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
eqv_descriptors: equivocator_descriptors IM
i: index
li: EquivocatorLabel (IM i)
s: composite_state (equivocator_IM IM)
e: equivocator_state (IM i)
Hproper: proper_descriptor (IM i) 
  (eqv_descriptors i) e
itemi': transition_item
Hex: existing_equivocator_label 
  (IM i)
  (eq_rect_r
     (λ n : index, EquivocatorLabel (IM n)) li
     eq_refl)
Hli: l itemi' =
existing_equivocator_label_extract 
  (IM i)
  (eq_rect_r
     (λ n : index, EquivocatorLabel (IM n)) li
     eq_refl) Hex
Hproper': proper_descriptor (IM i)
  (equivocator_label_descriptor
     (eq_rect_r
        (λ n : index,
           EquivocatorLabel (IM n)) li
        eq_refl)) (s i)
Hex_new: previous_state_descriptor_prop 
  (IM i) (eqv_descriptors i) 
  (s i)
  (equivocator_label_descriptor
     (eq_rect_r
        (λ n : index,
           EquivocatorLabel (IM n)) li
        eq_refl))
Hdestinationi: equivocator_state_descriptor_project e
  (eqv_descriptors i) =
destination itemi'
Hv: equivocator_valid (IM i) li (s i, input itemi')
Ht': equivocator_transition 
  (IM i) li (s i, input itemi') =
(e, output itemi')
Hvx: valid (l itemi')
  (equivocator_state_descriptor_project 
     (s i)
     (equivocator_label_descriptor
        (eq_rect_r
           (λ n : index, EquivocatorLabel (IM n))
           li eq_refl)), 
   input itemi')
Htx: transition (l itemi')
  (equivocator_state_descriptor_project 
     (s i)
     (equivocator_label_descriptor
        (eq_rect_r
           (λ n : index, EquivocatorLabel (IM n))
           li eq_refl)), 
   input itemi') =
(destination itemi', output itemi')
(let (si', om') :=
   transition
     (existing_equivocator_label_extract (IM i)
        (eq_rect_r
           (λ n : index, EquivocatorLabel (IM n)) li
           eq_refl) Hex)
     (equivocator_state_descriptor_project (s i)
        (equivocator_label_descriptor
           (eq_rect_r
              (λ n : index, EquivocatorLabel (IM n))
              li eq_refl)), input itemi') in
 (state_update IM
    (λ eqv : index,
       equivocator_state_descriptor_project (s eqv)
         (equivocator_descriptors_update IM
            eqv_descriptors i
            (equivocator_label_descriptor
               (eq_rect_r
                  (λ n : index,
                     EquivocatorLabel (IM n)) li
                  eq_refl)) eqv)) i si', om')) =
(λ eqv : index,
   equivocator_state_descriptor_project
     (state_update (equivocator_IM IM) s i e eqv)
     (eqv_descriptors eqv), output itemi')
      simpl in *.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
eqv_descriptors: equivocator_descriptors IM
i: index
li: EquivocatorLabel (IM i)
s: composite_state (equivocator_IM IM)
e: equivocator_state (IM i)
Hproper: proper_descriptor (IM i) 
  (eqv_descriptors i) e
itemi': transition_item
Hex: existing_equivocator_label 
  (IM i)
  (eq_rect_r
     (λ n : index, EquivocatorLabel (IM n)) li
     eq_refl)
Hli: l itemi' =
existing_equivocator_label_extract 
  (IM i)
  (eq_rect_r
     (λ n : index, EquivocatorLabel (IM n)) li
     eq_refl) Hex
Hproper': proper_descriptor (IM i)
  (equivocator_label_descriptor
     (eq_rect_r
        (λ n : index,
           EquivocatorLabel (IM n)) li
        eq_refl)) (s i)
Hex_new: previous_state_descriptor_prop 
  (IM i) (eqv_descriptors i) 
  (s i)
  (equivocator_label_descriptor
     (eq_rect_r
        (λ n : index,
           EquivocatorLabel (IM n)) li
        eq_refl))
Hdestinationi: equivocator_state_descriptor_project e
  (eqv_descriptors i) =
destination itemi'
Hv: equivocator_valid (IM i) li (s i, input itemi')
Ht': equivocator_transition 
  (IM i) li (s i, input itemi') =
(e, output itemi')
Hvx: valid (l itemi')
  (equivocator_state_descriptor_project 
     (s i)
     (equivocator_label_descriptor
        (eq_rect_r
           (λ n : index, EquivocatorLabel (IM n))
           li eq_refl)), 
   input itemi')
Htx: transition (l itemi')
  (equivocator_state_descriptor_project 
     (s i)
     (equivocator_label_descriptor
        (eq_rect_r
           (λ n : index, EquivocatorLabel (IM n))
           li eq_refl)), 
   input itemi') =
(destination itemi', output itemi')
(let (si', om') :=
   transition
     (existing_equivocator_label_extract (IM i)
        (eq_rect_r
           (λ n : index, EquivocatorLabel (IM n)) li
           eq_refl) Hex)
     (equivocator_state_descriptor_project (s i)
        (equivocator_label_descriptor
           (eq_rect_r
              (λ n : index, EquivocatorLabel (IM n))
              li eq_refl)), input itemi') in
 (state_update IM
    (λ eqv : index,
       equivocator_state_descriptor_project (s eqv)
         (equivocator_descriptors_update IM
            eqv_descriptors i
            (equivocator_label_descriptor
               (eq_rect_r
                  (λ n : index,
                     EquivocatorLabel (IM n)) li
                  eq_refl)) eqv)) i si', om')) =
(λ eqv : index,
   equivocator_state_descriptor_project
     (state_update (equivocator_IM IM) s i e eqv)
     (eqv_descriptors eqv), output itemi') rewrite Hli in Htx.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
eqv_descriptors: equivocator_descriptors IM
i: index
li: EquivocatorLabel (IM i)
s: composite_state (equivocator_IM IM)
e: equivocator_state (IM i)
Hproper: proper_descriptor (IM i) 
  (eqv_descriptors i) e
itemi': transition_item
Hex: existing_equivocator_label 
  (IM i)
  (eq_rect_r
     (λ n : index, EquivocatorLabel (IM n)) li
     eq_refl)
Hli: l itemi' =
existing_equivocator_label_extract 
  (IM i)
  (eq_rect_r
     (λ n : index, EquivocatorLabel (IM n)) li
     eq_refl) Hex
Hproper': proper_descriptor (IM i)
  (equivocator_label_descriptor
     (eq_rect_r
        (λ n : index,
           EquivocatorLabel (IM n)) li
        eq_refl)) (s i)
Hex_new: previous_state_descriptor_prop 
  (IM i) (eqv_descriptors i) 
  (s i)
  (equivocator_label_descriptor
     (eq_rect_r
        (λ n : index,
           EquivocatorLabel (IM n)) li
        eq_refl))
Hdestinationi: equivocator_state_descriptor_project e
  (eqv_descriptors i) =
destination itemi'
Hv: equivocator_valid (IM i) li (s i, input itemi')
Ht': equivocator_transition 
  (IM i) li (s i, input itemi') =
(e, output itemi')
Hvx: valid (l itemi')
  (equivocator_state_descriptor_project 
     (s i)
     (equivocator_label_descriptor
        (eq_rect_r
           (λ n : index, EquivocatorLabel (IM n))
           li eq_refl)), 
   input itemi')
Htx: transition
  (existing_equivocator_label_extract 
     (IM i)
     (eq_rect_r
        (λ n : index, EquivocatorLabel (IM n))
        li eq_refl) Hex)
  (equivocator_state_descriptor_project 
     (s i)
     (equivocator_label_descriptor
        (eq_rect_r
           (λ n : index, EquivocatorLabel (IM n))
           li eq_refl)), 
   input itemi') =
(destination itemi', output itemi')
(let (si', om') :=
   transition
     (existing_equivocator_label_extract (IM i)
        (eq_rect_r
           (λ n : index, EquivocatorLabel (IM n)) li
           eq_refl) Hex)
     (equivocator_state_descriptor_project (s i)
        (equivocator_label_descriptor
           (eq_rect_r
              (λ n : index, EquivocatorLabel (IM n))
              li eq_refl)), input itemi') in
 (state_update IM
    (λ eqv : index,
       equivocator_state_descriptor_project (s eqv)
         (equivocator_descriptors_update IM
            eqv_descriptors i
            (equivocator_label_descriptor
               (eq_rect_r
                  (λ n : index,
                     EquivocatorLabel (IM n)) li
                  eq_refl)) eqv)) i si', om')) =
(λ eqv : index,
   equivocator_state_descriptor_project
     (state_update (equivocator_IM IM) s i e eqv)
     (eqv_descriptors eqv), output itemi') rewrite Htx.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
eqv_descriptors: equivocator_descriptors IM
i: index
li: EquivocatorLabel (IM i)
s: composite_state (equivocator_IM IM)
e: equivocator_state (IM i)
Hproper: proper_descriptor (IM i) 
  (eqv_descriptors i) e
itemi': transition_item
Hex: existing_equivocator_label 
  (IM i)
  (eq_rect_r
     (λ n : index, EquivocatorLabel (IM n)) li
     eq_refl)
Hli: l itemi' =
existing_equivocator_label_extract 
  (IM i)
  (eq_rect_r
     (λ n : index, EquivocatorLabel (IM n)) li
     eq_refl) Hex
Hproper': proper_descriptor (IM i)
  (equivocator_label_descriptor
     (eq_rect_r
        (λ n : index,
           EquivocatorLabel (IM n)) li
        eq_refl)) (s i)
Hex_new: previous_state_descriptor_prop 
  (IM i) (eqv_descriptors i) 
  (s i)
  (equivocator_label_descriptor
     (eq_rect_r
        (λ n : index,
           EquivocatorLabel (IM n)) li
        eq_refl))
Hdestinationi: equivocator_state_descriptor_project e
  (eqv_descriptors i) =
destination itemi'
Hv: equivocator_valid (IM i) li (s i, input itemi')
Ht': equivocator_transition 
  (IM i) li (s i, input itemi') =
(e, output itemi')
Hvx: valid (l itemi')
  (equivocator_state_descriptor_project 
     (s i)
     (equivocator_label_descriptor
        (eq_rect_r
           (λ n : index, EquivocatorLabel (IM n))
           li eq_refl)), 
   input itemi')
Htx: transition
  (existing_equivocator_label_extract 
     (IM i)
     (eq_rect_r
        (λ n : index, EquivocatorLabel (IM n))
        li eq_refl) Hex)
  (equivocator_state_descriptor_project 
     (s i)
     (equivocator_label_descriptor
        (eq_rect_r
           (λ n : index, EquivocatorLabel (IM n))
           li eq_refl)), 
   input itemi') =
(destination itemi', output itemi')
(state_update IM
   (λ eqv : index,
      equivocator_state_descriptor_project (s eqv)
        (equivocator_descriptors_update IM
           eqv_descriptors i
           (equivocator_label_descriptor
              (eq_rect_r
                 (λ n : index, EquivocatorLabel (IM n))
                 li eq_refl)) eqv)) i
   (destination itemi'), output itemi') =
(λ eqv : index,
   equivocator_state_descriptor_project
     (state_update (equivocator_IM IM) s i e eqv)
     (eqv_descriptors eqv), output itemi') f_equal.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
eqv_descriptors: equivocator_descriptors IM
i: index
li: EquivocatorLabel (IM i)
s: composite_state (equivocator_IM IM)
e: equivocator_state (IM i)
Hproper: proper_descriptor (IM i) 
  (eqv_descriptors i) e
itemi': transition_item
Hex: existing_equivocator_label 
  (IM i)
  (eq_rect_r
     (λ n : index, EquivocatorLabel (IM n)) li
     eq_refl)
Hli: l itemi' =
existing_equivocator_label_extract 
  (IM i)
  (eq_rect_r
     (λ n : index, EquivocatorLabel (IM n)) li
     eq_refl) Hex
Hproper': proper_descriptor (IM i)
  (equivocator_label_descriptor
     (eq_rect_r
        (λ n : index,
           EquivocatorLabel (IM n)) li
        eq_refl)) (s i)
Hex_new: previous_state_descriptor_prop 
  (IM i) (eqv_descriptors i) 
  (s i)
  (equivocator_label_descriptor
     (eq_rect_r
        (λ n : index,
           EquivocatorLabel (IM n)) li
        eq_refl))
Hdestinationi: equivocator_state_descriptor_project e
  (eqv_descriptors i) =
destination itemi'
Hv: equivocator_valid (IM i) li (s i, input itemi')
Ht': equivocator_transition 
  (IM i) li (s i, input itemi') =
(e, output itemi')
Hvx: valid (l itemi')
  (equivocator_state_descriptor_project 
     (s i)
     (equivocator_label_descriptor
        (eq_rect_r
           (λ n : index, EquivocatorLabel (IM n))
           li eq_refl)), 
   input itemi')
Htx: transition
  (existing_equivocator_label_extract 
     (IM i)
     (eq_rect_r
        (λ n : index, EquivocatorLabel (IM n))
        li eq_refl) Hex)
  (equivocator_state_descriptor_project 
     (s i)
     (equivocator_label_descriptor
        (eq_rect_r
           (λ n : index, EquivocatorLabel (IM n))
           li eq_refl)), 
   input itemi') =
(destination itemi', output itemi')
state_update IM
  (λ eqv : index,
     equivocator_state_descriptor_project (s eqv)
       (equivocator_descriptors_update IM
          eqv_descriptors i
          (equivocator_label_descriptor
             (eq_rect_r
                (λ n : index, EquivocatorLabel (IM n))
                li eq_refl)) eqv)) i
  (destination itemi') =
(λ eqv : index,
   equivocator_state_descriptor_project
     (state_update (equivocator_IM IM) s i e eqv)
     (eqv_descriptors eqv))
      by extensionality eqv; destruct (decide (i = eqv)); subst; state_update_simpl.
  -message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
eqv_descriptors: equivocator_descriptors IM
i: index
li: EquivocatorLabel (IM i)
input, output: option message
s: composite_state (equivocator_IM IM)
e: equivocator_state (IM i)
Hproper: proper_descriptor (IM i) 
  (eqv_descriptors i)
  (state_update (equivocator_IM IM) s i e i)
eqv_descriptorsi': MachineDescriptor (IM i)
Hitemx: True
Hproper': proper_descriptor (IM i) eqv_descriptorsi'
  (s i)
Hex_new: previous_state_descriptor_prop 
  (IM i) (eqv_descriptors i) 
  (s i) eqv_descriptorsi'
Hchar: equivocator_state_descriptor_project e
  (eqv_descriptors i) =
equivocator_state_descriptor_project 
  (s i) eqv_descriptorsi'
Hv: equivocator_valid (IM i) li (s i, input)
Ht': equivocator_transition (IM i) li (s i, input) =
(e, output)
proper_descriptor (IM i)
  (equivocator_descriptors_update IM eqv_descriptors i
     eqv_descriptorsi' i) (s i)
∧ equivocator_descriptors_update IM eqv_descriptors i
    eqv_descriptorsi' =
  equivocator_descriptors_update IM eqv_descriptors i
    (equivocator_descriptors_update IM eqv_descriptors
       i eqv_descriptorsi' i)
  ∧ s =
    state_update (equivocator_IM IM)
      (state_update (equivocator_IM IM) s i e) i (s i)
    ∧ previous_state_descriptor_prop (IM i)
        (eqv_descriptors i) (s i)
        (equivocator_descriptors_update IM
           eqv_descriptors i eqv_descriptorsi' i)
      ∧ equivocators_state_project IM eqv_descriptors
          (state_update (equivocator_IM IM) s i e) =
        equivocators_state_project IM
          (equivocator_descriptors_update IM
             eqv_descriptors i eqv_descriptorsi') s repeat split.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
eqv_descriptors: equivocator_descriptors IM
i: index
li: EquivocatorLabel (IM i)
input, output: option message
s: composite_state (equivocator_IM IM)
e: equivocator_state (IM i)
Hproper: proper_descriptor (IM i) 
  (eqv_descriptors i)
  (state_update (equivocator_IM IM) s i e i)
eqv_descriptorsi': MachineDescriptor (IM i)
Hitemx: True
Hproper': proper_descriptor (IM i) eqv_descriptorsi'
  (s i)
Hex_new: previous_state_descriptor_prop 
  (IM i) (eqv_descriptors i) 
  (s i) eqv_descriptorsi'
Hchar: equivocator_state_descriptor_project e
  (eqv_descriptors i) =
equivocator_state_descriptor_project 
  (s i) eqv_descriptorsi'
Hv: equivocator_valid (IM i) li (s i, input)
Ht': equivocator_transition (IM i) li (s i, input) =
(e, output)
proper_descriptor (IM i)
  (equivocator_descriptors_update IM eqv_descriptors i
     eqv_descriptorsi' i) (s i)
message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
eqv_descriptors: equivocator_descriptors IM
i: index
li: EquivocatorLabel (IM i)
input, output: option message
s: composite_state (equivocator_IM IM)
e: equivocator_state (IM i)
Hproper: proper_descriptor (IM i) 
  (eqv_descriptors i)
  (state_update (equivocator_IM IM) s i e i)
eqv_descriptorsi': MachineDescriptor (IM i)
Hitemx: True
Hproper': proper_descriptor (IM i) eqv_descriptorsi'
  (s i)
Hex_new: previous_state_descriptor_prop 
  (IM i) (eqv_descriptors i) 
  (s i) eqv_descriptorsi'
Hchar: equivocator_state_descriptor_project e
  (eqv_descriptors i) =
equivocator_state_descriptor_project 
  (s i) eqv_descriptorsi'
Hv: equivocator_valid (IM i) li (s i, input)
Ht': equivocator_transition (IM i) li (s i, input) =
(e, output)
equivocator_descriptors_update IM eqv_descriptors i
  eqv_descriptorsi' =
equivocator_descriptors_update IM eqv_descriptors i
  (equivocator_descriptors_update IM eqv_descriptors i
     eqv_descriptorsi' i)
message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
eqv_descriptors: equivocator_descriptors IM
i: index
li: EquivocatorLabel (IM i)
input, output: option message
s: composite_state (equivocator_IM IM)
e: equivocator_state (IM i)
Hproper: proper_descriptor (IM i) 
  (eqv_descriptors i)
  (state_update (equivocator_IM IM) s i e i)
eqv_descriptorsi': MachineDescriptor (IM i)
Hitemx: True
Hproper': proper_descriptor (IM i) eqv_descriptorsi'
  (s i)
Hex_new: previous_state_descriptor_prop 
  (IM i) (eqv_descriptors i) 
  (s i) eqv_descriptorsi'
Hchar: equivocator_state_descriptor_project e
  (eqv_descriptors i) =
equivocator_state_descriptor_project 
  (s i) eqv_descriptorsi'
Hv: equivocator_valid (IM i) li (s i, input)
Ht': equivocator_transition (IM i) li (s i, input) =
(e, output)
s =
state_update (equivocator_IM IM)
  (state_update (equivocator_IM IM) s i e) i 
  (s i)
message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
eqv_descriptors: equivocator_descriptors IM
i: index
li: EquivocatorLabel (IM i)
input, output: option message
s: composite_state (equivocator_IM IM)
e: equivocator_state (IM i)
Hproper: proper_descriptor (IM i) 
  (eqv_descriptors i)
  (state_update (equivocator_IM IM) s i e i)
eqv_descriptorsi': MachineDescriptor (IM i)
Hitemx: True
Hproper': proper_descriptor (IM i) eqv_descriptorsi'
  (s i)
Hex_new: previous_state_descriptor_prop 
  (IM i) (eqv_descriptors i) 
  (s i) eqv_descriptorsi'
Hchar: equivocator_state_descriptor_project e
  (eqv_descriptors i) =
equivocator_state_descriptor_project 
  (s i) eqv_descriptorsi'
Hv: equivocator_valid (IM i) li (s i, input)
Ht': equivocator_transition (IM i) li (s i, input) =
(e, output)
previous_state_descriptor_prop 
  (IM i) (eqv_descriptors i) 
  (s i)
  (equivocator_descriptors_update IM eqv_descriptors i
     eqv_descriptorsi' i)
message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
eqv_descriptors: equivocator_descriptors IM
i: index
li: EquivocatorLabel (IM i)
input, output: option message
s: composite_state (equivocator_IM IM)
e: equivocator_state (IM i)
Hproper: proper_descriptor (IM i) 
  (eqv_descriptors i)
  (state_update (equivocator_IM IM) s i e i)
eqv_descriptorsi': MachineDescriptor (IM i)
Hitemx: True
Hproper': proper_descriptor (IM i) eqv_descriptorsi'
  (s i)
Hex_new: previous_state_descriptor_prop 
  (IM i) (eqv_descriptors i) 
  (s i) eqv_descriptorsi'
Hchar: equivocator_state_descriptor_project e
  (eqv_descriptors i) =
equivocator_state_descriptor_project 
  (s i) eqv_descriptorsi'
Hv: equivocator_valid (IM i) li (s i, input)
Ht': equivocator_transition (IM i) li (s i, input) =
(e, output)
equivocators_state_project IM eqv_descriptors
  (state_update (equivocator_IM IM) s i e) =
equivocators_state_project IM
  (equivocator_descriptors_update IM eqv_descriptors i
     eqv_descriptorsi') s
    +message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
eqv_descriptors: equivocator_descriptors IM
i: index
li: EquivocatorLabel (IM i)
input, output: option message
s: composite_state (equivocator_IM IM)
e: equivocator_state (IM i)
Hproper: proper_descriptor (IM i) 
  (eqv_descriptors i)
  (state_update (equivocator_IM IM) s i e i)
eqv_descriptorsi': MachineDescriptor (IM i)
Hitemx: True
Hproper': proper_descriptor (IM i) eqv_descriptorsi'
  (s i)
Hex_new: previous_state_descriptor_prop 
  (IM i) (eqv_descriptors i) 
  (s i) eqv_descriptorsi'
Hchar: equivocator_state_descriptor_project e
  (eqv_descriptors i) =
equivocator_state_descriptor_project 
  (s i) eqv_descriptorsi'
Hv: equivocator_valid (IM i) li (s i, input)
Ht': equivocator_transition (IM i) li (s i, input) =
(e, output)
proper_descriptor (IM i)
  (equivocator_descriptors_update IM eqv_descriptors i
     eqv_descriptorsi' i) (s i) by state_update_simpl.
    +message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
eqv_descriptors: equivocator_descriptors IM
i: index
li: EquivocatorLabel (IM i)
input, output: option message
s: composite_state (equivocator_IM IM)
e: equivocator_state (IM i)
Hproper: proper_descriptor (IM i) 
  (eqv_descriptors i)
  (state_update (equivocator_IM IM) s i e i)
eqv_descriptorsi': MachineDescriptor (IM i)
Hitemx: True
Hproper': proper_descriptor (IM i) eqv_descriptorsi'
  (s i)
Hex_new: previous_state_descriptor_prop 
  (IM i) (eqv_descriptors i) 
  (s i) eqv_descriptorsi'
Hchar: equivocator_state_descriptor_project e
  (eqv_descriptors i) =
equivocator_state_descriptor_project 
  (s i) eqv_descriptorsi'
Hv: equivocator_valid (IM i) li (s i, input)
Ht': equivocator_transition (IM i) li (s i, input) =
(e, output)
equivocator_descriptors_update IM eqv_descriptors i
  eqv_descriptorsi' =
equivocator_descriptors_update IM eqv_descriptors i
  (equivocator_descriptors_update IM eqv_descriptors i
     eqv_descriptorsi' i) by state_update_simpl.
    +message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
eqv_descriptors: equivocator_descriptors IM
i: index
li: EquivocatorLabel (IM i)
input, output: option message
s: composite_state (equivocator_IM IM)
e: equivocator_state (IM i)
Hproper: proper_descriptor (IM i) 
  (eqv_descriptors i)
  (state_update (equivocator_IM IM) s i e i)
eqv_descriptorsi': MachineDescriptor (IM i)
Hitemx: True
Hproper': proper_descriptor (IM i) eqv_descriptorsi'
  (s i)
Hex_new: previous_state_descriptor_prop 
  (IM i) (eqv_descriptors i) 
  (s i) eqv_descriptorsi'
Hchar: equivocator_state_descriptor_project e
  (eqv_descriptors i) =
equivocator_state_descriptor_project 
  (s i) eqv_descriptorsi'
Hv: equivocator_valid (IM i) li (s i, input)
Ht': equivocator_transition (IM i) li (s i, input) =
(e, output)
s =
state_update (equivocator_IM IM)
  (state_update (equivocator_IM IM) s i e) i (s i) by extensionality j; destruct (decide (i = j)); subst; state_update_simpl.
    +message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
eqv_descriptors: equivocator_descriptors IM
i: index
li: EquivocatorLabel (IM i)
input, output: option message
s: composite_state (equivocator_IM IM)
e: equivocator_state (IM i)
Hproper: proper_descriptor (IM i) 
  (eqv_descriptors i)
  (state_update (equivocator_IM IM) s i e i)
eqv_descriptorsi': MachineDescriptor (IM i)
Hitemx: True
Hproper': proper_descriptor (IM i) eqv_descriptorsi'
  (s i)
Hex_new: previous_state_descriptor_prop 
  (IM i) (eqv_descriptors i) 
  (s i) eqv_descriptorsi'
Hchar: equivocator_state_descriptor_project e
  (eqv_descriptors i) =
equivocator_state_descriptor_project 
  (s i) eqv_descriptorsi'
Hv: equivocator_valid (IM i) li (s i, input)
Ht': equivocator_transition (IM i) li (s i, input) =
(e, output)
previous_state_descriptor_prop (IM i)
  (eqv_descriptors i) (s i)
  (equivocator_descriptors_update IM eqv_descriptors i
     eqv_descriptorsi' i) by state_update_simpl.
    +message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
eqv_descriptors: equivocator_descriptors IM
i: index
li: EquivocatorLabel (IM i)
input, output: option message
s: composite_state (equivocator_IM IM)
e: equivocator_state (IM i)
Hproper: proper_descriptor (IM i) 
  (eqv_descriptors i)
  (state_update (equivocator_IM IM) s i e i)
eqv_descriptorsi': MachineDescriptor (IM i)
Hitemx: True
Hproper': proper_descriptor (IM i) eqv_descriptorsi'
  (s i)
Hex_new: previous_state_descriptor_prop 
  (IM i) (eqv_descriptors i) 
  (s i) eqv_descriptorsi'
Hchar: equivocator_state_descriptor_project e
  (eqv_descriptors i) =
equivocator_state_descriptor_project 
  (s i) eqv_descriptorsi'
Hv: equivocator_valid (IM i) li (s i, input)
Ht': equivocator_transition (IM i) li (s i, input) =
(e, output)
equivocators_state_project IM eqv_descriptors
  (state_update (equivocator_IM IM) s i e) =
equivocators_state_project IM
  (equivocator_descriptors_update IM eqv_descriptors i
     eqv_descriptorsi') s unfold equivocators_state_project, EquivocatorsComposition.equivocators_state_project.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
eqv_descriptors: equivocator_descriptors IM
i: index
li: EquivocatorLabel (IM i)
input, output: option message
s: composite_state (equivocator_IM IM)
e: equivocator_state (IM i)
Hproper: proper_descriptor (IM i) 
  (eqv_descriptors i)
  (state_update (equivocator_IM IM) s i e i)
eqv_descriptorsi': MachineDescriptor (IM i)
Hitemx: True
Hproper': proper_descriptor (IM i) eqv_descriptorsi'
  (s i)
Hex_new: previous_state_descriptor_prop 
  (IM i) (eqv_descriptors i) 
  (s i) eqv_descriptorsi'
Hchar: equivocator_state_descriptor_project e
  (eqv_descriptors i) =
equivocator_state_descriptor_project 
  (s i) eqv_descriptorsi'
Hv: equivocator_valid (IM i) li (s i, input)
Ht': equivocator_transition (IM i) li (s i, input) =
(e, output)
(λ eqv : index,
   equivocator_state_descriptor_project
     (state_update (equivocator_IM IM) s i e eqv)
     (eqv_descriptors eqv)) =
(λ eqv : index,
   equivocator_state_descriptor_project (s eqv)
     (equivocator_descriptors_update IM
        eqv_descriptors i eqv_descriptorsi' eqv))
      by extensionality eqv; destruct (decide (i = eqv)); subst; state_update_simpl.
Qed.

Lemma equivocators_transition_item_project_proper_characterization
  (eqv_descriptors : equivocator_descriptors IM)
  (item : composite_transition_item (equivocator_IM IM))
  (Hproper : proper_equivocator_descriptors IM eqv_descriptors (destination item))
  : exists oitem eqv_descriptors',
    equivocators_transition_item_project eqv_descriptors item = Some (oitem, eqv_descriptors')
    /\ match oitem with
      | Some itemx =>
        (exists (Hex : existing_equivocator_label _ (projT2 (l item))),
          existT (projT1 (l item)) (existing_equivocator_label_extract _ _ Hex) = l itemx) /\
         input item = input itemx /\ output item = output itemx /\
        (equivocators_state_project IM eqv_descriptors (destination item) = destination itemx)
        /\ eqv_descriptors' (projT1 (l item)) = (equivocator_label_descriptor (projT2 (l item)))
      | None => True
      end
    /\ forall
      (s : composite_state (equivocator_IM IM))
      (Hv : composite_valid (equivocator_IM IM) (l item) (s, input item))
      (Ht : composite_transition (equivocator_IM IM) (l item) (s, input item) =
        (destination item, output item)),
      proper_equivocator_descriptors IM eqv_descriptors' s /\
      eqv_descriptors' = equivocator_descriptors_update IM eqv_descriptors
                          (projT1 (l item)) (eqv_descriptors' (projT1 (l item))) /\
      s = state_update (equivocator_IM IM) (destination item) (projT1 (l item)) (s (projT1 (l item))) /\
      previous_state_descriptor_prop (IM (projT1 (l item))) (eqv_descriptors (projT1 (l item)))
        (s (projT1 (l item))) (eqv_descriptors' (projT1 (l item))) /\
      match oitem with
      | Some itemx =>
        forall (sx : composite_state IM)
          (Hsx : sx = equivocators_state_project IM eqv_descriptors' s),
          composite_valid IM (l itemx) (sx, input itemx) /\
          composite_transition IM (l itemx) (sx, input itemx) = (destination itemx, output itemx)
      | None =>
        equivocators_state_project IM eqv_descriptors (destination item) =
        equivocators_state_project IM eqv_descriptors' s
      end.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
eqv_descriptors: equivocator_descriptors IM
item: composite_transition_item (equivocator_IM IM)
Hproper: proper_equivocator_descriptors IM
  eqv_descriptors (destination item)
∃ (oitem : option (composite_transition_item IM)) 
  (eqv_descriptors' : equivocator_descriptors IM),
  equivocators_transition_item_project eqv_descriptors
    item = Some (oitem, eqv_descriptors')
  ∧ match oitem with
    | Some itemx =>
        (∃ Hex : existing_equivocator_label
                   (IM (projT1 (l item)))
                   (projT2 (l item)),
           existT (projT1 (l item))
             (existing_equivocator_label_extract
                (IM (projT1 (l item)))
                (projT2 (l item)) Hex) = l itemx)
        ∧ input item = input itemx
          ∧ output item = output itemx
            ∧ equivocators_state_project IM
                eqv_descriptors (destination item) =
              destination itemx
              ∧ eqv_descriptors' (projT1 (l item)) =
                equivocator_label_descriptor
                  (projT2 (l item))
    | None => True
    end
    ∧ (∀ s : composite_state (equivocator_IM IM),
         composite_valid (equivocator_IM IM) (l item)
           (s, input item)
         → composite_transition (equivocator_IM IM)
             (l item) (s, input item) =
           (destination item, output item)
           → proper_equivocator_descriptors IM
               eqv_descriptors' s
             ∧ eqv_descriptors' =
               equivocator_descriptors_update IM
                 eqv_descriptors (projT1 (l item))
                 (eqv_descriptors' (projT1 (l item)))
               ∧ s =
                 state_update (equivocator_IM IM)
                   (destination item)
                   (projT1 (l item))
                   (s (projT1 (l item)))
                 ∧ previous_state_descriptor_prop
                     (IM (projT1 (l item)))
                     (eqv_descriptors
                        (projT1 (l item)))
                     (s (projT1 (l item)))
                     (eqv_descriptors'
                        (projT1 (l item)))
                   ∧ match oitem with
                     | Some itemx =>
                         ∀ sx : composite_state IM,
                           sx =
                           equivocators_state_project
                             IM eqv_descriptors' s
                           → composite_valid IM
                               (l itemx)
                               (sx, input itemx)
                             ∧ composite_transition IM
                                 (l itemx)
                                 (sx, input itemx) =
                               (destination itemx,
                                output itemx)
                     | None =>
                         equivocators_state_project IM
                           eqv_descriptors
                           (destination item) =
                         equivocators_state_project IM
                           eqv_descriptors' s
                     end)
Proof.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
eqv_descriptors: equivocator_descriptors IM
item: composite_transition_item (equivocator_IM IM)
Hproper: proper_equivocator_descriptors IM
  eqv_descriptors (destination item)
∃ (oitem : option (composite_transition_item IM)) 
  (eqv_descriptors' : equivocator_descriptors IM),
  equivocators_transition_item_project eqv_descriptors
    item = Some (oitem, eqv_descriptors')
  ∧ match oitem with
    | Some itemx =>
        (∃ Hex : existing_equivocator_label
                   (IM (projT1 (l item)))
                   (projT2 (l item)),
           existT (projT1 (l item))
             (existing_equivocator_label_extract
                (IM (projT1 (l item)))
                (projT2 (l item)) Hex) = l itemx)
        ∧ input item = input itemx
          ∧ output item = output itemx
            ∧ equivocators_state_project IM
                eqv_descriptors (destination item) =
              destination itemx
              ∧ eqv_descriptors' (projT1 (l item)) =
                equivocator_label_descriptor
                  (projT2 (l item))
    | None => True
    end
    ∧ (∀ s : composite_state (equivocator_IM IM),
         composite_valid (equivocator_IM IM) (l item)
           (s, input item)
         → composite_transition (equivocator_IM IM)
             (l item) (s, input item) =
           (destination item, output item)
           → proper_equivocator_descriptors IM
               eqv_descriptors' s
             ∧ eqv_descriptors' =
               equivocator_descriptors_update IM
                 eqv_descriptors (projT1 (l item))
                 (eqv_descriptors' (projT1 (l item)))
               ∧ s =
                 state_update (equivocator_IM IM)
                   (destination item)
                   (projT1 (l item))
                   (s (projT1 (l item)))
                 ∧ previous_state_descriptor_prop
                     (IM (projT1 (l item)))
                     (eqv_descriptors
                        (projT1 (l item)))
                     (s (projT1 (l item)))
                     (eqv_descriptors'
                        (projT1 (l item)))
                   ∧ match oitem with
                     | Some itemx =>
                         ∀ sx : composite_state IM,
                           sx =
                           equivocators_state_project
                             IM eqv_descriptors' s
                           → composite_valid IM
                               (l itemx)
                               (sx, input itemx)
                             ∧ composite_transition IM
                                 (l itemx)
                                 (sx, input itemx) =
                               (destination itemx,
                                output itemx)
                     | None =>
                         equivocators_state_project IM
                           eqv_descriptors
                           (destination item) =
                         equivocators_state_project IM
                           eqv_descriptors' s
                     end)
  destruct (equivocators_transition_item_project_proper_descriptor_characterization
    eqv_descriptors item (Hproper (projT1 (l item))))
    as [oitem [eqv_descriptors' [Hoitem [Hitemx Hchar]]]].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
eqv_descriptors: equivocator_descriptors IM
item: composite_transition_item (equivocator_IM IM)
Hproper: proper_equivocator_descriptors IM
  eqv_descriptors (destination item)
oitem: option (composite_transition_item IM)
eqv_descriptors': equivocator_descriptors IM
Hoitem: equivocators_transition_item_project
  eqv_descriptors item =
Some (oitem, eqv_descriptors')
Hitemx: match oitem with
| Some itemx =>
    (∃ Hex : existing_equivocator_label
               (IM (projT1 (l item)))
               (projT2 (l item)),
       existT (projT1 (l item))
         (existing_equivocator_label_extract
            (IM (projT1 (l item)))
            (projT2 (l item)) Hex) = 
       l itemx)
    ∧ input item = input itemx
      ∧ output item = output itemx
        ∧ equivocators_state_project IM
            eqv_descriptors
            (destination item) =
          destination itemx
          ∧ eqv_descriptors'
              (projT1 (l item)) =
            equivocator_label_descriptor
              (projT2 (l item))
| None => True
end
Hchar: ∀ s : composite_state (equivocator_IM IM),
  composite_valid (equivocator_IM IM) 
    (l item) (s, input item)
  → composite_transition 
      (equivocator_IM IM) 
      (l item) (s, input item) =
    (destination item, output item)
    → proper_descriptor 
        (IM (projT1 (l item)))
        (eqv_descriptors' (projT1 (l item)))
        (s (projT1 (l item)))
      ∧ eqv_descriptors' =
        equivocator_descriptors_update IM
          eqv_descriptors 
          (projT1 (l item))
          (eqv_descriptors' (projT1 (l item)))
        ∧ s =
          state_update 
            (equivocator_IM IM)
            (destination item)
            (projT1 (l item))
            (s (projT1 (l item)))
          ∧ previous_state_descriptor_prop
              (IM (projT1 (l item)))
              (eqv_descriptors
                 (projT1 (l item)))
              (s (projT1 (l item)))
              (eqv_descriptors'
                 (projT1 (l item)))
            ∧ match oitem with
              | Some itemx =>
                  ∀ sx : composite_state IM,
                    sx =
                    equivocators_state_project
                      IM eqv_descriptors' s
                    → composite_valid IM
                      (l itemx)
                      (sx, input itemx)
                      ∧ 
                      composite_transition IM
                      (l itemx)
                      (sx, input itemx) =
                      (
                      destination itemx,
                      output itemx)
              | None =>
                  equivocators_state_project
                    IM eqv_descriptors
                    (destination item) =
                  equivocators_state_project
                    IM eqv_descriptors' s
              end
∃ (oitem : option (composite_transition_item IM)) 
  (eqv_descriptors' : equivocator_descriptors IM),
  equivocators_transition_item_project eqv_descriptors
    item = Some (oitem, eqv_descriptors')
  ∧ match oitem with
    | Some itemx =>
        (∃ Hex : existing_equivocator_label
                   (IM (projT1 (l item)))
                   (projT2 (l item)),
           existT (projT1 (l item))
             (existing_equivocator_label_extract
                (IM (projT1 (l item)))
                (projT2 (l item)) Hex) = l itemx)
        ∧ input item = input itemx
          ∧ output item = output itemx
            ∧ equivocators_state_project IM
                eqv_descriptors (destination item) =
              destination itemx
              ∧ eqv_descriptors' (projT1 (l item)) =
                equivocator_label_descriptor
                  (projT2 (l item))
    | None => True
    end
    ∧ (∀ s : composite_state (equivocator_IM IM),
         composite_valid (equivocator_IM IM) (l item)
           (s, input item)
         → composite_transition (equivocator_IM IM)
             (l item) (s, input item) =
           (destination item, output item)
           → proper_equivocator_descriptors IM
               eqv_descriptors' s
             ∧ eqv_descriptors' =
               equivocator_descriptors_update IM
                 eqv_descriptors (projT1 (l item))
                 (eqv_descriptors' (projT1 (l item)))
               ∧ s =
                 state_update (equivocator_IM IM)
                   (destination item)
                   (projT1 (l item))
                   (s (projT1 (l item)))
                 ∧ previous_state_descriptor_prop
                     (IM (projT1 (l item)))
                     (eqv_descriptors
                        (projT1 (l item)))
                     (s (projT1 (l item)))
                     (eqv_descriptors'
                        (projT1 (l item)))
                   ∧ match oitem with
                     | Some itemx =>
                         ∀ sx : composite_state IM,
                           sx =
                           equivocators_state_project
                             IM eqv_descriptors' s
                           → composite_valid IM
                               (l itemx)
                               (sx, input itemx)
                             ∧ composite_transition IM
                                 (l itemx)
                                 (sx, input itemx) =
                               (destination itemx,
                                output itemx)
                     | None =>
                         equivocators_state_project IM
                           eqv_descriptors
                           (destination item) =
                         equivocators_state_project IM
                           eqv_descriptors' s
                     end)
  exists oitem, eqv_descriptors'.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
eqv_descriptors: equivocator_descriptors IM
item: composite_transition_item (equivocator_IM IM)
Hproper: proper_equivocator_descriptors IM
  eqv_descriptors (destination item)
oitem: option (composite_transition_item IM)
eqv_descriptors': equivocator_descriptors IM
Hoitem: equivocators_transition_item_project
  eqv_descriptors item =
Some (oitem, eqv_descriptors')
Hitemx: match oitem with
| Some itemx =>
    (∃ Hex : existing_equivocator_label
               (IM (projT1 (l item)))
               (projT2 (l item)),
       existT (projT1 (l item))
         (existing_equivocator_label_extract
            (IM (projT1 (l item)))
            (projT2 (l item)) Hex) = 
       l itemx)
    ∧ input item = input itemx
      ∧ output item = output itemx
        ∧ equivocators_state_project IM
            eqv_descriptors
            (destination item) =
          destination itemx
          ∧ eqv_descriptors'
              (projT1 (l item)) =
            equivocator_label_descriptor
              (projT2 (l item))
| None => True
end
Hchar: ∀ s : composite_state (equivocator_IM IM),
  composite_valid (equivocator_IM IM) 
    (l item) (s, input item)
  → composite_transition 
      (equivocator_IM IM) 
      (l item) (s, input item) =
    (destination item, output item)
    → proper_descriptor 
        (IM (projT1 (l item)))
        (eqv_descriptors' (projT1 (l item)))
        (s (projT1 (l item)))
      ∧ eqv_descriptors' =
        equivocator_descriptors_update IM
          eqv_descriptors 
          (projT1 (l item))
          (eqv_descriptors' (projT1 (l item)))
        ∧ s =
          state_update 
            (equivocator_IM IM)
            (destination item)
            (projT1 (l item))
            (s (projT1 (l item)))
          ∧ previous_state_descriptor_prop
              (IM (projT1 (l item)))
              (eqv_descriptors
                 (projT1 (l item)))
              (s (projT1 (l item)))
              (eqv_descriptors'
                 (projT1 (l item)))
            ∧ match oitem with
              | Some itemx =>
                  ∀ sx : composite_state IM,
                    sx =
                    equivocators_state_project
                      IM eqv_descriptors' s
                    → composite_valid IM
                      (l itemx)
                      (sx, input itemx)
                      ∧ 
                      composite_transition IM
                      (l itemx)
                      (sx, input itemx) =
                      (
                      destination itemx,
                      output itemx)
              | None =>
                  equivocators_state_project
                    IM eqv_descriptors
                    (destination item) =
                  equivocators_state_project
                    IM eqv_descriptors' s
              end
equivocators_transition_item_project eqv_descriptors
  item = Some (oitem, eqv_descriptors')
∧ match oitem with
  | Some itemx =>
      (∃ Hex : existing_equivocator_label
                 (IM (projT1 (l item)))
                 (projT2 (l item)),
         existT (projT1 (l item))
           (existing_equivocator_label_extract
              (IM (projT1 (l item))) (projT2 (l item))
              Hex) = l itemx)
      ∧ input item = input itemx
        ∧ output item = output itemx
          ∧ equivocators_state_project IM
              eqv_descriptors (destination item) =
            destination itemx
            ∧ eqv_descriptors' (projT1 (l item)) =
              equivocator_label_descriptor
                (projT2 (l item))
  | None => True
  end
  ∧ (∀ s : composite_state (equivocator_IM IM),
       composite_valid (equivocator_IM IM) (l item)
         (s, input item)
       → composite_transition (equivocator_IM IM)
           (l item) (s, input item) =
         (destination item, output item)
         → proper_equivocator_descriptors IM
             eqv_descriptors' s
           ∧ eqv_descriptors' =
             equivocator_descriptors_update IM
               eqv_descriptors (projT1 (l item))
               (eqv_descriptors' (projT1 (l item)))
             ∧ s =
               state_update (equivocator_IM IM)
                 (destination item) (projT1 (l item))
                 (s (projT1 (l item)))
               ∧ previous_state_descriptor_prop
                   (IM (projT1 (l item)))
                   (eqv_descriptors (projT1 (l item)))
                   (s (projT1 (l item)))
                   (eqv_descriptors' (projT1 (l item)))
                 ∧ match oitem with
                   | Some itemx =>
                       ∀ sx : composite_state IM,
                         sx =
                         equivocators_state_project IM
                           eqv_descriptors' s
                         → composite_valid IM
                             (l itemx)
                             (sx, input itemx)
                           ∧ composite_transition IM
                               (l itemx)
                               (sx, input itemx) =
                             (destination itemx,
                              output itemx)
                   | None =>
                       equivocators_state_project IM
                         eqv_descriptors
                         (destination item) =
                       equivocators_state_project IM
                         eqv_descriptors' s
                   end) split; [done |].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
eqv_descriptors: equivocator_descriptors IM
item: composite_transition_item (equivocator_IM IM)
Hproper: proper_equivocator_descriptors IM
  eqv_descriptors (destination item)
oitem: option (composite_transition_item IM)
eqv_descriptors': equivocator_descriptors IM
Hoitem: equivocators_transition_item_project
  eqv_descriptors item =
Some (oitem, eqv_descriptors')
Hitemx: match oitem with
| Some itemx =>
    (∃ Hex : existing_equivocator_label
               (IM (projT1 (l item)))
               (projT2 (l item)),
       existT (projT1 (l item))
         (existing_equivocator_label_extract
            (IM (projT1 (l item)))
            (projT2 (l item)) Hex) = 
       l itemx)
    ∧ input item = input itemx
      ∧ output item = output itemx
        ∧ equivocators_state_project IM
            eqv_descriptors
            (destination item) =
          destination itemx
          ∧ eqv_descriptors'
              (projT1 (l item)) =
            equivocator_label_descriptor
              (projT2 (l item))
| None => True
end
Hchar: ∀ s : composite_state (equivocator_IM IM),
  composite_valid (equivocator_IM IM) 
    (l item) (s, input item)
  → composite_transition 
      (equivocator_IM IM) 
      (l item) (s, input item) =
    (destination item, output item)
    → proper_descriptor 
        (IM (projT1 (l item)))
        (eqv_descriptors' (projT1 (l item)))
        (s (projT1 (l item)))
      ∧ eqv_descriptors' =
        equivocator_descriptors_update IM
          eqv_descriptors 
          (projT1 (l item))
          (eqv_descriptors' (projT1 (l item)))
        ∧ s =
          state_update 
            (equivocator_IM IM)
            (destination item)
            (projT1 (l item))
            (s (projT1 (l item)))
          ∧ previous_state_descriptor_prop
              (IM (projT1 (l item)))
              (eqv_descriptors
                 (projT1 (l item)))
              (s (projT1 (l item)))
              (eqv_descriptors'
                 (projT1 (l item)))
            ∧ match oitem with
              | Some itemx =>
                  ∀ sx : composite_state IM,
                    sx =
                    equivocators_state_project
                      IM eqv_descriptors' s
                    → composite_valid IM
                      (l itemx)
                      (sx, input itemx)
                      ∧ 
                      composite_transition IM
                      (l itemx)
                      (sx, input itemx) =
                      (
                      destination itemx,
                      output itemx)
              | None =>
                  equivocators_state_project
                    IM eqv_descriptors
                    (destination item) =
                  equivocators_state_project
                    IM eqv_descriptors' s
              end
match oitem with
| Some itemx =>
    (∃ Hex : existing_equivocator_label
               (IM (projT1 (l item)))
               (projT2 (l item)),
       existT (projT1 (l item))
         (existing_equivocator_label_extract
            (IM (projT1 (l item))) (projT2 (l item))
            Hex) = l itemx)
    ∧ input item = input itemx
      ∧ output item = output itemx
        ∧ equivocators_state_project IM
            eqv_descriptors (destination item) =
          destination itemx
          ∧ eqv_descriptors' (projT1 (l item)) =
            equivocator_label_descriptor
              (projT2 (l item))
| None => True
end
∧ (∀ s : composite_state (equivocator_IM IM),
     composite_valid (equivocator_IM IM) (l item)
       (s, input item)
     → composite_transition (equivocator_IM IM)
         (l item) (s, input item) =
       (destination item, output item)
       → proper_equivocator_descriptors IM
           eqv_descriptors' s
         ∧ eqv_descriptors' =
           equivocator_descriptors_update IM
             eqv_descriptors (projT1 (l item))
             (eqv_descriptors' (projT1 (l item)))
           ∧ s =
             state_update (equivocator_IM IM)
               (destination item) (projT1 (l item))
               (s (projT1 (l item)))
             ∧ previous_state_descriptor_prop
                 (IM (projT1 (l item)))
                 (eqv_descriptors (projT1 (l item)))
                 (s (projT1 (l item)))
                 (eqv_descriptors' (projT1 (l item)))
               ∧ match oitem with
                 | Some itemx =>
                     ∀ sx : composite_state IM,
                       sx =
                       equivocators_state_project IM
                         eqv_descriptors' s
                       → composite_valid IM (l itemx)
                           (sx, input itemx)
                         ∧ composite_transition IM
                             (l itemx)
                             (sx, input itemx) =
                           (destination itemx,
                            output itemx)
                 | None =>
                     equivocators_state_project IM
                       eqv_descriptors
                       (destination item) =
                     equivocators_state_project IM
                       eqv_descriptors' s
                 end)
  split; [done |].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
eqv_descriptors: equivocator_descriptors IM
item: composite_transition_item (equivocator_IM IM)
Hproper: proper_equivocator_descriptors IM
  eqv_descriptors (destination item)
oitem: option (composite_transition_item IM)
eqv_descriptors': equivocator_descriptors IM
Hoitem: equivocators_transition_item_project
  eqv_descriptors item =
Some (oitem, eqv_descriptors')
Hitemx: match oitem with
| Some itemx =>
    (∃ Hex : existing_equivocator_label
               (IM (projT1 (l item)))
               (projT2 (l item)),
       existT (projT1 (l item))
         (existing_equivocator_label_extract
            (IM (projT1 (l item)))
            (projT2 (l item)) Hex) = 
       l itemx)
    ∧ input item = input itemx
      ∧ output item = output itemx
        ∧ equivocators_state_project IM
            eqv_descriptors
            (destination item) =
          destination itemx
          ∧ eqv_descriptors'
              (projT1 (l item)) =
            equivocator_label_descriptor
              (projT2 (l item))
| None => True
end
Hchar: ∀ s : composite_state (equivocator_IM IM),
  composite_valid (equivocator_IM IM) 
    (l item) (s, input item)
  → composite_transition 
      (equivocator_IM IM) 
      (l item) (s, input item) =
    (destination item, output item)
    → proper_descriptor 
        (IM (projT1 (l item)))
        (eqv_descriptors' (projT1 (l item)))
        (s (projT1 (l item)))
      ∧ eqv_descriptors' =
        equivocator_descriptors_update IM
          eqv_descriptors 
          (projT1 (l item))
          (eqv_descriptors' (projT1 (l item)))
        ∧ s =
          state_update 
            (equivocator_IM IM)
            (destination item)
            (projT1 (l item))
            (s (projT1 (l item)))
          ∧ previous_state_descriptor_prop
              (IM (projT1 (l item)))
              (eqv_descriptors
                 (projT1 (l item)))
              (s (projT1 (l item)))
              (eqv_descriptors'
                 (projT1 (l item)))
            ∧ match oitem with
              | Some itemx =>
                  ∀ sx : composite_state IM,
                    sx =
                    equivocators_state_project
                      IM eqv_descriptors' s
                    → composite_valid IM
                      (l itemx)
                      (sx, input itemx)
                      ∧ 
                      composite_transition IM
                      (l itemx)
                      (sx, input itemx) =
                      (
                      destination itemx,
                      output itemx)
              | None =>
                  equivocators_state_project
                    IM eqv_descriptors
                    (destination item) =
                  equivocators_state_project
                    IM eqv_descriptors' s
              end
∀ s : composite_state (equivocator_IM IM),
  composite_valid (equivocator_IM IM) (l item)
    (s, input item)
  → composite_transition (equivocator_IM IM) (l item)
      (s, input item) =
    (destination item, output item)
    → proper_equivocator_descriptors IM
        eqv_descriptors' s
      ∧ eqv_descriptors' =
        equivocator_descriptors_update IM
          eqv_descriptors (projT1 (l item))
          (eqv_descriptors' (projT1 (l item)))
        ∧ s =
          state_update (equivocator_IM IM)
            (destination item) (projT1 (l item))
            (s (projT1 (l item)))
          ∧ previous_state_descriptor_prop
              (IM (projT1 (l item)))
              (eqv_descriptors (projT1 (l item)))
              (s (projT1 (l item)))
              (eqv_descriptors' (projT1 (l item)))
            ∧ match oitem with
              | Some itemx =>
                  ∀ sx : composite_state IM,
                    sx =
                    equivocators_state_project IM
                      eqv_descriptors' s
                    → composite_valid IM (l itemx)
                        (sx, input itemx)
                      ∧ composite_transition IM
                          (l itemx) (sx, input itemx) =
                        (destination itemx,
                         output itemx)
              | None =>
                  equivocators_state_project IM
                    eqv_descriptors (destination item) =
                  equivocators_state_project IM
                    eqv_descriptors' s
              end
  intros.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
eqv_descriptors: equivocator_descriptors IM
item: composite_transition_item (equivocator_IM IM)
Hproper: proper_equivocator_descriptors IM
  eqv_descriptors (destination item)
oitem: option (composite_transition_item IM)
eqv_descriptors': equivocator_descriptors IM
Hoitem: equivocators_transition_item_project
  eqv_descriptors item =
Some (oitem, eqv_descriptors')
Hitemx: match oitem with
| Some itemx =>
    (∃ Hex : existing_equivocator_label
               (IM (projT1 (l item)))
               (projT2 (l item)),
       existT (projT1 (l item))
         (existing_equivocator_label_extract
            (IM (projT1 (l item)))
            (projT2 (l item)) Hex) = 
       l itemx)
    ∧ input item = input itemx
      ∧ output item = output itemx
        ∧ equivocators_state_project IM
            eqv_descriptors
            (destination item) =
          destination itemx
          ∧ eqv_descriptors'
              (projT1 (l item)) =
            equivocator_label_descriptor
              (projT2 (l item))
| None => True
end
Hchar: ∀ s : composite_state (equivocator_IM IM),
  composite_valid (equivocator_IM IM) 
    (l item) (s, input item)
  → composite_transition 
      (equivocator_IM IM) 
      (l item) (s, input item) =
    (destination item, output item)
    → proper_descriptor 
        (IM (projT1 (l item)))
        (eqv_descriptors' (projT1 (l item)))
        (s (projT1 (l item)))
      ∧ eqv_descriptors' =
        equivocator_descriptors_update IM
          eqv_descriptors 
          (projT1 (l item))
          (eqv_descriptors' (projT1 (l item)))
        ∧ s =
          state_update 
            (equivocator_IM IM)
            (destination item)
            (projT1 (l item))
            (s (projT1 (l item)))
          ∧ previous_state_descriptor_prop
              (IM (projT1 (l item)))
              (eqv_descriptors
                 (projT1 (l item)))
              (s (projT1 (l item)))
              (eqv_descriptors'
                 (projT1 (l item)))
            ∧ match oitem with
              | Some itemx =>
                  ∀ sx : composite_state IM,
                    sx =
                    equivocators_state_project
                      IM eqv_descriptors' s
                    → composite_valid IM
                      (l itemx)
                      (sx, input itemx)
                      ∧ 
                      composite_transition IM
                      (l itemx)
                      (sx, input itemx) =
                      (
                      destination itemx,
                      output itemx)
              | None =>
                  equivocators_state_project
                    IM eqv_descriptors
                    (destination item) =
                  equivocators_state_project
                    IM eqv_descriptors' s
              end
s: composite_state (equivocator_IM IM)
Hv: composite_valid (equivocator_IM IM) 
  (l item) (s, input item)
Ht: composite_transition (equivocator_IM IM) 
  (l item) (s, input item) =
(destination item, output item)
proper_equivocator_descriptors IM eqv_descriptors' s
∧ eqv_descriptors' =
  equivocator_descriptors_update IM eqv_descriptors
    (projT1 (l item))
    (eqv_descriptors' (projT1 (l item)))
  ∧ s =
    state_update (equivocator_IM IM)
      (destination item) (projT1 (l item))
      (s (projT1 (l item)))
    ∧ previous_state_descriptor_prop
        (IM (projT1 (l item)))
        (eqv_descriptors (projT1 (l item)))
        (s (projT1 (l item)))
        (eqv_descriptors' (projT1 (l item)))
      ∧ match oitem with
        | Some itemx =>
            ∀ sx : composite_state IM,
              sx =
              equivocators_state_project IM
                eqv_descriptors' s
              → composite_valid IM (l itemx)
                  (sx, input itemx)
                ∧ composite_transition IM (l itemx)
                    (sx, input itemx) =
                  (destination itemx, output itemx)
        | None =>
            equivocators_state_project IM
              eqv_descriptors (destination item) =
            equivocators_state_project IM
              eqv_descriptors' s
        end
  specialize (Hchar s Hv Ht) as (Hproperi' & Heqv' & Hs & Hex_new & Hchar).message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
eqv_descriptors: equivocator_descriptors IM
item: composite_transition_item (equivocator_IM IM)
Hproper: proper_equivocator_descriptors IM
  eqv_descriptors (destination item)
oitem: option (composite_transition_item IM)
eqv_descriptors': equivocator_descriptors IM
Hoitem: equivocators_transition_item_project
  eqv_descriptors item =
Some (oitem, eqv_descriptors')
Hitemx: match oitem with
| Some itemx =>
    (∃ Hex : existing_equivocator_label
               (IM (projT1 (l item)))
               (projT2 (l item)),
       existT (projT1 (l item))
         (existing_equivocator_label_extract
            (IM (projT1 (l item)))
            (projT2 (l item)) Hex) = 
       l itemx)
    ∧ input item = input itemx
      ∧ output item = output itemx
        ∧ equivocators_state_project IM
            eqv_descriptors
            (destination item) =
          destination itemx
          ∧ eqv_descriptors'
              (projT1 (l item)) =
            equivocator_label_descriptor
              (projT2 (l item))
| None => True
end
s: composite_state (equivocator_IM IM)
Hv: composite_valid (equivocator_IM IM) 
  (l item) (s, input item)
Ht: composite_transition (equivocator_IM IM) 
  (l item) (s, input item) =
(destination item, output item)
Hproperi': proper_descriptor (IM (projT1 (l item)))
  (eqv_descriptors' (projT1 (l item)))
  (s (projT1 (l item)))
Heqv': eqv_descriptors' =
equivocator_descriptors_update IM
  eqv_descriptors (projT1 (l item))
  (eqv_descriptors' (projT1 (l item)))
Hs: s =
state_update (equivocator_IM IM)
  (destination item) (projT1 (l item))
  (s (projT1 (l item)))
Hex_new: previous_state_descriptor_prop
  (IM (projT1 (l item)))
  (eqv_descriptors (projT1 (l item)))
  (s (projT1 (l item)))
  (eqv_descriptors' (projT1 (l item)))
Hchar: match oitem with
| Some itemx =>
    ∀ sx : composite_state IM,
      sx =
      equivocators_state_project IM
        eqv_descriptors' s
      → composite_valid IM 
          (l itemx) (sx, input itemx)
        ∧ composite_transition IM 
            (l itemx) (
            sx, input itemx) =
          (destination itemx, output itemx)
| None =>
    equivocators_state_project IM
      eqv_descriptors (destination item) =
    equivocators_state_project IM
      eqv_descriptors' s
end
proper_equivocator_descriptors IM eqv_descriptors' s
∧ eqv_descriptors' =
  equivocator_descriptors_update IM eqv_descriptors
    (projT1 (l item))
    (eqv_descriptors' (projT1 (l item)))
  ∧ s =
    state_update (equivocator_IM IM)
      (destination item) (projT1 (l item))
      (s (projT1 (l item)))
    ∧ previous_state_descriptor_prop
        (IM (projT1 (l item)))
        (eqv_descriptors (projT1 (l item)))
        (s (projT1 (l item)))
        (eqv_descriptors' (projT1 (l item)))
      ∧ match oitem with
        | Some itemx =>
            ∀ sx : composite_state IM,
              sx =
              equivocators_state_project IM
                eqv_descriptors' s
              → composite_valid IM (l itemx)
                  (sx, input itemx)
                ∧ composite_transition IM (l itemx)
                    (sx, input itemx) =
                  (destination itemx, output itemx)
        | None =>
            equivocators_state_project IM
              eqv_descriptors (destination item) =
            equivocators_state_project IM
              eqv_descriptors' s
        end
  clear Hv Ht Hoitem.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
eqv_descriptors: equivocator_descriptors IM
item: composite_transition_item (equivocator_IM IM)
Hproper: proper_equivocator_descriptors IM
  eqv_descriptors (destination item)
oitem: option (composite_transition_item IM)
eqv_descriptors': equivocator_descriptors IM
Hitemx: match oitem with
| Some itemx =>
    (∃ Hex : existing_equivocator_label
               (IM (projT1 (l item)))
               (projT2 (l item)),
       existT (projT1 (l item))
         (existing_equivocator_label_extract
            (IM (projT1 (l item)))
            (projT2 (l item)) Hex) = 
       l itemx)
    ∧ input item = input itemx
      ∧ output item = output itemx
        ∧ equivocators_state_project IM
            eqv_descriptors
            (destination item) =
          destination itemx
          ∧ eqv_descriptors'
              (projT1 (l item)) =
            equivocator_label_descriptor
              (projT2 (l item))
| None => True
end
s: composite_state (equivocator_IM IM)
Hproperi': proper_descriptor (IM (projT1 (l item)))
  (eqv_descriptors' (projT1 (l item)))
  (s (projT1 (l item)))
Heqv': eqv_descriptors' =
equivocator_descriptors_update IM
  eqv_descriptors (projT1 (l item))
  (eqv_descriptors' (projT1 (l item)))
Hs: s =
state_update (equivocator_IM IM)
  (destination item) (projT1 (l item))
  (s (projT1 (l item)))
Hex_new: previous_state_descriptor_prop
  (IM (projT1 (l item)))
  (eqv_descriptors (projT1 (l item)))
  (s (projT1 (l item)))
  (eqv_descriptors' (projT1 (l item)))
Hchar: match oitem with
| Some itemx =>
    ∀ sx : composite_state IM,
      sx =
      equivocators_state_project IM
        eqv_descriptors' s
      → composite_valid IM 
          (l itemx) (sx, input itemx)
        ∧ composite_transition IM 
            (l itemx) (
            sx, input itemx) =
          (destination itemx, output itemx)
| None =>
    equivocators_state_project IM
      eqv_descriptors (destination item) =
    equivocators_state_project IM
      eqv_descriptors' s
end
proper_equivocator_descriptors IM eqv_descriptors' s
∧ eqv_descriptors' =
  equivocator_descriptors_update IM eqv_descriptors
    (projT1 (l item))
    (eqv_descriptors' (projT1 (l item)))
  ∧ s =
    state_update (equivocator_IM IM)
      (destination item) (projT1 (l item))
      (s (projT1 (l item)))
    ∧ previous_state_descriptor_prop
        (IM (projT1 (l item)))
        (eqv_descriptors (projT1 (l item)))
        (s (projT1 (l item)))
        (eqv_descriptors' (projT1 (l item)))
      ∧ match oitem with
        | Some itemx =>
            ∀ sx : composite_state IM,
              sx =
              equivocators_state_project IM
                eqv_descriptors' s
              → composite_valid IM (l itemx)
                  (sx, input itemx)
                ∧ composite_transition IM (l itemx)
                    (sx, input itemx) =
                  (destination itemx, output itemx)
        | None =>
            equivocators_state_project IM
              eqv_descriptors (destination item) =
            equivocators_state_project IM
              eqv_descriptors' s
        end
  split; [| by repeat split]; clear Hchar.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
eqv_descriptors: equivocator_descriptors IM
item: composite_transition_item (equivocator_IM IM)
Hproper: proper_equivocator_descriptors IM
  eqv_descriptors (destination item)
oitem: option (composite_transition_item IM)
eqv_descriptors': equivocator_descriptors IM
Hitemx: match oitem with
| Some itemx =>
    (∃ Hex : existing_equivocator_label
               (IM (projT1 (l item)))
               (projT2 (l item)),
       existT (projT1 (l item))
         (existing_equivocator_label_extract
            (IM (projT1 (l item)))
            (projT2 (l item)) Hex) = 
       l itemx)
    ∧ input item = input itemx
      ∧ output item = output itemx
        ∧ equivocators_state_project IM
            eqv_descriptors
            (destination item) =
          destination itemx
          ∧ eqv_descriptors'
              (projT1 (l item)) =
            equivocator_label_descriptor
              (projT2 (l item))
| None => True
end
s: composite_state (equivocator_IM IM)
Hproperi': proper_descriptor (IM (projT1 (l item)))
  (eqv_descriptors' (projT1 (l item)))
  (s (projT1 (l item)))
Heqv': eqv_descriptors' =
equivocator_descriptors_update IM
  eqv_descriptors (projT1 (l item))
  (eqv_descriptors' (projT1 (l item)))
Hs: s =
state_update (equivocator_IM IM)
  (destination item) (projT1 (l item))
  (s (projT1 (l item)))
Hex_new: previous_state_descriptor_prop
  (IM (projT1 (l item)))
  (eqv_descriptors (projT1 (l item)))
  (s (projT1 (l item)))
  (eqv_descriptors' (projT1 (l item)))
proper_equivocator_descriptors IM eqv_descriptors' s
  intro eqv.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
eqv_descriptors: equivocator_descriptors IM
item: composite_transition_item (equivocator_IM IM)
Hproper: proper_equivocator_descriptors IM
  eqv_descriptors (destination item)
oitem: option (composite_transition_item IM)
eqv_descriptors': equivocator_descriptors IM
Hitemx: match oitem with
| Some itemx =>
    (∃ Hex : existing_equivocator_label
               (IM (projT1 (l item)))
               (projT2 (l item)),
       existT (projT1 (l item))
         (existing_equivocator_label_extract
            (IM (projT1 (l item)))
            (projT2 (l item)) Hex) = 
       l itemx)
    ∧ input item = input itemx
      ∧ output item = output itemx
        ∧ equivocators_state_project IM
            eqv_descriptors
            (destination item) =
          destination itemx
          ∧ eqv_descriptors'
              (projT1 (l item)) =
            equivocator_label_descriptor
              (projT2 (l item))
| None => True
end
s: composite_state (equivocator_IM IM)
Hproperi': proper_descriptor (IM (projT1 (l item)))
  (eqv_descriptors' (projT1 (l item)))
  (s (projT1 (l item)))
Heqv': eqv_descriptors' =
equivocator_descriptors_update IM
  eqv_descriptors (projT1 (l item))
  (eqv_descriptors' (projT1 (l item)))
Hs: s =
state_update (equivocator_IM IM)
  (destination item) (projT1 (l item))
  (s (projT1 (l item)))
Hex_new: previous_state_descriptor_prop
  (IM (projT1 (l item)))
  (eqv_descriptors (projT1 (l item)))
  (s (projT1 (l item)))
  (eqv_descriptors' (projT1 (l item)))
eqv: index
proper_descriptor (IM eqv) (eqv_descriptors' eqv)
  (s eqv)
  rewrite Heqv', Hs.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
eqv_descriptors: equivocator_descriptors IM
item: composite_transition_item (equivocator_IM IM)
Hproper: proper_equivocator_descriptors IM
  eqv_descriptors (destination item)
oitem: option (composite_transition_item IM)
eqv_descriptors': equivocator_descriptors IM
Hitemx: match oitem with
| Some itemx =>
    (∃ Hex : existing_equivocator_label
               (IM (projT1 (l item)))
               (projT2 (l item)),
       existT (projT1 (l item))
         (existing_equivocator_label_extract
            (IM (projT1 (l item)))
            (projT2 (l item)) Hex) = 
       l itemx)
    ∧ input item = input itemx
      ∧ output item = output itemx
        ∧ equivocators_state_project IM
            eqv_descriptors
            (destination item) =
          destination itemx
          ∧ eqv_descriptors'
              (projT1 (l item)) =
            equivocator_label_descriptor
              (projT2 (l item))
| None => True
end
s: composite_state (equivocator_IM IM)
Hproperi': proper_descriptor (IM (projT1 (l item)))
  (eqv_descriptors' (projT1 (l item)))
  (s (projT1 (l item)))
Heqv': eqv_descriptors' =
equivocator_descriptors_update IM
  eqv_descriptors (projT1 (l item))
  (eqv_descriptors' (projT1 (l item)))
Hs: s =
state_update (equivocator_IM IM)
  (destination item) (projT1 (l item))
  (s (projT1 (l item)))
Hex_new: previous_state_descriptor_prop
  (IM (projT1 (l item)))
  (eqv_descriptors (projT1 (l item)))
  (s (projT1 (l item)))
  (eqv_descriptors' (projT1 (l item)))
eqv: index
proper_descriptor (IM eqv)
  (equivocator_descriptors_update IM eqv_descriptors
     (projT1 (l item))
     (eqv_descriptors' (projT1 (l item))) eqv)
  (state_update (equivocator_IM IM) (destination item)
     (projT1 (l item)) (s (projT1 (l item))) eqv)
  by destruct (decide (eqv = projT1 (l item))); subst; state_update_simpl.
Qed.

Lemma equivocators_transition_item_project_inv_characterization
  (eqv_descriptors eqv_descriptors' : equivocator_descriptors IM)
  (item : composite_transition_item (equivocator_IM IM))
  (itemx : composite_transition_item IM)
  (Hpr_item : equivocators_transition_item_project eqv_descriptors item =
              Some (Some itemx, eqv_descriptors'))
  : (exists (Hex : existing_equivocator_label _ (projT2 (l item))),
      existT (projT1 (l item)) (existing_equivocator_label_extract _ _ Hex) = l itemx) /\
    input item = input itemx /\ output item = output itemx /\
    equivocators_state_project IM eqv_descriptors (destination item) = destination itemx.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
eqv_descriptors, eqv_descriptors': equivocator_descriptors IM
item: composite_transition_item (equivocator_IM IM)
itemx: composite_transition_item IM
Hpr_item: equivocators_transition_item_project
  eqv_descriptors item =
Some (Some itemx, eqv_descriptors')
(∃ Hex : existing_equivocator_label
           (IM (projT1 (l item))) (projT2 (l item)),
   existT (projT1 (l item))
     (existing_equivocator_label_extract
        (IM (projT1 (l item))) (projT2 (l item)) Hex) =
   l itemx)
∧ input item = input itemx
  ∧ output item = output itemx
    ∧ equivocators_state_project IM eqv_descriptors
        (destination item) = destination itemx
Proof.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
eqv_descriptors, eqv_descriptors': equivocator_descriptors IM
item: composite_transition_item (equivocator_IM IM)
itemx: composite_transition_item IM
Hpr_item: equivocators_transition_item_project
  eqv_descriptors item =
Some (Some itemx, eqv_descriptors')
(∃ Hex : existing_equivocator_label
           (IM (projT1 (l item))) (projT2 (l item)),
   existT (projT1 (l item))
     (existing_equivocator_label_extract
        (IM (projT1 (l item))) (projT2 (l item)) Hex) =
   l itemx)
∧ input item = input itemx
  ∧ output item = output itemx
    ∧ equivocators_state_project IM eqv_descriptors
        (destination item) = destination itemx
  unfold equivocators_transition_item_project in Hpr_item.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
eqv_descriptors, eqv_descriptors': equivocator_descriptors IM
item: composite_transition_item (equivocator_IM IM)
itemx: composite_transition_item IM
Hpr_item: match
  equivocator_vlsm_transition_item_project
    (IM (projT1 (l item)))
    (composite_transition_item_projection
       (equivocator_IM IM) item)
    (eqv_descriptors (projT1 (l item)))
with
| Some (Some item', deqv') =>
    Some
      (Some
         {|
           l :=
             existT (projT1 (l item))
               (l item');
           input := input item;
           destination :=
             equivocators_state_project IM
               eqv_descriptors
               (destination item);
           output := output item
         |},
       equivocator_descriptors_update IM
         eqv_descriptors 
         (projT1 (l item)) deqv')
| Some (None, deqv') =>
    Some
      (None,
       equivocator_descriptors_update IM
         eqv_descriptors 
         (projT1 (l item)) deqv')
| None => None
end = Some (Some itemx, eqv_descriptors')
(∃ Hex : existing_equivocator_label
           (IM (projT1 (l item))) (projT2 (l item)),
   existT (projT1 (l item))
     (existing_equivocator_label_extract
        (IM (projT1 (l item))) (projT2 (l item)) Hex) =
   l itemx)
∧ input item = input itemx
  ∧ output item = output itemx
    ∧ equivocators_state_project IM eqv_descriptors
        (destination item) = destination itemx
  destruct
    (equivocator_vlsm_transition_item_project
      (IM (projT1 (l item)))
      (composite_transition_item_projection (equivocator_IM IM) item)
      (eqv_descriptors (projT1 (l item))))
    as [([itemi |], descriptori) |] eqn: Hpr_itemi
  ; [| by congruence..].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
eqv_descriptors, eqv_descriptors': equivocator_descriptors IM
item: composite_transition_item (equivocator_IM IM)
itemx: composite_transition_item IM
itemi: transition_item
descriptori: MachineDescriptor (IM (projT1 (l item)))
Hpr_itemi: equivocator_vlsm_transition_item_project
  (IM (projT1 (l item)))
  (composite_transition_item_projection
     (equivocator_IM IM) item)
  (eqv_descriptors (projT1 (l item))) =
Some (Some itemi, descriptori)
Hpr_item: Some
  (Some
     {|
       l :=
         existT (projT1 (l item)) (l itemi);
       input := input item;
       destination :=
         equivocators_state_project IM
           eqv_descriptors
           (destination item);
       output := output item
     |},
   equivocator_descriptors_update IM
     eqv_descriptors (projT1 (l item))
     descriptori) =
Some (Some itemx, eqv_descriptors')
(∃ Hex : existing_equivocator_label
           (IM (projT1 (l item))) (projT2 (l item)),
   existT (projT1 (l item))
     (existing_equivocator_label_extract
        (IM (projT1 (l item))) (projT2 (l item)) Hex) =
   l itemx)
∧ input item = input itemx
  ∧ output item = output itemx
    ∧ equivocators_state_project IM eqv_descriptors
        (destination item) = destination itemx
  inversion Hpr_item.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
eqv_descriptors, eqv_descriptors': equivocator_descriptors IM
item: composite_transition_item (equivocator_IM IM)
itemx: composite_transition_item IM
itemi: transition_item
descriptori: MachineDescriptor (IM (projT1 (l item)))
Hpr_itemi: equivocator_vlsm_transition_item_project
  (IM (projT1 (l item)))
  (composite_transition_item_projection
     (equivocator_IM IM) item)
  (eqv_descriptors (projT1 (l item))) =
Some (Some itemi, descriptori)
Hpr_item: Some
  (Some
     {|
       l :=
         existT (projT1 (l item)) (l itemi);
       input := input item;
       destination :=
         equivocators_state_project IM
           eqv_descriptors
           (destination item);
       output := output item
     |},
   equivocator_descriptors_update IM
     eqv_descriptors (projT1 (l item))
     descriptori) =
Some (Some itemx, eqv_descriptors')
H2: {|
  l := existT (projT1 (l item)) (l itemi);
  input := input item;
  destination :=
    equivocators_state_project IM eqv_descriptors
      (destination item);
  output := output item
|} = itemx
H3: equivocator_descriptors_update IM eqv_descriptors
  (projT1 (l item)) descriptori =
eqv_descriptors'
(∃ Hex : existing_equivocator_label
           (IM (projT1 (l item))) (projT2 (l item)),
   existT (projT1 (l item))
     (existing_equivocator_label_extract
        (IM (projT1 (l item))) (projT2 (l item)) Hex) =
   l
     {|
       l := existT (projT1 (l item)) (l itemi);
       input := input item;
       destination :=
         equivocators_state_project IM eqv_descriptors
           (destination item);
       output := output item
     |})
∧ input item =
  input
    {|
      l := existT (projT1 (l item)) (l itemi);
      input := input item;
      destination :=
        equivocators_state_project IM eqv_descriptors
          (destination item);
      output := output item
    |}
  ∧ output item =
    output
      {|
        l := existT (projT1 (l item)) (l itemi);
        input := input item;
        destination :=
          equivocators_state_project IM
            eqv_descriptors (destination item);
        output := output item
      |}
    ∧ equivocators_state_project IM eqv_descriptors
        (destination item) =
      destination
        {|
          l := existT (projT1 (l item)) (l itemi);
          input := input item;
          destination :=
            equivocators_state_project IM
              eqv_descriptors (destination item);
          output := output item
        |} subst.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
eqv_descriptors: equivocator_descriptors IM
item: composite_transition_item (equivocator_IM IM)
itemi: transition_item
descriptori: MachineDescriptor (IM (projT1 (l item)))
Hpr_itemi: equivocator_vlsm_transition_item_project
  (IM (projT1 (l item)))
  (composite_transition_item_projection
     (equivocator_IM IM) item)
  (eqv_descriptors (projT1 (l item))) =
Some (Some itemi, descriptori)
Hpr_item: Some
  (Some
     {|
       l :=
         existT (projT1 (l item)) (l itemi);
       input := input item;
       destination :=
         equivocators_state_project IM
           eqv_descriptors
           (destination item);
       output := output item
     |},
   equivocator_descriptors_update IM
     eqv_descriptors (projT1 (l item))
     descriptori) =
Some
  (Some
     {|
       l :=
         existT (projT1 (l item)) (l itemi);
       input := input item;
       destination :=
         equivocators_state_project IM
           eqv_descriptors
           (destination item);
       output := output item
     |},
   equivocator_descriptors_update IM
     eqv_descriptors (projT1 (l item))
     descriptori)
(∃ Hex : existing_equivocator_label
           (IM (projT1 (l item))) (projT2 (l item)),
   existT (projT1 (l item))
     (existing_equivocator_label_extract
        (IM (projT1 (l item))) (projT2 (l item)) Hex) =
   l
     {|
       l := existT (projT1 (l item)) (l itemi);
       input := input item;
       destination :=
         equivocators_state_project IM eqv_descriptors
           (destination item);
       output := output item
     |})
∧ input item =
  input
    {|
      l := existT (projT1 (l item)) (l itemi);
      input := input item;
      destination :=
        equivocators_state_project IM eqv_descriptors
          (destination item);
      output := output item
    |}
  ∧ output item =
    output
      {|
        l := existT (projT1 (l item)) (l itemi);
        input := input item;
        destination :=
          equivocators_state_project IM
            eqv_descriptors (destination item);
        output := output item
      |}
    ∧ equivocators_state_project IM eqv_descriptors
        (destination item) =
      destination
        {|
          l := existT (projT1 (l item)) (l itemi);
          input := input item;
          destination :=
            equivocators_state_project IM
              eqv_descriptors (destination item);
          output := output item
        |} clear Hpr_item.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
eqv_descriptors: equivocator_descriptors IM
item: composite_transition_item (equivocator_IM IM)
itemi: transition_item
descriptori: MachineDescriptor (IM (projT1 (l item)))
Hpr_itemi: equivocator_vlsm_transition_item_project
  (IM (projT1 (l item)))
  (composite_transition_item_projection
     (equivocator_IM IM) item)
  (eqv_descriptors (projT1 (l item))) =
Some (Some itemi, descriptori)
(∃ Hex : existing_equivocator_label
           (IM (projT1 (l item))) (projT2 (l item)),
   existT (projT1 (l item))
     (existing_equivocator_label_extract
        (IM (projT1 (l item))) (projT2 (l item)) Hex) =
   l
     {|
       l := existT (projT1 (l item)) (l itemi);
       input := input item;
       destination :=
         equivocators_state_project IM eqv_descriptors
           (destination item);
       output := output item
     |})
∧ input item =
  input
    {|
      l := existT (projT1 (l item)) (l itemi);
      input := input item;
      destination :=
        equivocators_state_project IM eqv_descriptors
          (destination item);
      output := output item
    |}
  ∧ output item =
    output
      {|
        l := existT (projT1 (l item)) (l itemi);
        input := input item;
        destination :=
          equivocators_state_project IM
            eqv_descriptors (destination item);
        output := output item
      |}
    ∧ equivocators_state_project IM eqv_descriptors
        (destination item) =
      destination
        {|
          l := existT (projT1 (l item)) (l itemi);
          input := input item;
          destination :=
            equivocators_state_project IM
              eqv_descriptors (destination item);
          output := output item
        |} simpl.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
eqv_descriptors: equivocator_descriptors IM
item: composite_transition_item (equivocator_IM IM)
itemi: transition_item
descriptori: MachineDescriptor (IM (projT1 (l item)))
Hpr_itemi: equivocator_vlsm_transition_item_project
  (IM (projT1 (l item)))
  (composite_transition_item_projection
     (equivocator_IM IM) item)
  (eqv_descriptors (projT1 (l item))) =
Some (Some itemi, descriptori)
(∃ Hex : existing_equivocator_label
           (IM (projT1 (l item))) (projT2 (l item)),
   existT (projT1 (l item))
     (existing_equivocator_label_extract
        (IM (projT1 (l item))) (projT2 (l item)) Hex) =
   existT (projT1 (l item)) (l itemi))
∧ input item = input item
  ∧ output item = output item
    ∧ equivocators_state_project IM eqv_descriptors
        (destination item) =
      equivocators_state_project IM eqv_descriptors
        (destination item)
  repeat split.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
eqv_descriptors: equivocator_descriptors IM
item: composite_transition_item (equivocator_IM IM)
itemi: transition_item
descriptori: MachineDescriptor (IM (projT1 (l item)))
Hpr_itemi: equivocator_vlsm_transition_item_project
  (IM (projT1 (l item)))
  (composite_transition_item_projection
     (equivocator_IM IM) item)
  (eqv_descriptors (projT1 (l item))) =
Some (Some itemi, descriptori)
∃ Hex : existing_equivocator_label
          (IM (projT1 (l item))) (projT2 (l item)),
  existT (projT1 (l item))
    (existing_equivocator_label_extract
       (IM (projT1 (l item))) (projT2 (l item)) Hex) =
  existT (projT1 (l item)) (l itemi)
  apply equivocator_transition_item_project_inv_characterization in Hpr_itemi
    as [[Hex Hl]].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
eqv_descriptors: equivocator_descriptors IM
item: composite_transition_item (equivocator_IM IM)
itemi: transition_item
descriptori: MachineDescriptor (IM (projT1 (l item)))
Hex: existing_equivocator_label
  (IM (projT1 (l item)))
  (l
     (composite_transition_item_projection
        (equivocator_IM IM) item))
Hl: l itemi =
existing_equivocator_label_extract
  (IM (projT1 (l item)))
  (l
     (composite_transition_item_projection
        (equivocator_IM IM) item)) Hex
H1: input
  (composite_transition_item_projection
     (equivocator_IM IM) item) = 
input itemi
∧ output
    (composite_transition_item_projection
       (equivocator_IM IM) item) = 
  output itemi
  ∧ equivocator_state_descriptor_project
      (destination
         (composite_transition_item_projection
            (equivocator_IM IM) item))
      (eqv_descriptors (projT1 (l item))) =
    destination itemi
    ∧ descriptori =
      equivocator_label_descriptor
        (l
           (composite_transition_item_projection
              (equivocator_IM IM) item))
∃ Hex : existing_equivocator_label
          (IM (projT1 (l item))) (projT2 (l item)),
  existT (projT1 (l item))
    (existing_equivocator_label_extract
       (IM (projT1 (l item))) (projT2 (l item)) Hex) =
  existT (projT1 (l item)) (l itemi)
  exists Hex.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
eqv_descriptors: equivocator_descriptors IM
item: composite_transition_item (equivocator_IM IM)
itemi: transition_item
descriptori: MachineDescriptor (IM (projT1 (l item)))
Hex: existing_equivocator_label
  (IM (projT1 (l item)))
  (l
     (composite_transition_item_projection
        (equivocator_IM IM) item))
Hl: l itemi =
existing_equivocator_label_extract
  (IM (projT1 (l item)))
  (l
     (composite_transition_item_projection
        (equivocator_IM IM) item)) Hex
H1: input
  (composite_transition_item_projection
     (equivocator_IM IM) item) = 
input itemi
∧ output
    (composite_transition_item_projection
       (equivocator_IM IM) item) = 
  output itemi
  ∧ equivocator_state_descriptor_project
      (destination
         (composite_transition_item_projection
            (equivocator_IM IM) item))
      (eqv_descriptors (projT1 (l item))) =
    destination itemi
    ∧ descriptori =
      equivocator_label_descriptor
        (l
           (composite_transition_item_projection
              (equivocator_IM IM) item))
existT (projT1 (l item))
  (existing_equivocator_label_extract
     (IM (projT1 (l item))) (projT2 (l item)) Hex) =
existT (projT1 (l item)) (l itemi)
  by f_equal.
Qed.

Definition equivocators_trace_project_folder
  (item : composite_transition_item (equivocator_IM IM))
  (result : option (list (composite_transition_item IM) * equivocator_descriptors IM))
  : option (list (composite_transition_item IM) * equivocator_descriptors IM)
  :=
  match result with
  | None => None
  | Some (r, idescriptor) =>
    match equivocators_transition_item_project idescriptor item with
    | None => None
    | Some (None, odescriptor) => Some (r, odescriptor)
    | Some (Some item', odescriptor) => Some (item' :: r, odescriptor)
    end
  end.

Lemma equivocators_trace_project_fold_None
  (tr : list (composite_transition_item (equivocator_IM IM)))
  : fold_right equivocators_trace_project_folder None tr = None.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
tr: list
  (composite_transition_item (equivocator_IM IM))
foldr equivocators_trace_project_folder None tr = None
Proof.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
tr: list
  (composite_transition_item (equivocator_IM IM))
foldr equivocators_trace_project_folder None tr = None
  by induction tr; cbn; rewrite ?IHtr.
Qed.

Lemma equivocators_trace_project_folder_additive_iff
  (tr : list (composite_transition_item (equivocator_IM IM)))
  (itrX : list (composite_transition_item IM))
  (ieqv_descriptors eqv_descriptors : equivocator_descriptors IM)
  (trX' : list (composite_transition_item IM))
  : fold_right equivocators_trace_project_folder (Some (itrX, ieqv_descriptors)) tr
    = Some (trX', eqv_descriptors)
  <-> exists trX : list (composite_transition_item IM),
    fold_right equivocators_trace_project_folder (Some ([], ieqv_descriptors)) tr
      = Some (trX, eqv_descriptors)
    /\ trX' = trX ++ itrX.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
tr: list
  (composite_transition_item (equivocator_IM IM))
itrX: list (composite_transition_item IM)
ieqv_descriptors, eqv_descriptors: equivocator_descriptors IM
trX': list (composite_transition_item IM)
foldr equivocators_trace_project_folder
  (Some (itrX, ieqv_descriptors)) tr =
Some (trX', eqv_descriptors)
↔ (∃ trX : list (composite_transition_item IM),
     foldr equivocators_trace_project_folder
       (Some ([], ieqv_descriptors)) tr =
     Some (trX, eqv_descriptors) ∧ trX' = trX ++ itrX)
Proof.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
tr: list
  (composite_transition_item (equivocator_IM IM))
itrX: list (composite_transition_item IM)
ieqv_descriptors, eqv_descriptors: equivocator_descriptors IM
trX': list (composite_transition_item IM)
foldr equivocators_trace_project_folder
  (Some (itrX, ieqv_descriptors)) tr =
Some (trX', eqv_descriptors)
↔ (∃ trX : list (composite_transition_item IM),
     foldr equivocators_trace_project_folder
       (Some ([], ieqv_descriptors)) tr =
     Some (trX, eqv_descriptors) ∧ trX' = trX ++ itrX)
  revert trX' eqv_descriptors.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
tr: list
  (composite_transition_item (equivocator_IM IM))
itrX: list (composite_transition_item IM)
ieqv_descriptors: equivocator_descriptors IM
∀ (trX' : list (composite_transition_item IM)) (eqv_descriptors : 
                                                equivocator_descriptors
                                                 IM),
  foldr equivocators_trace_project_folder
    (Some (itrX, ieqv_descriptors)) tr =
  Some (trX', eqv_descriptors)
  ↔ (∃ trX : list (composite_transition_item IM),
       foldr equivocators_trace_project_folder
         (Some ([], ieqv_descriptors)) tr =
       Some (trX, eqv_descriptors)
       ∧ trX' = trX ++ itrX)
  induction tr; intros.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
itrX: list (composite_transition_item IM)
ieqv_descriptors: equivocator_descriptors IM
trX': list (composite_transition_item IM)
eqv_descriptors: equivocator_descriptors IM
foldr equivocators_trace_project_folder
  (Some (itrX, ieqv_descriptors)) [] =
Some (trX', eqv_descriptors)
↔ (∃ trX : list (composite_transition_item IM),
     foldr equivocators_trace_project_folder
       (Some ([], ieqv_descriptors)) [] =
     Some (trX, eqv_descriptors) ∧ trX' = trX ++ itrX)
message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
a: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
itrX: list (composite_transition_item IM)
ieqv_descriptors: equivocator_descriptors IM
IHtr: ∀ (trX' : list (composite_transition_item IM)) 
  (eqv_descriptors : equivocator_descriptors IM),
  foldr equivocators_trace_project_folder
    (Some (itrX, ieqv_descriptors)) tr =
  Some (trX', eqv_descriptors)
  ↔ (∃ trX : list
               (composite_transition_item IM),
       foldr equivocators_trace_project_folder
         (Some ([], ieqv_descriptors)) tr =
       Some (trX, eqv_descriptors)
       ∧ trX' = trX ++ itrX)
trX': list (composite_transition_item IM)
eqv_descriptors: equivocator_descriptors IM
foldr equivocators_trace_project_folder
  (Some (itrX, ieqv_descriptors)) 
  (a :: tr) = Some (trX', eqv_descriptors)
↔ (∃ trX : list (composite_transition_item IM),
     foldr equivocators_trace_project_folder
       (Some ([], ieqv_descriptors)) 
       (a :: tr) = Some (trX, eqv_descriptors)
     ∧ trX' = trX ++ itrX)
  -message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
itrX: list (composite_transition_item IM)
ieqv_descriptors: equivocator_descriptors IM
trX': list (composite_transition_item IM)
eqv_descriptors: equivocator_descriptors IM
foldr equivocators_trace_project_folder
  (Some (itrX, ieqv_descriptors)) [] =
Some (trX', eqv_descriptors)
↔ (∃ trX : list (composite_transition_item IM),
     foldr equivocators_trace_project_folder
       (Some ([], ieqv_descriptors)) [] =
     Some (trX, eqv_descriptors) ∧ trX' = trX ++ itrX) simpl.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
itrX: list (composite_transition_item IM)
ieqv_descriptors: equivocator_descriptors IM
trX': list (composite_transition_item IM)
eqv_descriptors: equivocator_descriptors IM
Some (itrX, ieqv_descriptors) =
Some (trX', eqv_descriptors)
↔ (∃ trX : list (composite_transition_item IM),
     Some ([], ieqv_descriptors) =
     Some (trX, eqv_descriptors) ∧ trX' = trX ++ itrX) split; intro Htr.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
itrX: list (composite_transition_item IM)
ieqv_descriptors: equivocator_descriptors IM
trX': list (composite_transition_item IM)
eqv_descriptors: equivocator_descriptors IM
Htr: Some (itrX, ieqv_descriptors) =
Some (trX', eqv_descriptors)
∃ trX : list (composite_transition_item IM),
  Some ([], ieqv_descriptors) =
  Some (trX, eqv_descriptors) ∧ trX' = trX ++ itrX
message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
itrX: list (composite_transition_item IM)
ieqv_descriptors: equivocator_descriptors IM
trX': list (composite_transition_item IM)
eqv_descriptors: equivocator_descriptors IM
Htr: ∃ trX : list (composite_transition_item IM),
  Some ([], ieqv_descriptors) =
  Some (trX, eqv_descriptors)
  ∧ trX' = trX ++ itrX
Some (itrX, ieqv_descriptors) =
Some (trX', eqv_descriptors)
    +message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
itrX: list (composite_transition_item IM)
ieqv_descriptors: equivocator_descriptors IM
trX': list (composite_transition_item IM)
eqv_descriptors: equivocator_descriptors IM
Htr: Some (itrX, ieqv_descriptors) =
Some (trX', eqv_descriptors)
∃ trX : list (composite_transition_item IM),
  Some ([], ieqv_descriptors) =
  Some (trX, eqv_descriptors) ∧ trX' = trX ++ itrX by inversion Htr; subst; exists [].
    +message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
itrX: list (composite_transition_item IM)
ieqv_descriptors: equivocator_descriptors IM
trX': list (composite_transition_item IM)
eqv_descriptors: equivocator_descriptors IM
Htr: ∃ trX : list (composite_transition_item IM),
  Some ([], ieqv_descriptors) =
  Some (trX, eqv_descriptors)
  ∧ trX' = trX ++ itrX
Some (itrX, ieqv_descriptors) =
Some (trX', eqv_descriptors) by destruct Htr as [trX [[= <-] ?]]; subst.
  -message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
a: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
itrX: list (composite_transition_item IM)
ieqv_descriptors: equivocator_descriptors IM
IHtr: ∀ (trX' : list (composite_transition_item IM)) 
  (eqv_descriptors : equivocator_descriptors IM),
  foldr equivocators_trace_project_folder
    (Some (itrX, ieqv_descriptors)) tr =
  Some (trX', eqv_descriptors)
  ↔ (∃ trX : list
               (composite_transition_item IM),
       foldr equivocators_trace_project_folder
         (Some ([], ieqv_descriptors)) tr =
       Some (trX, eqv_descriptors)
       ∧ trX' = trX ++ itrX)
trX': list (composite_transition_item IM)
eqv_descriptors: equivocator_descriptors IM
foldr equivocators_trace_project_folder
  (Some (itrX, ieqv_descriptors)) (a :: tr) =
Some (trX', eqv_descriptors)
↔ (∃ trX : list (composite_transition_item IM),
     foldr equivocators_trace_project_folder
       (Some ([], ieqv_descriptors)) (a :: tr) =
     Some (trX, eqv_descriptors) ∧ trX' = trX ++ itrX) simpl.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
a: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
itrX: list (composite_transition_item IM)
ieqv_descriptors: equivocator_descriptors IM
IHtr: ∀ (trX' : list (composite_transition_item IM)) 
  (eqv_descriptors : equivocator_descriptors IM),
  foldr equivocators_trace_project_folder
    (Some (itrX, ieqv_descriptors)) tr =
  Some (trX', eqv_descriptors)
  ↔ (∃ trX : list
               (composite_transition_item IM),
       foldr equivocators_trace_project_folder
         (Some ([], ieqv_descriptors)) tr =
       Some (trX, eqv_descriptors)
       ∧ trX' = trX ++ itrX)
trX': list (composite_transition_item IM)
eqv_descriptors: equivocator_descriptors IM
equivocators_trace_project_folder a
  (foldr equivocators_trace_project_folder
     (Some (itrX, ieqv_descriptors)) tr) =
Some (trX', eqv_descriptors)
↔ (∃ trX : list (composite_transition_item IM),
     equivocators_trace_project_folder a
       (foldr equivocators_trace_project_folder
          (Some ([], ieqv_descriptors)) tr) =
     Some (trX, eqv_descriptors) ∧ trX' = trX ++ itrX)
    remember (fold_right equivocators_trace_project_folder (Some (itrX, ieqv_descriptors)) tr)
      as pr_itrX_tr.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
a: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
itrX: list (composite_transition_item IM)
ieqv_descriptors: equivocator_descriptors IM
pr_itrX_tr: option
  (list (composite_transition_item IM) *
   equivocator_descriptors IM)
Heqpr_itrX_tr: pr_itrX_tr =
foldr
  equivocators_trace_project_folder
  (Some (itrX, ieqv_descriptors)) tr
IHtr: ∀ (trX' : list (composite_transition_item IM)) 
  (eqv_descriptors : equivocator_descriptors IM),
  pr_itrX_tr = Some (trX', eqv_descriptors)
  ↔ (∃ trX : list
               (composite_transition_item IM),
       foldr equivocators_trace_project_folder
         (Some ([], ieqv_descriptors)) tr =
       Some (trX, eqv_descriptors)
       ∧ trX' = trX ++ itrX)
trX': list (composite_transition_item IM)
eqv_descriptors: equivocator_descriptors IM
equivocators_trace_project_folder a pr_itrX_tr =
Some (trX', eqv_descriptors)
↔ (∃ trX : list (composite_transition_item IM),
     equivocators_trace_project_folder a
       (foldr equivocators_trace_project_folder
          (Some ([], ieqv_descriptors)) tr) =
     Some (trX, eqv_descriptors) ∧ trX' = trX ++ itrX)
    remember (fold_right equivocators_trace_project_folder (Some ([], ieqv_descriptors)) tr)
      as pr_tr.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
a: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
itrX: list (composite_transition_item IM)
ieqv_descriptors: equivocator_descriptors IM
pr_itrX_tr: option
  (list (composite_transition_item IM) *
   equivocator_descriptors IM)
Heqpr_itrX_tr: pr_itrX_tr =
foldr
  equivocators_trace_project_folder
  (Some (itrX, ieqv_descriptors)) tr
pr_tr: option
  (list (composite_transition_item IM) *
   equivocator_descriptors IM)
Heqpr_tr: pr_tr =
foldr equivocators_trace_project_folder
  (Some ([], ieqv_descriptors)) tr
IHtr: ∀ (trX' : list (composite_transition_item IM)) 
  (eqv_descriptors : equivocator_descriptors IM),
  pr_itrX_tr = Some (trX', eqv_descriptors)
  ↔ (∃ trX : list
               (composite_transition_item IM),
       pr_tr = Some (trX, eqv_descriptors)
       ∧ trX' = trX ++ itrX)
trX': list (composite_transition_item IM)
eqv_descriptors: equivocator_descriptors IM
equivocators_trace_project_folder a pr_itrX_tr =
Some (trX', eqv_descriptors)
↔ (∃ trX : list (composite_transition_item IM),
     equivocators_trace_project_folder a pr_tr =
     Some (trX, eqv_descriptors) ∧ trX' = trX ++ itrX)
    split.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
a: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
itrX: list (composite_transition_item IM)
ieqv_descriptors: equivocator_descriptors IM
pr_itrX_tr: option
  (list (composite_transition_item IM) *
   equivocator_descriptors IM)
Heqpr_itrX_tr: pr_itrX_tr =
foldr
  equivocators_trace_project_folder
  (Some (itrX, ieqv_descriptors)) tr
pr_tr: option
  (list (composite_transition_item IM) *
   equivocator_descriptors IM)
Heqpr_tr: pr_tr =
foldr equivocators_trace_project_folder
  (Some ([], ieqv_descriptors)) tr
IHtr: ∀ (trX' : list (composite_transition_item IM)) 
  (eqv_descriptors : equivocator_descriptors IM),
  pr_itrX_tr = Some (trX', eqv_descriptors)
  ↔ (∃ trX : list
               (composite_transition_item IM),
       pr_tr = Some (trX, eqv_descriptors)
       ∧ trX' = trX ++ itrX)
trX': list (composite_transition_item IM)
eqv_descriptors: equivocator_descriptors IM
equivocators_trace_project_folder a pr_itrX_tr =
Some (trX', eqv_descriptors)
→ ∃ trX : list (composite_transition_item IM),
    equivocators_trace_project_folder a pr_tr =
    Some (trX, eqv_descriptors) ∧ trX' = trX ++ itrX
message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
a: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
itrX: list (composite_transition_item IM)
ieqv_descriptors: equivocator_descriptors IM
pr_itrX_tr: option
  (list (composite_transition_item IM) *
   equivocator_descriptors IM)
Heqpr_itrX_tr: pr_itrX_tr =
foldr
  equivocators_trace_project_folder
  (Some (itrX, ieqv_descriptors)) tr
pr_tr: option
  (list (composite_transition_item IM) *
   equivocator_descriptors IM)
Heqpr_tr: pr_tr =
foldr equivocators_trace_project_folder
  (Some ([], ieqv_descriptors)) tr
IHtr: ∀ (trX' : list (composite_transition_item IM)) 
  (eqv_descriptors : equivocator_descriptors IM),
  pr_itrX_tr = Some (trX', eqv_descriptors)
  ↔ (∃ trX : list
               (composite_transition_item IM),
       pr_tr = Some (trX, eqv_descriptors)
       ∧ trX' = trX ++ itrX)
trX': list (composite_transition_item IM)
eqv_descriptors: equivocator_descriptors IM
(∃ trX : list (composite_transition_item IM),
   equivocators_trace_project_folder a pr_tr =
   Some (trX, eqv_descriptors) ∧ 
   trX' = trX ++ itrX)
→ equivocators_trace_project_folder a pr_itrX_tr =
  Some (trX', eqv_descriptors)
    +message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
a: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
itrX: list (composite_transition_item IM)
ieqv_descriptors: equivocator_descriptors IM
pr_itrX_tr: option
  (list (composite_transition_item IM) *
   equivocator_descriptors IM)
Heqpr_itrX_tr: pr_itrX_tr =
foldr
  equivocators_trace_project_folder
  (Some (itrX, ieqv_descriptors)) tr
pr_tr: option
  (list (composite_transition_item IM) *
   equivocator_descriptors IM)
Heqpr_tr: pr_tr =
foldr equivocators_trace_project_folder
  (Some ([], ieqv_descriptors)) tr
IHtr: ∀ (trX' : list (composite_transition_item IM)) 
  (eqv_descriptors : equivocator_descriptors IM),
  pr_itrX_tr = Some (trX', eqv_descriptors)
  ↔ (∃ trX : list
               (composite_transition_item IM),
       pr_tr = Some (trX, eqv_descriptors)
       ∧ trX' = trX ++ itrX)
trX': list (composite_transition_item IM)
eqv_descriptors: equivocator_descriptors IM
equivocators_trace_project_folder a pr_itrX_tr =
Some (trX', eqv_descriptors)
→ ∃ trX : list (composite_transition_item IM),
    equivocators_trace_project_folder a pr_tr =
    Some (trX, eqv_descriptors) ∧ trX' = trX ++ itrX intro Htr.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
a: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
itrX: list (composite_transition_item IM)
ieqv_descriptors: equivocator_descriptors IM
pr_itrX_tr: option
  (list (composite_transition_item IM) *
   equivocator_descriptors IM)
Heqpr_itrX_tr: pr_itrX_tr =
foldr
  equivocators_trace_project_folder
  (Some (itrX, ieqv_descriptors)) tr
pr_tr: option
  (list (composite_transition_item IM) *
   equivocator_descriptors IM)
Heqpr_tr: pr_tr =
foldr equivocators_trace_project_folder
  (Some ([], ieqv_descriptors)) tr
IHtr: ∀ (trX' : list (composite_transition_item IM)) 
  (eqv_descriptors : equivocator_descriptors IM),
  pr_itrX_tr = Some (trX', eqv_descriptors)
  ↔ (∃ trX : list
               (composite_transition_item IM),
       pr_tr = Some (trX, eqv_descriptors)
       ∧ trX' = trX ++ itrX)
trX': list (composite_transition_item IM)
eqv_descriptors: equivocator_descriptors IM
Htr: equivocators_trace_project_folder a pr_itrX_tr =
Some (trX', eqv_descriptors)
∃ trX : list (composite_transition_item IM),
  equivocators_trace_project_folder a pr_tr =
  Some (trX, eqv_descriptors) ∧ trX' = trX ++ itrX
      destruct pr_itrX_tr as [(tr1, e1) |] ; [| by inversion Htr].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
a: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
itrX: list (composite_transition_item IM)
ieqv_descriptors: equivocator_descriptors IM
tr1: list (composite_transition_item IM)
e1: equivocator_descriptors IM
Heqpr_itrX_tr: Some (tr1, e1) =
foldr
  equivocators_trace_project_folder
  (Some (itrX, ieqv_descriptors)) tr
pr_tr: option
  (list (composite_transition_item IM) *
   equivocator_descriptors IM)
Heqpr_tr: pr_tr =
foldr equivocators_trace_project_folder
  (Some ([], ieqv_descriptors)) tr
IHtr: ∀ (trX' : list (composite_transition_item IM)) 
  (eqv_descriptors : equivocator_descriptors IM),
  Some (tr1, e1) = Some (trX', eqv_descriptors)
  ↔ (∃ trX : list
               (composite_transition_item IM),
       pr_tr = Some (trX, eqv_descriptors)
       ∧ trX' = trX ++ itrX)
trX': list (composite_transition_item IM)
eqv_descriptors: equivocator_descriptors IM
Htr: equivocators_trace_project_folder a
  (Some (tr1, e1)) =
Some (trX', eqv_descriptors)
∃ trX : list (composite_transition_item IM),
  equivocators_trace_project_folder a pr_tr =
  Some (trX, eqv_descriptors) ∧ trX' = trX ++ itrX
      specialize (IHtr tr1 e1).message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
a: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
itrX: list (composite_transition_item IM)
ieqv_descriptors: equivocator_descriptors IM
tr1: list (composite_transition_item IM)
e1: equivocator_descriptors IM
Heqpr_itrX_tr: Some (tr1, e1) =
foldr
  equivocators_trace_project_folder
  (Some (itrX, ieqv_descriptors)) tr
pr_tr: option
  (list (composite_transition_item IM) *
   equivocator_descriptors IM)
Heqpr_tr: pr_tr =
foldr equivocators_trace_project_folder
  (Some ([], ieqv_descriptors)) tr
IHtr: Some (tr1, e1) = Some (tr1, e1)
↔ (∃ trX : list (composite_transition_item IM),
     pr_tr = Some (trX, e1) ∧ tr1 = trX ++ itrX)
trX': list (composite_transition_item IM)
eqv_descriptors: equivocator_descriptors IM
Htr: equivocators_trace_project_folder a
  (Some (tr1, e1)) =
Some (trX', eqv_descriptors)
∃ trX : list (composite_transition_item IM),
  equivocators_trace_project_folder a pr_tr =
  Some (trX, eqv_descriptors) ∧ trX' = trX ++ itrX apply proj1 in IHtr.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
a: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
itrX: list (composite_transition_item IM)
ieqv_descriptors: equivocator_descriptors IM
tr1: list (composite_transition_item IM)
e1: equivocator_descriptors IM
Heqpr_itrX_tr: Some (tr1, e1) =
foldr
  equivocators_trace_project_folder
  (Some (itrX, ieqv_descriptors)) tr
pr_tr: option
  (list (composite_transition_item IM) *
   equivocator_descriptors IM)
Heqpr_tr: pr_tr =
foldr equivocators_trace_project_folder
  (Some ([], ieqv_descriptors)) tr
IHtr: Some (tr1, e1) = Some (tr1, e1)
→ ∃ trX : list (composite_transition_item IM),
    pr_tr = Some (trX, e1) ∧ tr1 = trX ++ itrX
trX': list (composite_transition_item IM)
eqv_descriptors: equivocator_descriptors IM
Htr: equivocators_trace_project_folder a
  (Some (tr1, e1)) =
Some (trX', eqv_descriptors)
∃ trX : list (composite_transition_item IM),
  equivocators_trace_project_folder a pr_tr =
  Some (trX, eqv_descriptors) ∧ trX' = trX ++ itrX specialize (IHtr eq_refl).message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
a: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
itrX: list (composite_transition_item IM)
ieqv_descriptors: equivocator_descriptors IM
tr1: list (composite_transition_item IM)
e1: equivocator_descriptors IM
Heqpr_itrX_tr: Some (tr1, e1) =
foldr
  equivocators_trace_project_folder
  (Some (itrX, ieqv_descriptors)) tr
pr_tr: option
  (list (composite_transition_item IM) *
   equivocator_descriptors IM)
Heqpr_tr: pr_tr =
foldr equivocators_trace_project_folder
  (Some ([], ieqv_descriptors)) tr
IHtr: ∃ trX : list (composite_transition_item IM),
  pr_tr = Some (trX, e1) ∧ tr1 = trX ++ itrX
trX': list (composite_transition_item IM)
eqv_descriptors: equivocator_descriptors IM
Htr: equivocators_trace_project_folder a
  (Some (tr1, e1)) =
Some (trX', eqv_descriptors)
∃ trX : list (composite_transition_item IM),
  equivocators_trace_project_folder a pr_tr =
  Some (trX, eqv_descriptors) ∧ trX' = trX ++ itrX
      destruct IHtr as [trX [Hpr_tr Htr1]].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
a: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
itrX: list (composite_transition_item IM)
ieqv_descriptors: equivocator_descriptors IM
tr1: list (composite_transition_item IM)
e1: equivocator_descriptors IM
Heqpr_itrX_tr: Some (tr1, e1) =
foldr
  equivocators_trace_project_folder
  (Some (itrX, ieqv_descriptors)) tr
pr_tr: option
  (list (composite_transition_item IM) *
   equivocator_descriptors IM)
Heqpr_tr: pr_tr =
foldr equivocators_trace_project_folder
  (Some ([], ieqv_descriptors)) tr
trX: list (composite_transition_item IM)
Hpr_tr: pr_tr = Some (trX, e1)
Htr1: tr1 = trX ++ itrX
trX': list (composite_transition_item IM)
eqv_descriptors: equivocator_descriptors IM
Htr: equivocators_trace_project_folder a
  (Some (tr1, e1)) =
Some (trX', eqv_descriptors)
∃ trX : list (composite_transition_item IM),
  equivocators_trace_project_folder a pr_tr =
  Some (trX, eqv_descriptors) ∧ trX' = trX ++ itrX
      rewrite Hpr_tr in *.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
a: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
itrX: list (composite_transition_item IM)
ieqv_descriptors: equivocator_descriptors IM
tr1: list (composite_transition_item IM)
e1: equivocator_descriptors IM
Heqpr_itrX_tr: Some (tr1, e1) =
foldr
  equivocators_trace_project_folder
  (Some (itrX, ieqv_descriptors)) tr
pr_tr: option
  (list (composite_transition_item IM) *
   equivocator_descriptors IM)
trX: list (composite_transition_item IM)
Heqpr_tr: Some (trX, e1) =
foldr equivocators_trace_project_folder
  (Some ([], ieqv_descriptors)) tr
Hpr_tr: pr_tr = Some (trX, e1)
Htr1: tr1 = trX ++ itrX
trX': list (composite_transition_item IM)
eqv_descriptors: equivocator_descriptors IM
Htr: equivocators_trace_project_folder a
  (Some (tr1, e1)) =
Some (trX', eqv_descriptors)
∃ trX0 : list (composite_transition_item IM),
  equivocators_trace_project_folder a (Some (trX, e1)) =
  Some (trX0, eqv_descriptors) ∧ trX' = trX0 ++ itrX rewrite Htr1 in *.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
a: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
itrX: list (composite_transition_item IM)
ieqv_descriptors: equivocator_descriptors IM
tr1: list (composite_transition_item IM)
e1: equivocator_descriptors IM
trX: list (composite_transition_item IM)
Heqpr_itrX_tr: Some (trX ++ itrX, e1) =
foldr
  equivocators_trace_project_folder
  (Some (itrX, ieqv_descriptors)) tr
pr_tr: option
  (list (composite_transition_item IM) *
   equivocator_descriptors IM)
Heqpr_tr: Some (trX, e1) =
foldr equivocators_trace_project_folder
  (Some ([], ieqv_descriptors)) tr
Hpr_tr: pr_tr = Some (trX, e1)
Htr1: tr1 = trX ++ itrX
trX': list (composite_transition_item IM)
eqv_descriptors: equivocator_descriptors IM
Htr: equivocators_trace_project_folder a
  (Some (trX ++ itrX, e1)) =
Some (trX', eqv_descriptors)
∃ trX0 : list (composite_transition_item IM),
  equivocators_trace_project_folder a (Some (trX, e1)) =
  Some (trX0, eqv_descriptors) ∧ trX' = trX0 ++ itrX
      simpl in Htr.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
a: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
itrX: list (composite_transition_item IM)
ieqv_descriptors: equivocator_descriptors IM
tr1: list (composite_transition_item IM)
e1: equivocator_descriptors IM
trX: list (composite_transition_item IM)
Heqpr_itrX_tr: Some (trX ++ itrX, e1) =
foldr
  equivocators_trace_project_folder
  (Some (itrX, ieqv_descriptors)) tr
pr_tr: option
  (list (composite_transition_item IM) *
   equivocator_descriptors IM)
Heqpr_tr: Some (trX, e1) =
foldr equivocators_trace_project_folder
  (Some ([], ieqv_descriptors)) tr
Hpr_tr: pr_tr = Some (trX, e1)
Htr1: tr1 = trX ++ itrX
trX': list (composite_transition_item IM)
eqv_descriptors: equivocator_descriptors IM
Htr: match
  equivocators_transition_item_project e1 a
with
| Some (Some item', odescriptor) =>
    Some (item' :: trX ++ itrX, odescriptor)
| Some (None, odescriptor) =>
    Some (trX ++ itrX, odescriptor)
| None => None
end = Some (trX', eqv_descriptors)
∃ trX0 : list (composite_transition_item IM),
  equivocators_trace_project_folder a (Some (trX, e1)) =
  Some (trX0, eqv_descriptors) ∧ trX' = trX0 ++ itrX simpl.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
a: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
itrX: list (composite_transition_item IM)
ieqv_descriptors: equivocator_descriptors IM
tr1: list (composite_transition_item IM)
e1: equivocator_descriptors IM
trX: list (composite_transition_item IM)
Heqpr_itrX_tr: Some (trX ++ itrX, e1) =
foldr
  equivocators_trace_project_folder
  (Some (itrX, ieqv_descriptors)) tr
pr_tr: option
  (list (composite_transition_item IM) *
   equivocator_descriptors IM)
Heqpr_tr: Some (trX, e1) =
foldr equivocators_trace_project_folder
  (Some ([], ieqv_descriptors)) tr
Hpr_tr: pr_tr = Some (trX, e1)
Htr1: tr1 = trX ++ itrX
trX': list (composite_transition_item IM)
eqv_descriptors: equivocator_descriptors IM
Htr: match
  equivocators_transition_item_project e1 a
with
| Some (Some item', odescriptor) =>
    Some (item' :: trX ++ itrX, odescriptor)
| Some (None, odescriptor) =>
    Some (trX ++ itrX, odescriptor)
| None => None
end = Some (trX', eqv_descriptors)
∃ trX0 : list (composite_transition_item IM),
  match equivocators_transition_item_project e1 a with
  | Some (Some item', odescriptor) =>
      Some (item' :: trX, odescriptor)
  | Some (None, odescriptor) =>
      Some (trX, odescriptor)
  | None => None
  end = Some (trX0, eqv_descriptors)
  ∧ trX' = trX0 ++ itrX
      destruct (equivocators_transition_item_project e1 a)
        as [(oitem, eqv_descriptors'') |] eqn: Ha; [| by congruence].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
a: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
itrX: list (composite_transition_item IM)
ieqv_descriptors: equivocator_descriptors IM
tr1: list (composite_transition_item IM)
e1: equivocator_descriptors IM
trX: list (composite_transition_item IM)
Heqpr_itrX_tr: Some (trX ++ itrX, e1) =
foldr
  equivocators_trace_project_folder
  (Some (itrX, ieqv_descriptors)) tr
pr_tr: option
  (list (composite_transition_item IM) *
   equivocator_descriptors IM)
Heqpr_tr: Some (trX, e1) =
foldr equivocators_trace_project_folder
  (Some ([], ieqv_descriptors)) tr
Hpr_tr: pr_tr = Some (trX, e1)
Htr1: tr1 = trX ++ itrX
trX': list (composite_transition_item IM)
eqv_descriptors: equivocator_descriptors IM
oitem: option (composite_transition_item IM)
eqv_descriptors'': equivocator_descriptors IM
Ha: equivocators_transition_item_project e1 a =
Some (oitem, eqv_descriptors'')
Htr: match oitem with
| Some item' =>
    Some
      (item' :: trX ++ itrX, eqv_descriptors'')
| None => Some (trX ++ itrX, eqv_descriptors'')
end = Some (trX', eqv_descriptors)
∃ trX0 : list (composite_transition_item IM),
  match oitem with
  | Some item' =>
      Some (item' :: trX, eqv_descriptors'')
  | None => Some (trX, eqv_descriptors'')
  end = Some (trX0, eqv_descriptors)
  ∧ trX' = trX0 ++ itrX
      by destruct oitem; inversion Htr; eexists _.
    +message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
a: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
itrX: list (composite_transition_item IM)
ieqv_descriptors: equivocator_descriptors IM
pr_itrX_tr: option
  (list (composite_transition_item IM) *
   equivocator_descriptors IM)
Heqpr_itrX_tr: pr_itrX_tr =
foldr
  equivocators_trace_project_folder
  (Some (itrX, ieqv_descriptors)) tr
pr_tr: option
  (list (composite_transition_item IM) *
   equivocator_descriptors IM)
Heqpr_tr: pr_tr =
foldr equivocators_trace_project_folder
  (Some ([], ieqv_descriptors)) tr
IHtr: ∀ (trX' : list (composite_transition_item IM)) 
  (eqv_descriptors : equivocator_descriptors IM),
  pr_itrX_tr = Some (trX', eqv_descriptors)
  ↔ (∃ trX : list
               (composite_transition_item IM),
       pr_tr = Some (trX, eqv_descriptors)
       ∧ trX' = trX ++ itrX)
trX': list (composite_transition_item IM)
eqv_descriptors: equivocator_descriptors IM
(∃ trX : list (composite_transition_item IM),
   equivocators_trace_project_folder a pr_tr =
   Some (trX, eqv_descriptors) ∧ trX' = trX ++ itrX)
→ equivocators_trace_project_folder a pr_itrX_tr =
  Some (trX', eqv_descriptors) intros [trX [Htr HtrX']].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
a: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
itrX: list (composite_transition_item IM)
ieqv_descriptors: equivocator_descriptors IM
pr_itrX_tr: option
  (list (composite_transition_item IM) *
   equivocator_descriptors IM)
Heqpr_itrX_tr: pr_itrX_tr =
foldr
  equivocators_trace_project_folder
  (Some (itrX, ieqv_descriptors)) tr
pr_tr: option
  (list (composite_transition_item IM) *
   equivocator_descriptors IM)
Heqpr_tr: pr_tr =
foldr equivocators_trace_project_folder
  (Some ([], ieqv_descriptors)) tr
IHtr: ∀ (trX' : list (composite_transition_item IM)) 
  (eqv_descriptors : equivocator_descriptors IM),
  pr_itrX_tr = Some (trX', eqv_descriptors)
  ↔ (∃ trX : list
               (composite_transition_item IM),
       pr_tr = Some (trX, eqv_descriptors)
       ∧ trX' = trX ++ itrX)
trX': list (composite_transition_item IM)
eqv_descriptors: equivocator_descriptors IM
trX: list (composite_transition_item IM)
Htr: equivocators_trace_project_folder a pr_tr =
Some (trX, eqv_descriptors)
HtrX': trX' = trX ++ itrX
equivocators_trace_project_folder a pr_itrX_tr =
Some (trX', eqv_descriptors)
      subst trX'.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
a: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
itrX: list (composite_transition_item IM)
ieqv_descriptors: equivocator_descriptors IM
pr_itrX_tr: option
  (list (composite_transition_item IM) *
   equivocator_descriptors IM)
Heqpr_itrX_tr: pr_itrX_tr =
foldr
  equivocators_trace_project_folder
  (Some (itrX, ieqv_descriptors)) tr
pr_tr: option
  (list (composite_transition_item IM) *
   equivocator_descriptors IM)
Heqpr_tr: pr_tr =
foldr equivocators_trace_project_folder
  (Some ([], ieqv_descriptors)) tr
IHtr: ∀ (trX' : list (composite_transition_item IM)) 
  (eqv_descriptors : equivocator_descriptors IM),
  pr_itrX_tr = Some (trX', eqv_descriptors)
  ↔ (∃ trX : list
               (composite_transition_item IM),
       pr_tr = Some (trX, eqv_descriptors)
       ∧ trX' = trX ++ itrX)
eqv_descriptors: equivocator_descriptors IM
trX: list (composite_transition_item IM)
Htr: equivocators_trace_project_folder a pr_tr =
Some (trX, eqv_descriptors)
equivocators_trace_project_folder a pr_itrX_tr =
Some (trX ++ itrX, eqv_descriptors)
      destruct pr_tr as [(tr1, e1) |]; [| by inversion Htr].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
a: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
itrX: list (composite_transition_item IM)
ieqv_descriptors: equivocator_descriptors IM
pr_itrX_tr: option
  (list (composite_transition_item IM) *
   equivocator_descriptors IM)
Heqpr_itrX_tr: pr_itrX_tr =
foldr
  equivocators_trace_project_folder
  (Some (itrX, ieqv_descriptors)) tr
tr1: list (composite_transition_item IM)
e1: equivocator_descriptors IM
Heqpr_tr: Some (tr1, e1) =
foldr equivocators_trace_project_folder
  (Some ([], ieqv_descriptors)) tr
IHtr: ∀ (trX' : list (composite_transition_item IM)) 
  (eqv_descriptors : equivocator_descriptors IM),
  pr_itrX_tr = Some (trX', eqv_descriptors)
  ↔ (∃ trX : list
               (composite_transition_item IM),
       Some (tr1, e1) =
       Some (trX, eqv_descriptors)
       ∧ trX' = trX ++ itrX)
eqv_descriptors: equivocator_descriptors IM
trX: list (composite_transition_item IM)
Htr: equivocators_trace_project_folder a
  (Some (tr1, e1)) = Some (trX, eqv_descriptors)
equivocators_trace_project_folder a pr_itrX_tr =
Some (trX ++ itrX, eqv_descriptors)
      specialize (IHtr (tr1 ++ itrX) e1).message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
a: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
itrX: list (composite_transition_item IM)
ieqv_descriptors: equivocator_descriptors IM
pr_itrX_tr: option
  (list (composite_transition_item IM) *
   equivocator_descriptors IM)
Heqpr_itrX_tr: pr_itrX_tr =
foldr
  equivocators_trace_project_folder
  (Some (itrX, ieqv_descriptors)) tr
tr1: list (composite_transition_item IM)
e1: equivocator_descriptors IM
Heqpr_tr: Some (tr1, e1) =
foldr equivocators_trace_project_folder
  (Some ([], ieqv_descriptors)) tr
IHtr: pr_itrX_tr = Some (tr1 ++ itrX, e1)
↔ (∃ trX : list (composite_transition_item IM),
     Some (tr1, e1) = Some (trX, e1)
     ∧ tr1 ++ itrX = trX ++ itrX)
eqv_descriptors: equivocator_descriptors IM
trX: list (composite_transition_item IM)
Htr: equivocators_trace_project_folder a
  (Some (tr1, e1)) = Some (trX, eqv_descriptors)
equivocators_trace_project_folder a pr_itrX_tr =
Some (trX ++ itrX, eqv_descriptors) apply proj2 in IHtr.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
a: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
itrX: list (composite_transition_item IM)
ieqv_descriptors: equivocator_descriptors IM
pr_itrX_tr: option
  (list (composite_transition_item IM) *
   equivocator_descriptors IM)
Heqpr_itrX_tr: pr_itrX_tr =
foldr
  equivocators_trace_project_folder
  (Some (itrX, ieqv_descriptors)) tr
tr1: list (composite_transition_item IM)
e1: equivocator_descriptors IM
Heqpr_tr: Some (tr1, e1) =
foldr equivocators_trace_project_folder
  (Some ([], ieqv_descriptors)) tr
IHtr: (∃ trX : list (composite_transition_item IM),
   Some (tr1, e1) = Some (trX, e1)
   ∧ tr1 ++ itrX = trX ++ itrX)
→ pr_itrX_tr = Some (tr1 ++ itrX, e1)
eqv_descriptors: equivocator_descriptors IM
trX: list (composite_transition_item IM)
Htr: equivocators_trace_project_folder a
  (Some (tr1, e1)) = Some (trX, eqv_descriptors)
equivocators_trace_project_folder a pr_itrX_tr =
Some (trX ++ itrX, eqv_descriptors)
      rewrite IHtr by (eexists _; done).message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
a: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
itrX: list (composite_transition_item IM)
ieqv_descriptors: equivocator_descriptors IM
pr_itrX_tr: option
  (list (composite_transition_item IM) *
   equivocator_descriptors IM)
Heqpr_itrX_tr: pr_itrX_tr =
foldr
  equivocators_trace_project_folder
  (Some (itrX, ieqv_descriptors)) tr
tr1: list (composite_transition_item IM)
e1: equivocator_descriptors IM
Heqpr_tr: Some (tr1, e1) =
foldr equivocators_trace_project_folder
  (Some ([], ieqv_descriptors)) tr
IHtr: (∃ trX : list (composite_transition_item IM),
   Some (tr1, e1) = Some (trX, e1)
   ∧ tr1 ++ itrX = trX ++ itrX)
→ pr_itrX_tr = Some (tr1 ++ itrX, e1)
eqv_descriptors: equivocator_descriptors IM
trX: list (composite_transition_item IM)
Htr: equivocators_trace_project_folder a
  (Some (tr1, e1)) = Some (trX, eqv_descriptors)
equivocators_trace_project_folder a
  (Some (tr1 ++ itrX, e1)) =
Some (trX ++ itrX, eqv_descriptors)
      simpl in *.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
a: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
itrX: list (composite_transition_item IM)
ieqv_descriptors: equivocator_descriptors IM
pr_itrX_tr: option
  (list (composite_transition_item IM) *
   equivocator_descriptors IM)
Heqpr_itrX_tr: pr_itrX_tr =
foldr
  equivocators_trace_project_folder
  (Some (itrX, ieqv_descriptors)) tr
tr1: list (composite_transition_item IM)
e1: equivocator_descriptors IM
Heqpr_tr: Some (tr1, e1) =
foldr equivocators_trace_project_folder
  (Some ([], ieqv_descriptors)) tr
IHtr: (∃ trX : list (composite_transition_item IM),
   Some (tr1, e1) = Some (trX, e1)
   ∧ tr1 ++ itrX = trX ++ itrX)
→ pr_itrX_tr = Some (tr1 ++ itrX, e1)
eqv_descriptors: equivocator_descriptors IM
trX: list (composite_transition_item IM)
Htr: match
  equivocators_transition_item_project e1 a
with
| Some (Some item', odescriptor) =>
    Some (item' :: tr1, odescriptor)
| Some (None, odescriptor) =>
    Some (tr1, odescriptor)
| None => None
end = Some (trX, eqv_descriptors)
match equivocators_transition_item_project e1 a with
| Some (Some item', odescriptor) =>
    Some (item' :: tr1 ++ itrX, odescriptor)
| Some (None, odescriptor) =>
    Some (tr1 ++ itrX, odescriptor)
| None => None
end = Some (trX ++ itrX, eqv_descriptors)
      destruct (equivocators_transition_item_project e1 a)
        as [(oitem, odescriptor) |] eqn: Ha
      ; [| done].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
a: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
itrX: list (composite_transition_item IM)
ieqv_descriptors: equivocator_descriptors IM
pr_itrX_tr: option
  (list (composite_transition_item IM) *
   equivocator_descriptors IM)
Heqpr_itrX_tr: pr_itrX_tr =
foldr
  equivocators_trace_project_folder
  (Some (itrX, ieqv_descriptors)) tr
tr1: list (composite_transition_item IM)
e1: equivocator_descriptors IM
Heqpr_tr: Some (tr1, e1) =
foldr equivocators_trace_project_folder
  (Some ([], ieqv_descriptors)) tr
IHtr: (∃ trX : list (composite_transition_item IM),
   Some (tr1, e1) = Some (trX, e1)
   ∧ tr1 ++ itrX = trX ++ itrX)
→ pr_itrX_tr = Some (tr1 ++ itrX, e1)
eqv_descriptors: equivocator_descriptors IM
trX: list (composite_transition_item IM)
oitem: option (composite_transition_item IM)
odescriptor: equivocator_descriptors IM
Ha: equivocators_transition_item_project e1 a =
Some (oitem, odescriptor)
Htr: match oitem with
| Some item' => Some (item' :: tr1, odescriptor)
| None => Some (tr1, odescriptor)
end = Some (trX, eqv_descriptors)
match oitem with
| Some item' =>
    Some (item' :: tr1 ++ itrX, odescriptor)
| None => Some (tr1 ++ itrX, odescriptor)
end = Some (trX ++ itrX, eqv_descriptors)
      by destruct oitem as [item' |]; inversion Htr.
Qed.

Lemma equivocators_trace_project_folder_additive
  (tr : list (composite_transition_item (equivocator_IM IM)))
  (itrX trX : list (composite_transition_item IM))
  (ieqv_descriptors eqv_descriptors : equivocator_descriptors IM)
  (Htr : fold_right equivocators_trace_project_folder (Some ([], ieqv_descriptors)) tr
    = Some (trX, eqv_descriptors))
  : fold_right equivocators_trace_project_folder (Some (itrX, ieqv_descriptors)) tr
    = Some (trX ++ itrX, eqv_descriptors).message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
tr: list
  (composite_transition_item (equivocator_IM IM))
itrX, trX: list (composite_transition_item IM)
ieqv_descriptors, eqv_descriptors: equivocator_descriptors IM
Htr: foldr equivocators_trace_project_folder
  (Some ([], ieqv_descriptors)) tr =
Some (trX, eqv_descriptors)
foldr equivocators_trace_project_folder
  (Some (itrX, ieqv_descriptors)) tr =
Some (trX ++ itrX, eqv_descriptors)
Proof.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
tr: list
  (composite_transition_item (equivocator_IM IM))
itrX, trX: list (composite_transition_item IM)
ieqv_descriptors, eqv_descriptors: equivocator_descriptors IM
Htr: foldr equivocators_trace_project_folder
  (Some ([], ieqv_descriptors)) tr =
Some (trX, eqv_descriptors)
foldr equivocators_trace_project_folder
  (Some (itrX, ieqv_descriptors)) tr =
Some (trX ++ itrX, eqv_descriptors)
  apply equivocators_trace_project_folder_additive_iff.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
tr: list
  (composite_transition_item (equivocator_IM IM))
itrX, trX: list (composite_transition_item IM)
ieqv_descriptors, eqv_descriptors: equivocator_descriptors IM
Htr: foldr equivocators_trace_project_folder
  (Some ([], ieqv_descriptors)) tr =
Some (trX, eqv_descriptors)
∃ trX0 : list (composite_transition_item IM),
  foldr equivocators_trace_project_folder
    (Some ([], ieqv_descriptors)) tr =
  Some (trX0, eqv_descriptors)
  ∧ trX ++ itrX = trX0 ++ itrX
  by exists trX.
Qed.

The projection of an equivocators trace is obtained by traversing the trace from right to left guided by the descriptors produced by equivocators_transition_item_project and gathering all non-empty transition_items it produces.

Definition equivocators_trace_project
  (eqv_descriptors : equivocator_descriptors IM)
  (tr : list (composite_transition_item (equivocator_IM IM)))
  : option (list (composite_transition_item IM) * equivocator_descriptors IM)
  :=
  fold_right
    equivocators_trace_project_folder
    (Some ([], eqv_descriptors))
    tr.

Lemma equivocators_trace_project_app_iff
  (pre suf : list (composite_transition_item (equivocator_IM IM)))
  (ieqv_descriptors eqv_descriptors : equivocator_descriptors IM)
  (trX : list (composite_transition_item IM))
  : equivocators_trace_project eqv_descriptors (pre ++ suf)
    = Some (trX, ieqv_descriptors)
  <-> exists
    (preX sufX : list (composite_transition_item IM))
    (eqv_descriptors' : equivocator_descriptors IM),
    equivocators_trace_project eqv_descriptors suf = Some (sufX, eqv_descriptors') /\
    equivocators_trace_project eqv_descriptors' pre = Some (preX, ieqv_descriptors) /\
    trX = preX ++ sufX.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
ieqv_descriptors, eqv_descriptors: equivocator_descriptors IM
trX: list (composite_transition_item IM)
equivocators_trace_project eqv_descriptors
  (pre ++ suf) = Some (trX, ieqv_descriptors)
↔ (∃ (preX sufX : list (composite_transition_item IM)) 
     (eqv_descriptors' : equivocator_descriptors IM),
     equivocators_trace_project eqv_descriptors suf =
     Some (sufX, eqv_descriptors')
     ∧ equivocators_trace_project eqv_descriptors' pre =
       Some (preX, ieqv_descriptors)
       ∧ trX = preX ++ sufX)
Proof.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
ieqv_descriptors, eqv_descriptors: equivocator_descriptors IM
trX: list (composite_transition_item IM)
equivocators_trace_project eqv_descriptors
  (pre ++ suf) = Some (trX, ieqv_descriptors)
↔ (∃ (preX sufX : list (composite_transition_item IM)) 
     (eqv_descriptors' : equivocator_descriptors IM),
     equivocators_trace_project eqv_descriptors suf =
     Some (sufX, eqv_descriptors')
     ∧ equivocators_trace_project eqv_descriptors' pre =
       Some (preX, ieqv_descriptors)
       ∧ trX = preX ++ sufX)
  unfold equivocators_trace_project.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
ieqv_descriptors, eqv_descriptors: equivocator_descriptors IM
trX: list (composite_transition_item IM)
foldr equivocators_trace_project_folder
  (Some ([], eqv_descriptors)) (pre ++ suf) =
Some (trX, ieqv_descriptors)
↔ (∃ (preX sufX : list (composite_transition_item IM)) 
     (eqv_descriptors' : equivocator_descriptors IM),
     foldr equivocators_trace_project_folder
       (Some ([], eqv_descriptors)) suf =
     Some (sufX, eqv_descriptors')
     ∧ foldr equivocators_trace_project_folder
         (Some ([], eqv_descriptors')) pre =
       Some (preX, ieqv_descriptors)
       ∧ trX = preX ++ sufX)
  rewrite fold_right_app.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
ieqv_descriptors, eqv_descriptors: equivocator_descriptors IM
trX: list (composite_transition_item IM)
foldr equivocators_trace_project_folder
  (foldr equivocators_trace_project_folder
     (Some ([], eqv_descriptors)) suf) pre =
Some (trX, ieqv_descriptors)
↔ (∃ (preX sufX : list (composite_transition_item IM)) 
     (eqv_descriptors' : equivocator_descriptors IM),
     foldr equivocators_trace_project_folder
       (Some ([], eqv_descriptors)) suf =
     Some (sufX, eqv_descriptors')
     ∧ foldr equivocators_trace_project_folder
         (Some ([], eqv_descriptors')) pre =
       Some (preX, ieqv_descriptors)
       ∧ trX = preX ++ sufX)
  simpl.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
ieqv_descriptors, eqv_descriptors: equivocator_descriptors IM
trX: list (composite_transition_item IM)
foldr equivocators_trace_project_folder
  (foldr equivocators_trace_project_folder
     (Some ([], eqv_descriptors)) suf) pre =
Some (trX, ieqv_descriptors)
↔ (∃ (preX sufX : list (composite_transition_item IM)) 
     (eqv_descriptors' : equivocator_descriptors IM),
     foldr equivocators_trace_project_folder
       (Some ([], eqv_descriptors)) suf =
     Some (sufX, eqv_descriptors')
     ∧ foldr equivocators_trace_project_folder
         (Some ([], eqv_descriptors')) pre =
       Some (preX, ieqv_descriptors)
       ∧ trX = preX ++ sufX)
  match goal with
  |- fold_right _ ?r _ = _ <-> _ => remember r as r_sufX
  end.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
ieqv_descriptors, eqv_descriptors: equivocator_descriptors IM
trX: list (composite_transition_item IM)
r_sufX: option
  (list (composite_transition_item IM) *
   equivocator_descriptors IM)
Heqr_sufX: r_sufX =
foldr equivocators_trace_project_folder
  (Some ([], eqv_descriptors)) suf
foldr equivocators_trace_project_folder r_sufX pre =
Some (trX, ieqv_descriptors)
↔ (∃ (preX sufX : list (composite_transition_item IM)) 
     (eqv_descriptors' : equivocator_descriptors IM),
     r_sufX = Some (sufX, eqv_descriptors')
     ∧ foldr equivocators_trace_project_folder
         (Some ([], eqv_descriptors')) pre =
       Some (preX, ieqv_descriptors)
       ∧ trX = preX ++ sufX)
  destruct r_sufX as [(sufX, eqv_descriptors') |]
  ; [| by rewrite equivocators_trace_project_fold_None; split;
      [intro contra; congruence | intros [preX [sufX [eqv_descriptors' [contra _]]]]; congruence]].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
ieqv_descriptors, eqv_descriptors: equivocator_descriptors IM
trX, sufX: list (composite_transition_item IM)
eqv_descriptors': equivocator_descriptors IM
Heqr_sufX: Some (sufX, eqv_descriptors') =
foldr equivocators_trace_project_folder
  (Some ([], eqv_descriptors)) suf
foldr equivocators_trace_project_folder
  (Some (sufX, eqv_descriptors')) pre =
Some (trX, ieqv_descriptors)
↔ (∃ (preX
      sufX0 : list (composite_transition_item IM)) 
     (eqv_descriptors'0 : equivocator_descriptors IM),
     Some (sufX, eqv_descriptors') =
     Some (sufX0, eqv_descriptors'0)
     ∧ foldr equivocators_trace_project_folder
         (Some ([], eqv_descriptors'0)) pre =
       Some (preX, ieqv_descriptors)
       ∧ trX = preX ++ sufX0)
  rewrite (equivocators_trace_project_folder_additive_iff
    pre sufX eqv_descriptors' ieqv_descriptors trX).message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
ieqv_descriptors, eqv_descriptors: equivocator_descriptors IM
trX, sufX: list (composite_transition_item IM)
eqv_descriptors': equivocator_descriptors IM
Heqr_sufX: Some (sufX, eqv_descriptors') =
foldr equivocators_trace_project_folder
  (Some ([], eqv_descriptors)) suf
(∃ trX0 : list (composite_transition_item IM),
   foldr equivocators_trace_project_folder
     (Some ([], eqv_descriptors')) pre =
   Some (trX0, ieqv_descriptors) ∧ trX = trX0 ++ sufX)
↔ (∃ (preX
      sufX0 : list (composite_transition_item IM)) 
     (eqv_descriptors'0 : equivocator_descriptors IM),
     Some (sufX, eqv_descriptors') =
     Some (sufX0, eqv_descriptors'0)
     ∧ foldr equivocators_trace_project_folder
         (Some ([], eqv_descriptors'0)) pre =
       Some (preX, ieqv_descriptors)
       ∧ trX = preX ++ sufX0)
  split.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
ieqv_descriptors, eqv_descriptors: equivocator_descriptors IM
trX, sufX: list (composite_transition_item IM)
eqv_descriptors': equivocator_descriptors IM
Heqr_sufX: Some (sufX, eqv_descriptors') =
foldr equivocators_trace_project_folder
  (Some ([], eqv_descriptors)) suf
(∃ trX0 : list (composite_transition_item IM),
   foldr equivocators_trace_project_folder
     (Some ([], eqv_descriptors')) pre =
   Some (trX0, ieqv_descriptors) ∧ trX = trX0 ++ sufX)
→ ∃ (preX sufX0 : list (composite_transition_item IM)) 
    (eqv_descriptors'0 : equivocator_descriptors IM),
    Some (sufX, eqv_descriptors') =
    Some (sufX0, eqv_descriptors'0)
    ∧ foldr equivocators_trace_project_folder
        (Some ([], eqv_descriptors'0)) pre =
      Some (preX, ieqv_descriptors)
      ∧ trX = preX ++ sufX0
message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
ieqv_descriptors, eqv_descriptors: equivocator_descriptors IM
trX, sufX: list (composite_transition_item IM)
eqv_descriptors': equivocator_descriptors IM
Heqr_sufX: Some (sufX, eqv_descriptors') =
foldr equivocators_trace_project_folder
  (Some ([], eqv_descriptors)) suf
(∃ (preX sufX0 : list (composite_transition_item IM)) 
   (eqv_descriptors'0 : 
    equivocator_descriptors IM),
   Some (sufX, eqv_descriptors') =
   Some (sufX0, eqv_descriptors'0)
   ∧ foldr equivocators_trace_project_folder
       (Some ([], eqv_descriptors'0)) pre =
     Some (preX, ieqv_descriptors)
     ∧ trX = preX ++ sufX0)
→ ∃ trX0 : list (composite_transition_item IM),
    foldr equivocators_trace_project_folder
      (Some ([], eqv_descriptors')) pre =
    Some (trX0, ieqv_descriptors) ∧ 
    trX = trX0 ++ sufX
  -message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
ieqv_descriptors, eqv_descriptors: equivocator_descriptors IM
trX, sufX: list (composite_transition_item IM)
eqv_descriptors': equivocator_descriptors IM
Heqr_sufX: Some (sufX, eqv_descriptors') =
foldr equivocators_trace_project_folder
  (Some ([], eqv_descriptors)) suf
(∃ trX0 : list (composite_transition_item IM),
   foldr equivocators_trace_project_folder
     (Some ([], eqv_descriptors')) pre =
   Some (trX0, ieqv_descriptors) ∧ trX = trX0 ++ sufX)
→ ∃ (preX sufX0 : list (composite_transition_item IM)) 
    (eqv_descriptors'0 : equivocator_descriptors IM),
    Some (sufX, eqv_descriptors') =
    Some (sufX0, eqv_descriptors'0)
    ∧ foldr equivocators_trace_project_folder
        (Some ([], eqv_descriptors'0)) pre =
      Some (preX, ieqv_descriptors)
      ∧ trX = preX ++ sufX0 by intros (preX & HpreX & HtrX); exists preX, sufX, eqv_descriptors'.
  -message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
ieqv_descriptors, eqv_descriptors: equivocator_descriptors IM
trX, sufX: list (composite_transition_item IM)
eqv_descriptors': equivocator_descriptors IM
Heqr_sufX: Some (sufX, eqv_descriptors') =
foldr equivocators_trace_project_folder
  (Some ([], eqv_descriptors)) suf
(∃ (preX sufX0 : list (composite_transition_item IM)) 
   (eqv_descriptors'0 : equivocator_descriptors IM),
   Some (sufX, eqv_descriptors') =
   Some (sufX0, eqv_descriptors'0)
   ∧ foldr equivocators_trace_project_folder
       (Some ([], eqv_descriptors'0)) pre =
     Some (preX, ieqv_descriptors)
     ∧ trX = preX ++ sufX0)
→ ∃ trX0 : list (composite_transition_item IM),
    foldr equivocators_trace_project_folder
      (Some ([], eqv_descriptors')) pre =
    Some (trX0, ieqv_descriptors) ∧ trX = trX0 ++ sufX intros [preX [_sufX [_eqv_descriptors' [Heq [Hpre HtrX]]]]].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
ieqv_descriptors, eqv_descriptors: equivocator_descriptors IM
trX, sufX: list (composite_transition_item IM)
eqv_descriptors': equivocator_descriptors IM
Heqr_sufX: Some (sufX, eqv_descriptors') =
foldr equivocators_trace_project_folder
  (Some ([], eqv_descriptors)) suf
preX, _sufX: list (composite_transition_item IM)
_eqv_descriptors': equivocator_descriptors IM
Heq: Some (sufX, eqv_descriptors') =
Some (_sufX, _eqv_descriptors')
Hpre: foldr equivocators_trace_project_folder
  (Some ([], _eqv_descriptors')) pre =
Some (preX, ieqv_descriptors)
HtrX: trX = preX ++ _sufX
∃ trX0 : list (composite_transition_item IM),
  foldr equivocators_trace_project_folder
    (Some ([], eqv_descriptors')) pre =
  Some (trX0, ieqv_descriptors) ∧ trX = trX0 ++ sufX
    by exists preX; inversion Heq; subst.
Qed.

For every transition_item of the projection of a trace over the composition of equivocators, there exists a corresponding item in the original trace which projects to it.

Lemma equivocators_trace_project_app_inv_item
  (tr : list (composite_transition_item (equivocator_IM IM)))
  (ieqv_descriptors eqv_descriptors : equivocator_descriptors IM)
  (preX sufX : list (composite_transition_item IM))
  (itemX : composite_transition_item IM)
  : equivocators_trace_project eqv_descriptors tr
    = Some (preX ++ [itemX] ++ sufX, ieqv_descriptors) ->
  exists
    (pre suf : list (composite_transition_item (equivocator_IM IM)))
    (item : (composite_transition_item (equivocator_IM IM)))
    (item_descriptors pre_descriptors : equivocator_descriptors IM),
    equivocators_trace_project eqv_descriptors suf = Some (sufX, item_descriptors) /\
    equivocators_transition_item_project item_descriptors item = Some (Some itemX, pre_descriptors) /\
    equivocators_trace_project pre_descriptors pre = Some (preX, ieqv_descriptors) /\
    tr = pre ++ [item] ++ suf.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
tr: list
  (composite_transition_item (equivocator_IM IM))
ieqv_descriptors, eqv_descriptors: equivocator_descriptors IM
preX, sufX: list (composite_transition_item IM)
itemX: composite_transition_item IM
equivocators_trace_project eqv_descriptors tr =
Some (preX ++ [itemX] ++ sufX, ieqv_descriptors)
→ ∃ (pre
     suf : list
             (composite_transition_item
                (equivocator_IM IM))) (item : composite_transition_item
                                                (equivocator_IM
                                                 IM)) 
    (item_descriptors
     pre_descriptors : equivocator_descriptors IM),
    equivocators_trace_project eqv_descriptors suf =
    Some (sufX, item_descriptors)
    ∧ equivocators_transition_item_project
        item_descriptors item =
      Some (Some itemX, pre_descriptors)
      ∧ equivocators_trace_project pre_descriptors pre =
        Some (preX, ieqv_descriptors)
        ∧ tr = pre ++ [item] ++ suf
Proof.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
tr: list
  (composite_transition_item (equivocator_IM IM))
ieqv_descriptors, eqv_descriptors: equivocator_descriptors IM
preX, sufX: list (composite_transition_item IM)
itemX: composite_transition_item IM
equivocators_trace_project eqv_descriptors tr =
Some (preX ++ [itemX] ++ sufX, ieqv_descriptors)
→ ∃ (pre
     suf : list
             (composite_transition_item
                (equivocator_IM IM))) (item : composite_transition_item
                                                (equivocator_IM
                                                 IM)) 
    (item_descriptors
     pre_descriptors : equivocator_descriptors IM),
    equivocators_trace_project eqv_descriptors suf =
    Some (sufX, item_descriptors)
    ∧ equivocators_transition_item_project
        item_descriptors item =
      Some (Some itemX, pre_descriptors)
      ∧ equivocators_trace_project pre_descriptors pre =
        Some (preX, ieqv_descriptors)
        ∧ tr = pre ++ [item] ++ suf
  generalize dependent sufX.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
tr: list
  (composite_transition_item (equivocator_IM IM))
ieqv_descriptors, eqv_descriptors: equivocator_descriptors IM
preX: list (composite_transition_item IM)
itemX: composite_transition_item IM
∀ sufX : list (composite_transition_item IM),
  equivocators_trace_project eqv_descriptors tr =
  Some (preX ++ [itemX] ++ sufX, ieqv_descriptors)
  → ∃ (pre
       suf : list
               (composite_transition_item
                  (equivocator_IM IM))) (item : composite_transition_item
                                                 (equivocator_IM
                                                 IM)) 
      (item_descriptors
       pre_descriptors : equivocator_descriptors IM),
      equivocators_trace_project eqv_descriptors suf =
      Some (sufX, item_descriptors)
      ∧ equivocators_transition_item_project
          item_descriptors item =
        Some (Some itemX, pre_descriptors)
        ∧ equivocators_trace_project pre_descriptors
            pre = Some (preX, ieqv_descriptors)
          ∧ tr = pre ++ [item] ++ suf generalize dependent eqv_descriptors.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
tr: list
  (composite_transition_item (equivocator_IM IM))
ieqv_descriptors: equivocator_descriptors IM
preX: list (composite_transition_item IM)
itemX: composite_transition_item IM
∀ (eqv_descriptors : equivocator_descriptors IM) (sufX : 
                                                 list
                                                 (composite_transition_item
                                                 IM)),
  equivocators_trace_project eqv_descriptors tr =
  Some (preX ++ [itemX] ++ sufX, ieqv_descriptors)
  → ∃ (pre
       suf : list
               (composite_transition_item
                  (equivocator_IM IM))) (item : composite_transition_item
                                                 (equivocator_IM
                                                 IM)) 
      (item_descriptors
       pre_descriptors : equivocator_descriptors IM),
      equivocators_trace_project eqv_descriptors suf =
      Some (sufX, item_descriptors)
      ∧ equivocators_transition_item_project
          item_descriptors item =
        Some (Some itemX, pre_descriptors)
        ∧ equivocators_trace_project pre_descriptors
            pre = Some (preX, ieqv_descriptors)
          ∧ tr = pre ++ [item] ++ suf
  induction tr using rev_ind; intros eqv_descriptors sufX.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
ieqv_descriptors: equivocator_descriptors IM
preX: list (composite_transition_item IM)
itemX: composite_transition_item IM
eqv_descriptors: equivocator_descriptors IM
sufX: list (composite_transition_item IM)
equivocators_trace_project eqv_descriptors [] =
Some (preX ++ [itemX] ++ sufX, ieqv_descriptors)
→ ∃ (pre
     suf : list
             (composite_transition_item
                (equivocator_IM IM))) (item : composite_transition_item
                                                (equivocator_IM
                                                 IM)) 
    (item_descriptors
     pre_descriptors : equivocator_descriptors IM),
    equivocators_trace_project eqv_descriptors suf =
    Some (sufX, item_descriptors)
    ∧ equivocators_transition_item_project
        item_descriptors item =
      Some (Some itemX, pre_descriptors)
      ∧ equivocators_trace_project pre_descriptors pre =
        Some (preX, ieqv_descriptors)
        ∧ [] = pre ++ [item] ++ suf
message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
x: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
ieqv_descriptors: equivocator_descriptors IM
preX: list (composite_transition_item IM)
itemX: composite_transition_item IM
IHtr: ∀ (eqv_descriptors : equivocator_descriptors IM) 
  (sufX : list (composite_transition_item IM)),
  equivocators_trace_project eqv_descriptors tr =
  Some
    (preX ++ [itemX] ++ sufX, ieqv_descriptors)
  → ∃ (pre
       suf : list
               (composite_transition_item
                  (equivocator_IM IM))) 
      (item : composite_transition_item
                (equivocator_IM IM)) 
      (item_descriptors
       pre_descriptors : 
       equivocator_descriptors IM),
      equivocators_trace_project
        eqv_descriptors suf =
      Some (sufX, item_descriptors)
      ∧ equivocators_transition_item_project
          item_descriptors item =
        Some (Some itemX, pre_descriptors)
        ∧ equivocators_trace_project
            pre_descriptors pre =
          Some (preX, ieqv_descriptors)
          ∧ tr = pre ++ [item] ++ suf
eqv_descriptors: equivocator_descriptors IM
sufX: list (composite_transition_item IM)
equivocators_trace_project eqv_descriptors (tr ++ [x]) =
Some (preX ++ [itemX] ++ sufX, ieqv_descriptors)
→ ∃ (pre
     suf : list
             (composite_transition_item
                (equivocator_IM IM))) 
    (item : composite_transition_item
              (equivocator_IM IM)) 
    (item_descriptors
     pre_descriptors : 
     equivocator_descriptors IM),
    equivocators_trace_project eqv_descriptors suf =
    Some (sufX, item_descriptors)
    ∧ equivocators_transition_item_project
        item_descriptors item =
      Some (Some itemX, pre_descriptors)
      ∧ equivocators_trace_project pre_descriptors pre =
        Some (preX, ieqv_descriptors)
        ∧ tr ++ [x] = pre ++ [item] ++ suf
  -message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
ieqv_descriptors: equivocator_descriptors IM
preX: list (composite_transition_item IM)
itemX: composite_transition_item IM
eqv_descriptors: equivocator_descriptors IM
sufX: list (composite_transition_item IM)
equivocators_trace_project eqv_descriptors [] =
Some (preX ++ [itemX] ++ sufX, ieqv_descriptors)
→ ∃ (pre
     suf : list
             (composite_transition_item
                (equivocator_IM IM))) (item : composite_transition_item
                                                (equivocator_IM
                                                 IM)) 
    (item_descriptors
     pre_descriptors : equivocator_descriptors IM),
    equivocators_trace_project eqv_descriptors suf =
    Some (sufX, item_descriptors)
    ∧ equivocators_transition_item_project
        item_descriptors item =
      Some (Some itemX, pre_descriptors)
      ∧ equivocators_trace_project pre_descriptors pre =
        Some (preX, ieqv_descriptors)
        ∧ [] = pre ++ [item] ++ suf cbn; inversion 1 as [[Hnil Heq]]; clear -Hnil.message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
ieqv_descriptors: equivocator_descriptors IM
preX: list (composite_transition_item IM)
itemX: composite_transition_item IM
sufX: list (composite_transition_item IM)
Hnil: [] = preX ++ itemX :: sufX
∃ (pre
   suf : list
           (composite_transition_item
              (equivocator_IM IM))) (item : composite_transition_item
                                              (equivocator_IM
                                                 IM)) 
  (item_descriptors
   pre_descriptors : equivocator_descriptors IM),
  equivocators_trace_project ieqv_descriptors suf =
  Some (sufX, item_descriptors)
  ∧ equivocators_transition_item_project
      item_descriptors item =
    Some (Some itemX, pre_descriptors)
    ∧ equivocators_trace_project pre_descriptors pre =
      Some (preX, ieqv_descriptors)
      ∧ [] = pre ++ item :: suf
    by destruct preX; inversion Hnil.
  -message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
x: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
ieqv_descriptors: equivocator_descriptors IM
preX: list (composite_transition_item IM)
itemX: composite_transition_item IM
IHtr: ∀ (eqv_descriptors : equivocator_descriptors IM) 
  (sufX : list (composite_transition_item IM)),
  equivocators_trace_project eqv_descriptors tr =
  Some
    (preX ++ [itemX] ++ sufX, ieqv_descriptors)
  → ∃ (pre
       suf : list
               (composite_transition_item
                  (equivocator_IM IM))) 
      (item : composite_transition_item
                (equivocator_IM IM)) 
      (item_descriptors
       pre_descriptors : 
       equivocator_descriptors IM),
      equivocators_trace_project
        eqv_descriptors suf =
      Some (sufX, item_descriptors)
      ∧ equivocators_transition_item_project
          item_descriptors item =
        Some (Some itemX, pre_descriptors)
        ∧ equivocators_trace_project
            pre_descriptors pre =
          Some (preX, ieqv_descriptors)
          ∧ tr = pre ++ [item] ++ suf
eqv_descriptors: equivocator_descriptors IM
sufX: list (composite_transition_item IM)
equivocators_trace_project eqv_descriptors (tr ++ [x]) =
Some (preX ++ [itemX] ++ sufX, ieqv_descriptors)
→ ∃ (pre
     suf : list
             (composite_transition_item
                (equivocator_IM IM))) (item : composite_transition_item
                                                (equivocator_IM
                                                 IM)) 
    (item_descriptors
     pre_descriptors : equivocator_descriptors IM),
    equivocators_trace_project eqv_descriptors suf =
    Some (sufX, item_descriptors)
    ∧ equivocators_transition_item_project
        item_descriptors item =
      Some (Some itemX, pre_descriptors)
      ∧ equivocators_trace_project pre_descriptors pre =
        Some (preX, ieqv_descriptors)
        ∧ tr ++ [x] = pre ++ [item] ++ suf intro Hsome.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
x: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
ieqv_descriptors: equivocator_descriptors IM
preX: list (composite_transition_item IM)
itemX: composite_transition_item IM
IHtr: ∀ (eqv_descriptors : equivocator_descriptors IM) 
  (sufX : list (composite_transition_item IM)),
  equivocators_trace_project eqv_descriptors tr =
  Some
    (preX ++ [itemX] ++ sufX, ieqv_descriptors)
  → ∃ (pre
       suf : list
               (composite_transition_item
                  (equivocator_IM IM))) 
      (item : composite_transition_item
                (equivocator_IM IM)) 
      (item_descriptors
       pre_descriptors : 
       equivocator_descriptors IM),
      equivocators_trace_project
        eqv_descriptors suf =
      Some (sufX, item_descriptors)
      ∧ equivocators_transition_item_project
          item_descriptors item =
        Some (Some itemX, pre_descriptors)
        ∧ equivocators_trace_project
            pre_descriptors pre =
          Some (preX, ieqv_descriptors)
          ∧ tr = pre ++ [item] ++ suf
eqv_descriptors: equivocator_descriptors IM
sufX: list (composite_transition_item IM)
Hsome: equivocators_trace_project eqv_descriptors
  (tr ++ [x]) =
Some
  (preX ++ [itemX] ++ sufX, ieqv_descriptors)
∃ (pre
   suf : list
           (composite_transition_item
              (equivocator_IM IM))) (item : composite_transition_item
                                              (equivocator_IM
                                                 IM)) 
  (item_descriptors
   pre_descriptors : equivocator_descriptors IM),
  equivocators_trace_project eqv_descriptors suf =
  Some (sufX, item_descriptors)
  ∧ equivocators_transition_item_project
      item_descriptors item =
    Some (Some itemX, pre_descriptors)
    ∧ equivocators_trace_project pre_descriptors pre =
      Some (preX, ieqv_descriptors)
      ∧ tr ++ [x] = pre ++ [item] ++ suf
    apply equivocators_trace_project_app_iff in Hsome
      as (trX' & xX & eqv_descriptors' & Hpr_x & Hpr_tr & Heq).message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
x: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
ieqv_descriptors: equivocator_descriptors IM
preX: list (composite_transition_item IM)
itemX: composite_transition_item IM
IHtr: ∀ (eqv_descriptors : equivocator_descriptors IM) 
  (sufX : list (composite_transition_item IM)),
  equivocators_trace_project eqv_descriptors tr =
  Some
    (preX ++ [itemX] ++ sufX, ieqv_descriptors)
  → ∃ (pre
       suf : list
               (composite_transition_item
                  (equivocator_IM IM))) 
      (item : composite_transition_item
                (equivocator_IM IM)) 
      (item_descriptors
       pre_descriptors : 
       equivocator_descriptors IM),
      equivocators_trace_project
        eqv_descriptors suf =
      Some (sufX, item_descriptors)
      ∧ equivocators_transition_item_project
          item_descriptors item =
        Some (Some itemX, pre_descriptors)
        ∧ equivocators_trace_project
            pre_descriptors pre =
          Some (preX, ieqv_descriptors)
          ∧ tr = pre ++ [item] ++ suf
eqv_descriptors: equivocator_descriptors IM
sufX, trX', xX: list (composite_transition_item IM)
eqv_descriptors': equivocator_descriptors IM
Hpr_x: equivocators_trace_project eqv_descriptors [x] =
Some (xX, eqv_descriptors')
Hpr_tr: equivocators_trace_project eqv_descriptors'
  tr = Some (trX', ieqv_descriptors)
Heq: preX ++ [itemX] ++ sufX = trX' ++ xX
∃ (pre
   suf : list
           (composite_transition_item
              (equivocator_IM IM))) (item : composite_transition_item
                                              (equivocator_IM
                                                 IM)) 
  (item_descriptors
   pre_descriptors : equivocator_descriptors IM),
  equivocators_trace_project eqv_descriptors suf =
  Some (sufX, item_descriptors)
  ∧ equivocators_transition_item_project
      item_descriptors item =
    Some (Some itemX, pre_descriptors)
    ∧ equivocators_trace_project pre_descriptors pre =
      Some (preX, ieqv_descriptors)
      ∧ tr ++ [x] = pre ++ [item] ++ suf
    simpl in Hpr_x.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
x: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
ieqv_descriptors: equivocator_descriptors IM
preX: list (composite_transition_item IM)
itemX: composite_transition_item IM
IHtr: ∀ (eqv_descriptors : equivocator_descriptors IM) 
  (sufX : list (composite_transition_item IM)),
  equivocators_trace_project eqv_descriptors tr =
  Some
    (preX ++ [itemX] ++ sufX, ieqv_descriptors)
  → ∃ (pre
       suf : list
               (composite_transition_item
                  (equivocator_IM IM))) 
      (item : composite_transition_item
                (equivocator_IM IM)) 
      (item_descriptors
       pre_descriptors : 
       equivocator_descriptors IM),
      equivocators_trace_project
        eqv_descriptors suf =
      Some (sufX, item_descriptors)
      ∧ equivocators_transition_item_project
          item_descriptors item =
        Some (Some itemX, pre_descriptors)
        ∧ equivocators_trace_project
            pre_descriptors pre =
          Some (preX, ieqv_descriptors)
          ∧ tr = pre ++ [item] ++ suf
eqv_descriptors: equivocator_descriptors IM
sufX, trX', xX: list (composite_transition_item IM)
eqv_descriptors': equivocator_descriptors IM
Hpr_x: match
  equivocators_transition_item_project
    eqv_descriptors x
with
| Some (Some item', odescriptor) =>
    Some ([item'], odescriptor)
| Some (None, odescriptor) =>
    Some ([], odescriptor)
| None => None
end = Some (xX, eqv_descriptors')
Hpr_tr: equivocators_trace_project eqv_descriptors'
  tr = Some (trX', ieqv_descriptors)
Heq: preX ++ [itemX] ++ sufX = trX' ++ xX
∃ (pre
   suf : list
           (composite_transition_item
              (equivocator_IM IM))) (item : composite_transition_item
                                              (equivocator_IM
                                                 IM)) 
  (item_descriptors
   pre_descriptors : equivocator_descriptors IM),
  equivocators_trace_project eqv_descriptors suf =
  Some (sufX, item_descriptors)
  ∧ equivocators_transition_item_project
      item_descriptors item =
    Some (Some itemX, pre_descriptors)
    ∧ equivocators_trace_project pre_descriptors pre =
      Some (preX, ieqv_descriptors)
      ∧ tr ++ [x] = pre ++ [item] ++ suf
    destruct (equivocators_transition_item_project eqv_descriptors x)
      as [(ox, descriptorx) |] eqn: Hpr_x_item
    ; [| by congruence].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
x: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
ieqv_descriptors: equivocator_descriptors IM
preX: list (composite_transition_item IM)
itemX: composite_transition_item IM
IHtr: ∀ (eqv_descriptors : equivocator_descriptors IM) 
  (sufX : list (composite_transition_item IM)),
  equivocators_trace_project eqv_descriptors tr =
  Some
    (preX ++ [itemX] ++ sufX, ieqv_descriptors)
  → ∃ (pre
       suf : list
               (composite_transition_item
                  (equivocator_IM IM))) 
      (item : composite_transition_item
                (equivocator_IM IM)) 
      (item_descriptors
       pre_descriptors : 
       equivocator_descriptors IM),
      equivocators_trace_project
        eqv_descriptors suf =
      Some (sufX, item_descriptors)
      ∧ equivocators_transition_item_project
          item_descriptors item =
        Some (Some itemX, pre_descriptors)
        ∧ equivocators_trace_project
            pre_descriptors pre =
          Some (preX, ieqv_descriptors)
          ∧ tr = pre ++ [item] ++ suf
eqv_descriptors: equivocator_descriptors IM
sufX, trX', xX: list (composite_transition_item IM)
eqv_descriptors': equivocator_descriptors IM
ox: option (composite_transition_item IM)
descriptorx: equivocator_descriptors IM
Hpr_x_item: equivocators_transition_item_project
  eqv_descriptors x =
Some (ox, descriptorx)
Hpr_x: match ox with
| Some item' => Some ([item'], descriptorx)
| None => Some ([], descriptorx)
end = Some (xX, eqv_descriptors')
Hpr_tr: equivocators_trace_project eqv_descriptors'
  tr = Some (trX', ieqv_descriptors)
Heq: preX ++ [itemX] ++ sufX = trX' ++ xX
∃ (pre
   suf : list
           (composite_transition_item
              (equivocator_IM IM))) (item : composite_transition_item
                                              (equivocator_IM
                                                 IM)) 
  (item_descriptors
   pre_descriptors : equivocator_descriptors IM),
  equivocators_trace_project eqv_descriptors suf =
  Some (sufX, item_descriptors)
  ∧ equivocators_transition_item_project
      item_descriptors item =
    Some (Some itemX, pre_descriptors)
    ∧ equivocators_trace_project pre_descriptors pre =
      Some (preX, ieqv_descriptors)
      ∧ tr ++ [x] = pre ++ [item] ++ suf
    destruct xX as [| xX _empty].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
x: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
ieqv_descriptors: equivocator_descriptors IM
preX: list (composite_transition_item IM)
itemX: composite_transition_item IM
IHtr: ∀ (eqv_descriptors : equivocator_descriptors IM) 
  (sufX : list (composite_transition_item IM)),
  equivocators_trace_project eqv_descriptors tr =
  Some
    (preX ++ [itemX] ++ sufX, ieqv_descriptors)
  → ∃ (pre
       suf : list
               (composite_transition_item
                  (equivocator_IM IM))) 
      (item : composite_transition_item
                (equivocator_IM IM)) 
      (item_descriptors
       pre_descriptors : 
       equivocator_descriptors IM),
      equivocators_trace_project
        eqv_descriptors suf =
      Some (sufX, item_descriptors)
      ∧ equivocators_transition_item_project
          item_descriptors item =
        Some (Some itemX, pre_descriptors)
        ∧ equivocators_trace_project
            pre_descriptors pre =
          Some (preX, ieqv_descriptors)
          ∧ tr = pre ++ [item] ++ suf
eqv_descriptors: equivocator_descriptors IM
sufX, trX': list (composite_transition_item IM)
eqv_descriptors': equivocator_descriptors IM
ox: option (composite_transition_item IM)
descriptorx: equivocator_descriptors IM
Hpr_x_item: equivocators_transition_item_project
  eqv_descriptors x =
Some (ox, descriptorx)
Hpr_x: match ox with
| Some item' => Some ([item'], descriptorx)
| None => Some ([], descriptorx)
end = Some ([], eqv_descriptors')
Hpr_tr: equivocators_trace_project eqv_descriptors'
  tr = Some (trX', ieqv_descriptors)
Heq: preX ++ [itemX] ++ sufX = trX' ++ []
∃ (pre
   suf : list
           (composite_transition_item
              (equivocator_IM IM))) (item : composite_transition_item
                                              (equivocator_IM
                                                 IM)) 
  (item_descriptors
   pre_descriptors : equivocator_descriptors IM),
  equivocators_trace_project eqv_descriptors suf =
  Some (sufX, item_descriptors)
  ∧ equivocators_transition_item_project
      item_descriptors item =
    Some (Some itemX, pre_descriptors)
    ∧ equivocators_trace_project pre_descriptors pre =
      Some (preX, ieqv_descriptors)
      ∧ tr ++ [x] = pre ++ [item] ++ suf
message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
x: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
ieqv_descriptors: equivocator_descriptors IM
preX: list (composite_transition_item IM)
itemX: composite_transition_item IM
IHtr: ∀ (eqv_descriptors : equivocator_descriptors IM) 
  (sufX : list (composite_transition_item IM)),
  equivocators_trace_project eqv_descriptors tr =
  Some
    (preX ++ [itemX] ++ sufX, ieqv_descriptors)
  → ∃ (pre
       suf : list
               (composite_transition_item
                  (equivocator_IM IM))) 
      (item : composite_transition_item
                (equivocator_IM IM)) 
      (item_descriptors
       pre_descriptors : 
       equivocator_descriptors IM),
      equivocators_trace_project
        eqv_descriptors suf =
      Some (sufX, item_descriptors)
      ∧ equivocators_transition_item_project
          item_descriptors item =
        Some (Some itemX, pre_descriptors)
        ∧ equivocators_trace_project
            pre_descriptors pre =
          Some (preX, ieqv_descriptors)
          ∧ tr = pre ++ [item] ++ suf
eqv_descriptors: equivocator_descriptors IM
sufX, trX': list (composite_transition_item IM)
xX: composite_transition_item IM
_empty: list (composite_transition_item IM)
eqv_descriptors': equivocator_descriptors IM
ox: option (composite_transition_item IM)
descriptorx: equivocator_descriptors IM
Hpr_x_item: equivocators_transition_item_project
  eqv_descriptors x =
Some (ox, descriptorx)
Hpr_x: match ox with
| Some item' => Some ([item'], descriptorx)
| None => Some ([], descriptorx)
end = Some (xX :: _empty, eqv_descriptors')
Hpr_tr: equivocators_trace_project eqv_descriptors'
  tr = Some (trX', ieqv_descriptors)
Heq: preX ++ [itemX] ++ sufX = trX' ++ xX :: _empty
∃ (pre
   suf : list
           (composite_transition_item
              (equivocator_IM IM))) 
  (item : composite_transition_item
            (equivocator_IM IM)) 
  (item_descriptors
   pre_descriptors : equivocator_descriptors IM),
  equivocators_trace_project eqv_descriptors suf =
  Some (sufX, item_descriptors)
  ∧ equivocators_transition_item_project
      item_descriptors item =
    Some (Some itemX, pre_descriptors)
    ∧ equivocators_trace_project pre_descriptors pre =
      Some (preX, ieqv_descriptors)
      ∧ tr ++ [x] = pre ++ [item] ++ suf
    +message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
x: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
ieqv_descriptors: equivocator_descriptors IM
preX: list (composite_transition_item IM)
itemX: composite_transition_item IM
IHtr: ∀ (eqv_descriptors : equivocator_descriptors IM) 
  (sufX : list (composite_transition_item IM)),
  equivocators_trace_project eqv_descriptors tr =
  Some
    (preX ++ [itemX] ++ sufX, ieqv_descriptors)
  → ∃ (pre
       suf : list
               (composite_transition_item
                  (equivocator_IM IM))) 
      (item : composite_transition_item
                (equivocator_IM IM)) 
      (item_descriptors
       pre_descriptors : 
       equivocator_descriptors IM),
      equivocators_trace_project
        eqv_descriptors suf =
      Some (sufX, item_descriptors)
      ∧ equivocators_transition_item_project
          item_descriptors item =
        Some (Some itemX, pre_descriptors)
        ∧ equivocators_trace_project
            pre_descriptors pre =
          Some (preX, ieqv_descriptors)
          ∧ tr = pre ++ [item] ++ suf
eqv_descriptors: equivocator_descriptors IM
sufX, trX': list (composite_transition_item IM)
eqv_descriptors': equivocator_descriptors IM
ox: option (composite_transition_item IM)
descriptorx: equivocator_descriptors IM
Hpr_x_item: equivocators_transition_item_project
  eqv_descriptors x =
Some (ox, descriptorx)
Hpr_x: match ox with
| Some item' => Some ([item'], descriptorx)
| None => Some ([], descriptorx)
end = Some ([], eqv_descriptors')
Hpr_tr: equivocators_trace_project eqv_descriptors'
  tr = Some (trX', ieqv_descriptors)
Heq: preX ++ [itemX] ++ sufX = trX' ++ []
∃ (pre
   suf : list
           (composite_transition_item
              (equivocator_IM IM))) (item : composite_transition_item
                                              (equivocator_IM
                                                 IM)) 
  (item_descriptors
   pre_descriptors : equivocator_descriptors IM),
  equivocators_trace_project eqv_descriptors suf =
  Some (sufX, item_descriptors)
  ∧ equivocators_transition_item_project
      item_descriptors item =
    Some (Some itemX, pre_descriptors)
    ∧ equivocators_trace_project pre_descriptors pre =
      Some (preX, ieqv_descriptors)
      ∧ tr ++ [x] = pre ++ [item] ++ suf destruct ox; [by congruence |].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
x: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
ieqv_descriptors: equivocator_descriptors IM
preX: list (composite_transition_item IM)
itemX: composite_transition_item IM
IHtr: ∀ (eqv_descriptors : equivocator_descriptors IM) 
  (sufX : list (composite_transition_item IM)),
  equivocators_trace_project eqv_descriptors tr =
  Some
    (preX ++ [itemX] ++ sufX, ieqv_descriptors)
  → ∃ (pre
       suf : list
               (composite_transition_item
                  (equivocator_IM IM))) 
      (item : composite_transition_item
                (equivocator_IM IM)) 
      (item_descriptors
       pre_descriptors : 
       equivocator_descriptors IM),
      equivocators_trace_project
        eqv_descriptors suf =
      Some (sufX, item_descriptors)
      ∧ equivocators_transition_item_project
          item_descriptors item =
        Some (Some itemX, pre_descriptors)
        ∧ equivocators_trace_project
            pre_descriptors pre =
          Some (preX, ieqv_descriptors)
          ∧ tr = pre ++ [item] ++ suf
eqv_descriptors: equivocator_descriptors IM
sufX, trX': list (composite_transition_item IM)
eqv_descriptors', descriptorx: equivocator_descriptors IM
Hpr_x_item: equivocators_transition_item_project
  eqv_descriptors x =
Some (None, descriptorx)
Hpr_x: Some ([], descriptorx) =
Some ([], eqv_descriptors')
Hpr_tr: equivocators_trace_project eqv_descriptors'
  tr = Some (trX', ieqv_descriptors)
Heq: preX ++ [itemX] ++ sufX = trX' ++ []
∃ (pre
   suf : list
           (composite_transition_item
              (equivocator_IM IM))) (item : composite_transition_item
                                              (equivocator_IM
                                                 IM)) 
  (item_descriptors
   pre_descriptors : equivocator_descriptors IM),
  equivocators_trace_project eqv_descriptors suf =
  Some (sufX, item_descriptors)
  ∧ equivocators_transition_item_project
      item_descriptors item =
    Some (Some itemX, pre_descriptors)
    ∧ equivocators_trace_project pre_descriptors pre =
      Some (preX, ieqv_descriptors)
      ∧ tr ++ [x] = pre ++ [item] ++ suf
      inversion Hpr_x.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
x: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
ieqv_descriptors: equivocator_descriptors IM
preX: list (composite_transition_item IM)
itemX: composite_transition_item IM
IHtr: ∀ (eqv_descriptors : equivocator_descriptors IM) 
  (sufX : list (composite_transition_item IM)),
  equivocators_trace_project eqv_descriptors tr =
  Some
    (preX ++ [itemX] ++ sufX, ieqv_descriptors)
  → ∃ (pre
       suf : list
               (composite_transition_item
                  (equivocator_IM IM))) 
      (item : composite_transition_item
                (equivocator_IM IM)) 
      (item_descriptors
       pre_descriptors : 
       equivocator_descriptors IM),
      equivocators_trace_project
        eqv_descriptors suf =
      Some (sufX, item_descriptors)
      ∧ equivocators_transition_item_project
          item_descriptors item =
        Some (Some itemX, pre_descriptors)
        ∧ equivocators_trace_project
            pre_descriptors pre =
          Some (preX, ieqv_descriptors)
          ∧ tr = pre ++ [item] ++ suf
eqv_descriptors: equivocator_descriptors IM
sufX, trX': list (composite_transition_item IM)
eqv_descriptors', descriptorx: equivocator_descriptors IM
Hpr_x_item: equivocators_transition_item_project
  eqv_descriptors x =
Some (None, descriptorx)
Hpr_x: Some ([], descriptorx) =
Some ([], eqv_descriptors')
Hpr_tr: equivocators_trace_project eqv_descriptors'
  tr = Some (trX', ieqv_descriptors)
Heq: preX ++ [itemX] ++ sufX = trX' ++ []
H2: descriptorx = eqv_descriptors'
∃ (pre
   suf : list
           (composite_transition_item
              (equivocator_IM IM))) (item : composite_transition_item
                                              (equivocator_IM
                                                 IM)) 
  (item_descriptors
   pre_descriptors : equivocator_descriptors IM),
  equivocators_trace_project eqv_descriptors suf =
  Some (sufX, item_descriptors)
  ∧ equivocators_transition_item_project
      item_descriptors item =
    Some (Some itemX, pre_descriptors)
    ∧ equivocators_trace_project pre_descriptors pre =
      Some (preX, ieqv_descriptors)
      ∧ tr ++ [x] = pre ++ [item] ++ suf subst.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
x: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
ieqv_descriptors: equivocator_descriptors IM
preX: list (composite_transition_item IM)
itemX: composite_transition_item IM
IHtr: ∀ (eqv_descriptors : equivocator_descriptors IM) 
  (sufX : list (composite_transition_item IM)),
  equivocators_trace_project eqv_descriptors tr =
  Some
    (preX ++ [itemX] ++ sufX, ieqv_descriptors)
  → ∃ (pre
       suf : list
               (composite_transition_item
                  (equivocator_IM IM))) 
      (item : composite_transition_item
                (equivocator_IM IM)) 
      (item_descriptors
       pre_descriptors : 
       equivocator_descriptors IM),
      equivocators_trace_project
        eqv_descriptors suf =
      Some (sufX, item_descriptors)
      ∧ equivocators_transition_item_project
          item_descriptors item =
        Some (Some itemX, pre_descriptors)
        ∧ equivocators_trace_project
            pre_descriptors pre =
          Some (preX, ieqv_descriptors)
          ∧ tr = pre ++ [item] ++ suf
eqv_descriptors: equivocator_descriptors IM
sufX, trX': list (composite_transition_item IM)
eqv_descriptors': equivocator_descriptors IM
Hpr_x: Some ([], eqv_descriptors') =
Some ([], eqv_descriptors')
Hpr_x_item: equivocators_transition_item_project
  eqv_descriptors x =
Some (None, eqv_descriptors')
Hpr_tr: equivocators_trace_project eqv_descriptors'
  tr = Some (trX', ieqv_descriptors)
Heq: preX ++ [itemX] ++ sufX = trX' ++ []
∃ (pre
   suf : list
           (composite_transition_item
              (equivocator_IM IM))) (item : composite_transition_item
                                              (equivocator_IM
                                                 IM)) 
  (item_descriptors
   pre_descriptors : equivocator_descriptors IM),
  equivocators_trace_project eqv_descriptors suf =
  Some (sufX, item_descriptors)
  ∧ equivocators_transition_item_project
      item_descriptors item =
    Some (Some itemX, pre_descriptors)
    ∧ equivocators_trace_project pre_descriptors pre =
      Some (preX, ieqv_descriptors)
      ∧ tr ++ [x] = pre ++ [item] ++ suf clear Hpr_x.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
x: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
ieqv_descriptors: equivocator_descriptors IM
preX: list (composite_transition_item IM)
itemX: composite_transition_item IM
IHtr: ∀ (eqv_descriptors : equivocator_descriptors IM) 
  (sufX : list (composite_transition_item IM)),
  equivocators_trace_project eqv_descriptors tr =
  Some
    (preX ++ [itemX] ++ sufX, ieqv_descriptors)
  → ∃ (pre
       suf : list
               (composite_transition_item
                  (equivocator_IM IM))) 
      (item : composite_transition_item
                (equivocator_IM IM)) 
      (item_descriptors
       pre_descriptors : 
       equivocator_descriptors IM),
      equivocators_trace_project
        eqv_descriptors suf =
      Some (sufX, item_descriptors)
      ∧ equivocators_transition_item_project
          item_descriptors item =
        Some (Some itemX, pre_descriptors)
        ∧ equivocators_trace_project
            pre_descriptors pre =
          Some (preX, ieqv_descriptors)
          ∧ tr = pre ++ [item] ++ suf
eqv_descriptors: equivocator_descriptors IM
sufX, trX': list (composite_transition_item IM)
eqv_descriptors': equivocator_descriptors IM
Hpr_x_item: equivocators_transition_item_project
  eqv_descriptors x =
Some (None, eqv_descriptors')
Hpr_tr: equivocators_trace_project eqv_descriptors'
  tr = Some (trX', ieqv_descriptors)
Heq: preX ++ [itemX] ++ sufX = trX' ++ []
∃ (pre
   suf : list
           (composite_transition_item
              (equivocator_IM IM))) (item : composite_transition_item
                                              (equivocator_IM
                                                 IM)) 
  (item_descriptors
   pre_descriptors : equivocator_descriptors IM),
  equivocators_trace_project eqv_descriptors suf =
  Some (sufX, item_descriptors)
  ∧ equivocators_transition_item_project
      item_descriptors item =
    Some (Some itemX, pre_descriptors)
    ∧ equivocators_trace_project pre_descriptors pre =
      Some (preX, ieqv_descriptors)
      ∧ tr ++ [x] = pre ++ [item] ++ suf
      rewrite app_nil_r in  Heq.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
x: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
ieqv_descriptors: equivocator_descriptors IM
preX: list (composite_transition_item IM)
itemX: composite_transition_item IM
IHtr: ∀ (eqv_descriptors : equivocator_descriptors IM) 
  (sufX : list (composite_transition_item IM)),
  equivocators_trace_project eqv_descriptors tr =
  Some
    (preX ++ [itemX] ++ sufX, ieqv_descriptors)
  → ∃ (pre
       suf : list
               (composite_transition_item
                  (equivocator_IM IM))) 
      (item : composite_transition_item
                (equivocator_IM IM)) 
      (item_descriptors
       pre_descriptors : 
       equivocator_descriptors IM),
      equivocators_trace_project
        eqv_descriptors suf =
      Some (sufX, item_descriptors)
      ∧ equivocators_transition_item_project
          item_descriptors item =
        Some (Some itemX, pre_descriptors)
        ∧ equivocators_trace_project
            pre_descriptors pre =
          Some (preX, ieqv_descriptors)
          ∧ tr = pre ++ [item] ++ suf
eqv_descriptors: equivocator_descriptors IM
sufX, trX': list (composite_transition_item IM)
eqv_descriptors': equivocator_descriptors IM
Hpr_x_item: equivocators_transition_item_project
  eqv_descriptors x =
Some (None, eqv_descriptors')
Hpr_tr: equivocators_trace_project eqv_descriptors'
  tr = Some (trX', ieqv_descriptors)
Heq: preX ++ [itemX] ++ sufX = trX'
∃ (pre
   suf : list
           (composite_transition_item
              (equivocator_IM IM))) (item : composite_transition_item
                                              (equivocator_IM
                                                 IM)) 
  (item_descriptors
   pre_descriptors : equivocator_descriptors IM),
  equivocators_trace_project eqv_descriptors suf =
  Some (sufX, item_descriptors)
  ∧ equivocators_transition_item_project
      item_descriptors item =
    Some (Some itemX, pre_descriptors)
    ∧ equivocators_trace_project pre_descriptors pre =
      Some (preX, ieqv_descriptors)
      ∧ tr ++ [x] = pre ++ [item] ++ suf subst trX'.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
x: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
ieqv_descriptors: equivocator_descriptors IM
preX: list (composite_transition_item IM)
itemX: composite_transition_item IM
IHtr: ∀ (eqv_descriptors : equivocator_descriptors IM) 
  (sufX : list (composite_transition_item IM)),
  equivocators_trace_project eqv_descriptors tr =
  Some
    (preX ++ [itemX] ++ sufX, ieqv_descriptors)
  → ∃ (pre
       suf : list
               (composite_transition_item
                  (equivocator_IM IM))) 
      (item : composite_transition_item
                (equivocator_IM IM)) 
      (item_descriptors
       pre_descriptors : 
       equivocator_descriptors IM),
      equivocators_trace_project
        eqv_descriptors suf =
      Some (sufX, item_descriptors)
      ∧ equivocators_transition_item_project
          item_descriptors item =
        Some (Some itemX, pre_descriptors)
        ∧ equivocators_trace_project
            pre_descriptors pre =
          Some (preX, ieqv_descriptors)
          ∧ tr = pre ++ [item] ++ suf
eqv_descriptors: equivocator_descriptors IM
sufX: list (composite_transition_item IM)
eqv_descriptors': equivocator_descriptors IM
Hpr_x_item: equivocators_transition_item_project
  eqv_descriptors x =
Some (None, eqv_descriptors')
Hpr_tr: equivocators_trace_project eqv_descriptors'
  tr =
Some
  (preX ++ [itemX] ++ sufX, ieqv_descriptors)
∃ (pre
   suf : list
           (composite_transition_item
              (equivocator_IM IM))) (item : composite_transition_item
                                              (equivocator_IM
                                                 IM)) 
  (item_descriptors
   pre_descriptors : equivocator_descriptors IM),
  equivocators_trace_project eqv_descriptors suf =
  Some (sufX, item_descriptors)
  ∧ equivocators_transition_item_project
      item_descriptors item =
    Some (Some itemX, pre_descriptors)
    ∧ equivocators_trace_project pre_descriptors pre =
      Some (preX, ieqv_descriptors)
      ∧ tr ++ [x] = pre ++ [item] ++ suf
      specialize (IHtr eqv_descriptors' sufX Hpr_tr).message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
x: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
ieqv_descriptors: equivocator_descriptors IM
preX: list (composite_transition_item IM)
itemX: composite_transition_item IM
sufX: list (composite_transition_item IM)
eqv_descriptors': equivocator_descriptors IM
IHtr: ∃ (pre
   suf : list
           (composite_transition_item
              (equivocator_IM IM))) 
  (item : composite_transition_item
            (equivocator_IM IM)) 
  (item_descriptors
   pre_descriptors : equivocator_descriptors IM),
  equivocators_trace_project eqv_descriptors'
    suf = Some (sufX, item_descriptors)
  ∧ equivocators_transition_item_project
      item_descriptors item =
    Some (Some itemX, pre_descriptors)
    ∧ equivocators_trace_project
        pre_descriptors pre =
      Some (preX, ieqv_descriptors)
      ∧ tr = pre ++ [item] ++ suf
eqv_descriptors: equivocator_descriptors IM
Hpr_x_item: equivocators_transition_item_project
  eqv_descriptors x =
Some (None, eqv_descriptors')
Hpr_tr: equivocators_trace_project eqv_descriptors'
  tr =
Some
  (preX ++ [itemX] ++ sufX, ieqv_descriptors)
∃ (pre
   suf : list
           (composite_transition_item
              (equivocator_IM IM))) (item : composite_transition_item
                                              (equivocator_IM
                                                 IM)) 
  (item_descriptors
   pre_descriptors : equivocator_descriptors IM),
  equivocators_trace_project eqv_descriptors suf =
  Some (sufX, item_descriptors)
  ∧ equivocators_transition_item_project
      item_descriptors item =
    Some (Some itemX, pre_descriptors)
    ∧ equivocators_trace_project pre_descriptors pre =
      Some (preX, ieqv_descriptors)
      ∧ tr ++ [x] = pre ++ [item] ++ suf
      destruct IHtr as [pre [suf [item [item_descriptors [pre_descriptors [Hpr_suf [Hpr_item
        [Hpr_pre Heqtr]]]]]]]].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
x: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
ieqv_descriptors: equivocator_descriptors IM
preX: list (composite_transition_item IM)
itemX: composite_transition_item IM
sufX: list (composite_transition_item IM)
eqv_descriptors': equivocator_descriptors IM
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
item: composite_transition_item (equivocator_IM IM)
item_descriptors, pre_descriptors: equivocator_descriptors IM
Hpr_suf: equivocators_trace_project eqv_descriptors'
  suf = Some (sufX, item_descriptors)
Hpr_item: equivocators_transition_item_project
  item_descriptors item =
Some (Some itemX, pre_descriptors)
Hpr_pre: equivocators_trace_project pre_descriptors
  pre = Some (preX, ieqv_descriptors)
Heqtr: tr = pre ++ [item] ++ suf
eqv_descriptors: equivocator_descriptors IM
Hpr_x_item: equivocators_transition_item_project
  eqv_descriptors x =
Some (None, eqv_descriptors')
Hpr_tr: equivocators_trace_project eqv_descriptors'
  tr =
Some
  (preX ++ [itemX] ++ sufX, ieqv_descriptors)
∃ (pre
   suf : list
           (composite_transition_item
              (equivocator_IM IM))) (item : composite_transition_item
                                              (equivocator_IM
                                                 IM)) 
  (item_descriptors
   pre_descriptors : equivocator_descriptors IM),
  equivocators_trace_project eqv_descriptors suf =
  Some (sufX, item_descriptors)
  ∧ equivocators_transition_item_project
      item_descriptors item =
    Some (Some itemX, pre_descriptors)
    ∧ equivocators_trace_project pre_descriptors pre =
      Some (preX, ieqv_descriptors)
      ∧ tr ++ [x] = pre ++ [item] ++ suf
      exists pre, (suf ++ [x]), item, item_descriptors, pre_descriptors.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
x: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
ieqv_descriptors: equivocator_descriptors IM
preX: list (composite_transition_item IM)
itemX: composite_transition_item IM
sufX: list (composite_transition_item IM)
eqv_descriptors': equivocator_descriptors IM
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
item: composite_transition_item (equivocator_IM IM)
item_descriptors, pre_descriptors: equivocator_descriptors IM
Hpr_suf: equivocators_trace_project eqv_descriptors'
  suf = Some (sufX, item_descriptors)
Hpr_item: equivocators_transition_item_project
  item_descriptors item =
Some (Some itemX, pre_descriptors)
Hpr_pre: equivocators_trace_project pre_descriptors
  pre = Some (preX, ieqv_descriptors)
Heqtr: tr = pre ++ [item] ++ suf
eqv_descriptors: equivocator_descriptors IM
Hpr_x_item: equivocators_transition_item_project
  eqv_descriptors x =
Some (None, eqv_descriptors')
Hpr_tr: equivocators_trace_project eqv_descriptors'
  tr =
Some
  (preX ++ [itemX] ++ sufX, ieqv_descriptors)
equivocators_trace_project eqv_descriptors
  (suf ++ [x]) = Some (sufX, item_descriptors)
∧ equivocators_transition_item_project
    item_descriptors item =
  Some (Some itemX, pre_descriptors)
  ∧ equivocators_trace_project pre_descriptors pre =
    Some (preX, ieqv_descriptors)
    ∧ tr ++ [x] = pre ++ [item] ++ suf ++ [x]
      subst tr.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
x: composite_transition_item (equivocator_IM IM)
ieqv_descriptors: equivocator_descriptors IM
preX: list (composite_transition_item IM)
itemX: composite_transition_item IM
sufX: list (composite_transition_item IM)
eqv_descriptors': equivocator_descriptors IM
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
item: composite_transition_item (equivocator_IM IM)
item_descriptors, pre_descriptors: equivocator_descriptors IM
Hpr_suf: equivocators_trace_project eqv_descriptors'
  suf = Some (sufX, item_descriptors)
Hpr_item: equivocators_transition_item_project
  item_descriptors item =
Some (Some itemX, pre_descriptors)
Hpr_pre: equivocators_trace_project pre_descriptors
  pre = Some (preX, ieqv_descriptors)
eqv_descriptors: equivocator_descriptors IM
Hpr_x_item: equivocators_transition_item_project
  eqv_descriptors x =
Some (None, eqv_descriptors')
Hpr_tr: equivocators_trace_project eqv_descriptors'
  (pre ++ [item] ++ suf) =
Some
  (preX ++ [itemX] ++ sufX, ieqv_descriptors)
equivocators_trace_project eqv_descriptors
  (suf ++ [x]) = Some (sufX, item_descriptors)
∧ equivocators_transition_item_project
    item_descriptors item =
  Some (Some itemX, pre_descriptors)
  ∧ equivocators_trace_project pre_descriptors pre =
    Some (preX, ieqv_descriptors)
    ∧ (pre ++ [item] ++ suf) ++ [x] =
      pre ++ [item] ++ suf ++ [x] rewrite !app_assoc.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
x: composite_transition_item (equivocator_IM IM)
ieqv_descriptors: equivocator_descriptors IM
preX: list (composite_transition_item IM)
itemX: composite_transition_item IM
sufX: list (composite_transition_item IM)
eqv_descriptors': equivocator_descriptors IM
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
item: composite_transition_item (equivocator_IM IM)
item_descriptors, pre_descriptors: equivocator_descriptors IM
Hpr_suf: equivocators_trace_project eqv_descriptors'
  suf = Some (sufX, item_descriptors)
Hpr_item: equivocators_transition_item_project
  item_descriptors item =
Some (Some itemX, pre_descriptors)
Hpr_pre: equivocators_trace_project pre_descriptors
  pre = Some (preX, ieqv_descriptors)
eqv_descriptors: equivocator_descriptors IM
Hpr_x_item: equivocators_transition_item_project
  eqv_descriptors x =
Some (None, eqv_descriptors')
Hpr_tr: equivocators_trace_project eqv_descriptors'
  (pre ++ [item] ++ suf) =
Some
  (preX ++ [itemX] ++ sufX, ieqv_descriptors)
equivocators_trace_project eqv_descriptors
  (suf ++ [x]) = Some (sufX, item_descriptors)
∧ equivocators_transition_item_project
    item_descriptors item =
  Some (Some itemX, pre_descriptors)
  ∧ equivocators_trace_project pre_descriptors pre =
    Some (preX, ieqv_descriptors)
    ∧ ((pre ++ [item]) ++ suf) ++ [x] =
      ((pre ++ [item]) ++ suf) ++ [x]
      repeat split; [| done | done].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
x: composite_transition_item (equivocator_IM IM)
ieqv_descriptors: equivocator_descriptors IM
preX: list (composite_transition_item IM)
itemX: composite_transition_item IM
sufX: list (composite_transition_item IM)
eqv_descriptors': equivocator_descriptors IM
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
item: composite_transition_item (equivocator_IM IM)
item_descriptors, pre_descriptors: equivocator_descriptors IM
Hpr_suf: equivocators_trace_project eqv_descriptors'
  suf = Some (sufX, item_descriptors)
Hpr_item: equivocators_transition_item_project
  item_descriptors item =
Some (Some itemX, pre_descriptors)
Hpr_pre: equivocators_trace_project pre_descriptors
  pre = Some (preX, ieqv_descriptors)
eqv_descriptors: equivocator_descriptors IM
Hpr_x_item: equivocators_transition_item_project
  eqv_descriptors x =
Some (None, eqv_descriptors')
Hpr_tr: equivocators_trace_project eqv_descriptors'
  (pre ++ [item] ++ suf) =
Some
  (preX ++ [itemX] ++ sufX, ieqv_descriptors)
equivocators_trace_project eqv_descriptors
  (suf ++ [x]) = Some (sufX, item_descriptors)
      apply equivocators_trace_project_app_iff.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
x: composite_transition_item (equivocator_IM IM)
ieqv_descriptors: equivocator_descriptors IM
preX: list (composite_transition_item IM)
itemX: composite_transition_item IM
sufX: list (composite_transition_item IM)
eqv_descriptors': equivocator_descriptors IM
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
item: composite_transition_item (equivocator_IM IM)
item_descriptors, pre_descriptors: equivocator_descriptors IM
Hpr_suf: equivocators_trace_project eqv_descriptors'
  suf = Some (sufX, item_descriptors)
Hpr_item: equivocators_transition_item_project
  item_descriptors item =
Some (Some itemX, pre_descriptors)
Hpr_pre: equivocators_trace_project pre_descriptors
  pre = Some (preX, ieqv_descriptors)
eqv_descriptors: equivocator_descriptors IM
Hpr_x_item: equivocators_transition_item_project
  eqv_descriptors x =
Some (None, eqv_descriptors')
Hpr_tr: equivocators_trace_project eqv_descriptors'
  (pre ++ [item] ++ suf) =
Some
  (preX ++ [itemX] ++ sufX, ieqv_descriptors)
∃ (preX sufX0 : list (composite_transition_item IM)) 
  (eqv_descriptors' : equivocator_descriptors IM),
  equivocators_trace_project eqv_descriptors [x] =
  Some (sufX0, eqv_descriptors')
  ∧ equivocators_trace_project eqv_descriptors' suf =
    Some (preX, item_descriptors)
    ∧ sufX = preX ++ sufX0
      exists sufX, [], eqv_descriptors'.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
x: composite_transition_item (equivocator_IM IM)
ieqv_descriptors: equivocator_descriptors IM
preX: list (composite_transition_item IM)
itemX: composite_transition_item IM
sufX: list (composite_transition_item IM)
eqv_descriptors': equivocator_descriptors IM
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
item: composite_transition_item (equivocator_IM IM)
item_descriptors, pre_descriptors: equivocator_descriptors IM
Hpr_suf: equivocators_trace_project eqv_descriptors'
  suf = Some (sufX, item_descriptors)
Hpr_item: equivocators_transition_item_project
  item_descriptors item =
Some (Some itemX, pre_descriptors)
Hpr_pre: equivocators_trace_project pre_descriptors
  pre = Some (preX, ieqv_descriptors)
eqv_descriptors: equivocator_descriptors IM
Hpr_x_item: equivocators_transition_item_project
  eqv_descriptors x =
Some (None, eqv_descriptors')
Hpr_tr: equivocators_trace_project eqv_descriptors'
  (pre ++ [item] ++ suf) =
Some
  (preX ++ [itemX] ++ sufX, ieqv_descriptors)
equivocators_trace_project eqv_descriptors [x] =
Some ([], eqv_descriptors')
∧ equivocators_trace_project eqv_descriptors' suf =
  Some (sufX, item_descriptors) ∧ sufX = sufX ++ [] rewrite app_nil_r.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
x: composite_transition_item (equivocator_IM IM)
ieqv_descriptors: equivocator_descriptors IM
preX: list (composite_transition_item IM)
itemX: composite_transition_item IM
sufX: list (composite_transition_item IM)
eqv_descriptors': equivocator_descriptors IM
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
item: composite_transition_item (equivocator_IM IM)
item_descriptors, pre_descriptors: equivocator_descriptors IM
Hpr_suf: equivocators_trace_project eqv_descriptors'
  suf = Some (sufX, item_descriptors)
Hpr_item: equivocators_transition_item_project
  item_descriptors item =
Some (Some itemX, pre_descriptors)
Hpr_pre: equivocators_trace_project pre_descriptors
  pre = Some (preX, ieqv_descriptors)
eqv_descriptors: equivocator_descriptors IM
Hpr_x_item: equivocators_transition_item_project
  eqv_descriptors x =
Some (None, eqv_descriptors')
Hpr_tr: equivocators_trace_project eqv_descriptors'
  (pre ++ [item] ++ suf) =
Some
  (preX ++ [itemX] ++ sufX, ieqv_descriptors)
equivocators_trace_project eqv_descriptors [x] =
Some ([], eqv_descriptors')
∧ equivocators_trace_project eqv_descriptors' suf =
  Some (sufX, item_descriptors) ∧ sufX = sufX
      repeat split; [| done].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
x: composite_transition_item (equivocator_IM IM)
ieqv_descriptors: equivocator_descriptors IM
preX: list (composite_transition_item IM)
itemX: composite_transition_item IM
sufX: list (composite_transition_item IM)
eqv_descriptors': equivocator_descriptors IM
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
item: composite_transition_item (equivocator_IM IM)
item_descriptors, pre_descriptors: equivocator_descriptors IM
Hpr_suf: equivocators_trace_project eqv_descriptors'
  suf = Some (sufX, item_descriptors)
Hpr_item: equivocators_transition_item_project
  item_descriptors item =
Some (Some itemX, pre_descriptors)
Hpr_pre: equivocators_trace_project pre_descriptors
  pre = Some (preX, ieqv_descriptors)
eqv_descriptors: equivocator_descriptors IM
Hpr_x_item: equivocators_transition_item_project
  eqv_descriptors x =
Some (None, eqv_descriptors')
Hpr_tr: equivocators_trace_project eqv_descriptors'
  (pre ++ [item] ++ suf) =
Some
  (preX ++ [itemX] ++ sufX, ieqv_descriptors)
equivocators_trace_project eqv_descriptors [x] =
Some ([], eqv_descriptors')
      by simpl; rewrite Hpr_x_item.
    +message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
x: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
ieqv_descriptors: equivocator_descriptors IM
preX: list (composite_transition_item IM)
itemX: composite_transition_item IM
IHtr: ∀ (eqv_descriptors : equivocator_descriptors IM) 
  (sufX : list (composite_transition_item IM)),
  equivocators_trace_project eqv_descriptors tr =
  Some
    (preX ++ [itemX] ++ sufX, ieqv_descriptors)
  → ∃ (pre
       suf : list
               (composite_transition_item
                  (equivocator_IM IM))) 
      (item : composite_transition_item
                (equivocator_IM IM)) 
      (item_descriptors
       pre_descriptors : 
       equivocator_descriptors IM),
      equivocators_trace_project
        eqv_descriptors suf =
      Some (sufX, item_descriptors)
      ∧ equivocators_transition_item_project
          item_descriptors item =
        Some (Some itemX, pre_descriptors)
        ∧ equivocators_trace_project
            pre_descriptors pre =
          Some (preX, ieqv_descriptors)
          ∧ tr = pre ++ [item] ++ suf
eqv_descriptors: equivocator_descriptors IM
sufX, trX': list (composite_transition_item IM)
xX: composite_transition_item IM
_empty: list (composite_transition_item IM)
eqv_descriptors': equivocator_descriptors IM
ox: option (composite_transition_item IM)
descriptorx: equivocator_descriptors IM
Hpr_x_item: equivocators_transition_item_project
  eqv_descriptors x =
Some (ox, descriptorx)
Hpr_x: match ox with
| Some item' => Some ([item'], descriptorx)
| None => Some ([], descriptorx)
end = Some (xX :: _empty, eqv_descriptors')
Hpr_tr: equivocators_trace_project eqv_descriptors'
  tr = Some (trX', ieqv_descriptors)
Heq: preX ++ [itemX] ++ sufX = trX' ++ xX :: _empty
∃ (pre
   suf : list
           (composite_transition_item
              (equivocator_IM IM))) (item : composite_transition_item
                                              (equivocator_IM
                                                 IM)) 
  (item_descriptors
   pre_descriptors : equivocator_descriptors IM),
  equivocators_trace_project eqv_descriptors suf =
  Some (sufX, item_descriptors)
  ∧ equivocators_transition_item_project
      item_descriptors item =
    Some (Some itemX, pre_descriptors)
    ∧ equivocators_trace_project pre_descriptors pre =
      Some (preX, ieqv_descriptors)
      ∧ tr ++ [x] = pre ++ [item] ++ suf destruct ox; [| by congruence].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
x: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
ieqv_descriptors: equivocator_descriptors IM
preX: list (composite_transition_item IM)
itemX: composite_transition_item IM
IHtr: ∀ (eqv_descriptors : equivocator_descriptors IM) 
  (sufX : list (composite_transition_item IM)),
  equivocators_trace_project eqv_descriptors tr =
  Some
    (preX ++ [itemX] ++ sufX, ieqv_descriptors)
  → ∃ (pre
       suf : list
               (composite_transition_item
                  (equivocator_IM IM))) 
      (item : composite_transition_item
                (equivocator_IM IM)) 
      (item_descriptors
       pre_descriptors : 
       equivocator_descriptors IM),
      equivocators_trace_project
        eqv_descriptors suf =
      Some (sufX, item_descriptors)
      ∧ equivocators_transition_item_project
          item_descriptors item =
        Some (Some itemX, pre_descriptors)
        ∧ equivocators_trace_project
            pre_descriptors pre =
          Some (preX, ieqv_descriptors)
          ∧ tr = pre ++ [item] ++ suf
eqv_descriptors: equivocator_descriptors IM
sufX, trX': list (composite_transition_item IM)
xX: composite_transition_item IM
_empty: list (composite_transition_item IM)
eqv_descriptors': equivocator_descriptors IM
c: composite_transition_item IM
descriptorx: equivocator_descriptors IM
Hpr_x_item: equivocators_transition_item_project
  eqv_descriptors x =
Some (Some c, descriptorx)
Hpr_x: Some ([c], descriptorx) =
Some (xX :: _empty, eqv_descriptors')
Hpr_tr: equivocators_trace_project eqv_descriptors'
  tr = Some (trX', ieqv_descriptors)
Heq: preX ++ [itemX] ++ sufX = trX' ++ xX :: _empty
∃ (pre
   suf : list
           (composite_transition_item
              (equivocator_IM IM))) (item : composite_transition_item
                                              (equivocator_IM
                                                 IM)) 
  (item_descriptors
   pre_descriptors : equivocator_descriptors IM),
  equivocators_trace_project eqv_descriptors suf =
  Some (sufX, item_descriptors)
  ∧ equivocators_transition_item_project
      item_descriptors item =
    Some (Some itemX, pre_descriptors)
    ∧ equivocators_trace_project pre_descriptors pre =
      Some (preX, ieqv_descriptors)
      ∧ tr ++ [x] = pre ++ [item] ++ suf
      inversion Hpr_x.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
x: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
ieqv_descriptors: equivocator_descriptors IM
preX: list (composite_transition_item IM)
itemX: composite_transition_item IM
IHtr: ∀ (eqv_descriptors : equivocator_descriptors IM) 
  (sufX : list (composite_transition_item IM)),
  equivocators_trace_project eqv_descriptors tr =
  Some
    (preX ++ [itemX] ++ sufX, ieqv_descriptors)
  → ∃ (pre
       suf : list
               (composite_transition_item
                  (equivocator_IM IM))) 
      (item : composite_transition_item
                (equivocator_IM IM)) 
      (item_descriptors
       pre_descriptors : 
       equivocator_descriptors IM),
      equivocators_trace_project
        eqv_descriptors suf =
      Some (sufX, item_descriptors)
      ∧ equivocators_transition_item_project
          item_descriptors item =
        Some (Some itemX, pre_descriptors)
        ∧ equivocators_trace_project
            pre_descriptors pre =
          Some (preX, ieqv_descriptors)
          ∧ tr = pre ++ [item] ++ suf
eqv_descriptors: equivocator_descriptors IM
sufX, trX': list (composite_transition_item IM)
xX: composite_transition_item IM
_empty: list (composite_transition_item IM)
eqv_descriptors': equivocator_descriptors IM
c: composite_transition_item IM
descriptorx: equivocator_descriptors IM
Hpr_x_item: equivocators_transition_item_project
  eqv_descriptors x =
Some (Some c, descriptorx)
Hpr_x: Some ([c], descriptorx) =
Some (xX :: _empty, eqv_descriptors')
Hpr_tr: equivocators_trace_project eqv_descriptors'
  tr = Some (trX', ieqv_descriptors)
Heq: preX ++ [itemX] ++ sufX = trX' ++ xX :: _empty
H2: c = xX
H3: [] = _empty
H4: descriptorx = eqv_descriptors'
∃ (pre
   suf : list
           (composite_transition_item
              (equivocator_IM IM))) (item : composite_transition_item
                                              (equivocator_IM
                                                 IM)) 
  (item_descriptors
   pre_descriptors : equivocator_descriptors IM),
  equivocators_trace_project eqv_descriptors suf =
  Some (sufX, item_descriptors)
  ∧ equivocators_transition_item_project
      item_descriptors item =
    Some (Some itemX, pre_descriptors)
    ∧ equivocators_trace_project pre_descriptors pre =
      Some (preX, ieqv_descriptors)
      ∧ tr ++ [x] = pre ++ [item] ++ suf subst.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
x: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
ieqv_descriptors: equivocator_descriptors IM
preX: list (composite_transition_item IM)
itemX: composite_transition_item IM
IHtr: ∀ (eqv_descriptors : equivocator_descriptors IM) 
  (sufX : list (composite_transition_item IM)),
  equivocators_trace_project eqv_descriptors tr =
  Some
    (preX ++ [itemX] ++ sufX, ieqv_descriptors)
  → ∃ (pre
       suf : list
               (composite_transition_item
                  (equivocator_IM IM))) 
      (item : composite_transition_item
                (equivocator_IM IM)) 
      (item_descriptors
       pre_descriptors : 
       equivocator_descriptors IM),
      equivocators_trace_project
        eqv_descriptors suf =
      Some (sufX, item_descriptors)
      ∧ equivocators_transition_item_project
          item_descriptors item =
        Some (Some itemX, pre_descriptors)
        ∧ equivocators_trace_project
            pre_descriptors pre =
          Some (preX, ieqv_descriptors)
          ∧ tr = pre ++ [item] ++ suf
eqv_descriptors: equivocator_descriptors IM
sufX, trX': list (composite_transition_item IM)
xX: composite_transition_item IM
eqv_descriptors': equivocator_descriptors IM
Hpr_x_item: equivocators_transition_item_project
  eqv_descriptors x =
Some (Some xX, eqv_descriptors')
Hpr_x: Some ([xX], eqv_descriptors') =
Some ([xX], eqv_descriptors')
Hpr_tr: equivocators_trace_project eqv_descriptors'
  tr = Some (trX', ieqv_descriptors)
Heq: preX ++ [itemX] ++ sufX = trX' ++ [xX]
∃ (pre
   suf : list
           (composite_transition_item
              (equivocator_IM IM))) (item : composite_transition_item
                                              (equivocator_IM
                                                 IM)) 
  (item_descriptors
   pre_descriptors : equivocator_descriptors IM),
  equivocators_trace_project eqv_descriptors suf =
  Some (sufX, item_descriptors)
  ∧ equivocators_transition_item_project
      item_descriptors item =
    Some (Some itemX, pre_descriptors)
    ∧ equivocators_trace_project pre_descriptors pre =
      Some (preX, ieqv_descriptors)
      ∧ tr ++ [x] = pre ++ [item] ++ suf clear Hpr_x.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
x: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
ieqv_descriptors: equivocator_descriptors IM
preX: list (composite_transition_item IM)
itemX: composite_transition_item IM
IHtr: ∀ (eqv_descriptors : equivocator_descriptors IM) 
  (sufX : list (composite_transition_item IM)),
  equivocators_trace_project eqv_descriptors tr =
  Some
    (preX ++ [itemX] ++ sufX, ieqv_descriptors)
  → ∃ (pre
       suf : list
               (composite_transition_item
                  (equivocator_IM IM))) 
      (item : composite_transition_item
                (equivocator_IM IM)) 
      (item_descriptors
       pre_descriptors : 
       equivocator_descriptors IM),
      equivocators_trace_project
        eqv_descriptors suf =
      Some (sufX, item_descriptors)
      ∧ equivocators_transition_item_project
          item_descriptors item =
        Some (Some itemX, pre_descriptors)
        ∧ equivocators_trace_project
            pre_descriptors pre =
          Some (preX, ieqv_descriptors)
          ∧ tr = pre ++ [item] ++ suf
eqv_descriptors: equivocator_descriptors IM
sufX, trX': list (composite_transition_item IM)
xX: composite_transition_item IM
eqv_descriptors': equivocator_descriptors IM
Hpr_x_item: equivocators_transition_item_project
  eqv_descriptors x =
Some (Some xX, eqv_descriptors')
Hpr_tr: equivocators_trace_project eqv_descriptors'
  tr = Some (trX', ieqv_descriptors)
Heq: preX ++ [itemX] ++ sufX = trX' ++ [xX]
∃ (pre
   suf : list
           (composite_transition_item
              (equivocator_IM IM))) (item : composite_transition_item
                                              (equivocator_IM
                                                 IM)) 
  (item_descriptors
   pre_descriptors : equivocator_descriptors IM),
  equivocators_trace_project eqv_descriptors suf =
  Some (sufX, item_descriptors)
  ∧ equivocators_transition_item_project
      item_descriptors item =
    Some (Some itemX, pre_descriptors)
    ∧ equivocators_trace_project pre_descriptors pre =
      Some (preX, ieqv_descriptors)
      ∧ tr ++ [x] = pre ++ [item] ++ suf
      destruct_list_last sufX sufX' _xX Heq_sufX.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
x: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
ieqv_descriptors: equivocator_descriptors IM
preX: list (composite_transition_item IM)
itemX: composite_transition_item IM
IHtr: ∀ (eqv_descriptors : equivocator_descriptors IM) 
  (sufX : list (composite_transition_item IM)),
  equivocators_trace_project eqv_descriptors tr =
  Some
    (preX ++ [itemX] ++ sufX, ieqv_descriptors)
  → ∃ (pre
       suf : list
               (composite_transition_item
                  (equivocator_IM IM))) 
      (item : composite_transition_item
                (equivocator_IM IM)) 
      (item_descriptors
       pre_descriptors : 
       equivocator_descriptors IM),
      equivocators_trace_project
        eqv_descriptors suf =
      Some (sufX, item_descriptors)
      ∧ equivocators_transition_item_project
          item_descriptors item =
        Some (Some itemX, pre_descriptors)
        ∧ equivocators_trace_project
            pre_descriptors pre =
          Some (preX, ieqv_descriptors)
          ∧ tr = pre ++ [item] ++ suf
eqv_descriptors: equivocator_descriptors IM
sufX, trX': list (composite_transition_item IM)
xX: composite_transition_item IM
eqv_descriptors': equivocator_descriptors IM
Hpr_x_item: equivocators_transition_item_project
  eqv_descriptors x =
Some (Some xX, eqv_descriptors')
Hpr_tr: equivocators_trace_project eqv_descriptors'
  tr = Some (trX', ieqv_descriptors)
Heq: preX ++ [itemX] ++ [] = trX' ++ [xX]
Heq_sufX: sufX = []
∃ (pre
   suf : list
           (composite_transition_item
              (equivocator_IM IM))) (item : composite_transition_item
                                              (equivocator_IM
                                                 IM)) 
  (item_descriptors
   pre_descriptors : equivocator_descriptors IM),
  equivocators_trace_project eqv_descriptors suf =
  Some ([], item_descriptors)
  ∧ equivocators_transition_item_project
      item_descriptors item =
    Some (Some itemX, pre_descriptors)
    ∧ equivocators_trace_project pre_descriptors pre =
      Some (preX, ieqv_descriptors)
      ∧ tr ++ [x] = pre ++ [item] ++ suf
message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
x: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
ieqv_descriptors: equivocator_descriptors IM
preX: list (composite_transition_item IM)
itemX: composite_transition_item IM
IHtr: ∀ (eqv_descriptors : equivocator_descriptors IM) 
  (sufX : list (composite_transition_item IM)),
  equivocators_trace_project eqv_descriptors tr =
  Some
    (preX ++ [itemX] ++ sufX, ieqv_descriptors)
  → ∃ (pre
       suf : list
               (composite_transition_item
                  (equivocator_IM IM))) 
      (item : composite_transition_item
                (equivocator_IM IM)) 
      (item_descriptors
       pre_descriptors : 
       equivocator_descriptors IM),
      equivocators_trace_project
        eqv_descriptors suf =
      Some (sufX, item_descriptors)
      ∧ equivocators_transition_item_project
          item_descriptors item =
        Some (Some itemX, pre_descriptors)
        ∧ equivocators_trace_project
            pre_descriptors pre =
          Some (preX, ieqv_descriptors)
          ∧ tr = pre ++ [item] ++ suf
eqv_descriptors: equivocator_descriptors IM
sufX, trX': list (composite_transition_item IM)
xX: composite_transition_item IM
eqv_descriptors': equivocator_descriptors IM
Hpr_x_item: equivocators_transition_item_project
  eqv_descriptors x =
Some (Some xX, eqv_descriptors')
Hpr_tr: equivocators_trace_project eqv_descriptors'
  tr = Some (trX', ieqv_descriptors)
sufX': list (composite_transition_item IM)
_xX: composite_transition_item IM
Heq: preX ++ [itemX] ++ sufX' ++ [_xX] = trX' ++ [xX]
Heq_sufX: sufX = sufX' ++ [_xX]
∃ (pre
   suf : list
           (composite_transition_item
              (equivocator_IM IM))) 
  (item : composite_transition_item
            (equivocator_IM IM)) 
  (item_descriptors
   pre_descriptors : equivocator_descriptors IM),
  equivocators_trace_project eqv_descriptors suf =
  Some (sufX' ++ [_xX], item_descriptors)
  ∧ equivocators_transition_item_project
      item_descriptors item =
    Some (Some itemX, pre_descriptors)
    ∧ equivocators_trace_project pre_descriptors pre =
      Some (preX, ieqv_descriptors)
      ∧ tr ++ [x] = pre ++ [item] ++ suf
      *message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
x: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
ieqv_descriptors: equivocator_descriptors IM
preX: list (composite_transition_item IM)
itemX: composite_transition_item IM
IHtr: ∀ (eqv_descriptors : equivocator_descriptors IM) 
  (sufX : list (composite_transition_item IM)),
  equivocators_trace_project eqv_descriptors tr =
  Some
    (preX ++ [itemX] ++ sufX, ieqv_descriptors)
  → ∃ (pre
       suf : list
               (composite_transition_item
                  (equivocator_IM IM))) 
      (item : composite_transition_item
                (equivocator_IM IM)) 
      (item_descriptors
       pre_descriptors : 
       equivocator_descriptors IM),
      equivocators_trace_project
        eqv_descriptors suf =
      Some (sufX, item_descriptors)
      ∧ equivocators_transition_item_project
          item_descriptors item =
        Some (Some itemX, pre_descriptors)
        ∧ equivocators_trace_project
            pre_descriptors pre =
          Some (preX, ieqv_descriptors)
          ∧ tr = pre ++ [item] ++ suf
eqv_descriptors: equivocator_descriptors IM
sufX, trX': list (composite_transition_item IM)
xX: composite_transition_item IM
eqv_descriptors': equivocator_descriptors IM
Hpr_x_item: equivocators_transition_item_project
  eqv_descriptors x =
Some (Some xX, eqv_descriptors')
Hpr_tr: equivocators_trace_project eqv_descriptors'
  tr = Some (trX', ieqv_descriptors)
Heq: preX ++ [itemX] ++ [] = trX' ++ [xX]
Heq_sufX: sufX = []
∃ (pre
   suf : list
           (composite_transition_item
              (equivocator_IM IM))) (item : composite_transition_item
                                              (equivocator_IM
                                                 IM)) 
  (item_descriptors
   pre_descriptors : equivocator_descriptors IM),
  equivocators_trace_project eqv_descriptors suf =
  Some ([], item_descriptors)
  ∧ equivocators_transition_item_project
      item_descriptors item =
    Some (Some itemX, pre_descriptors)
    ∧ equivocators_trace_project pre_descriptors pre =
      Some (preX, ieqv_descriptors)
      ∧ tr ++ [x] = pre ++ [item] ++ suf subst.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
x: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
ieqv_descriptors: equivocator_descriptors IM
preX: list (composite_transition_item IM)
itemX: composite_transition_item IM
IHtr: ∀ (eqv_descriptors : equivocator_descriptors IM) 
  (sufX : list (composite_transition_item IM)),
  equivocators_trace_project eqv_descriptors tr =
  Some
    (preX ++ [itemX] ++ sufX, ieqv_descriptors)
  → ∃ (pre
       suf : list
               (composite_transition_item
                  (equivocator_IM IM))) 
      (item : composite_transition_item
                (equivocator_IM IM)) 
      (item_descriptors
       pre_descriptors : 
       equivocator_descriptors IM),
      equivocators_trace_project
        eqv_descriptors suf =
      Some (sufX, item_descriptors)
      ∧ equivocators_transition_item_project
          item_descriptors item =
        Some (Some itemX, pre_descriptors)
        ∧ equivocators_trace_project
            pre_descriptors pre =
          Some (preX, ieqv_descriptors)
          ∧ tr = pre ++ [item] ++ suf
eqv_descriptors: equivocator_descriptors IM
trX': list (composite_transition_item IM)
xX: composite_transition_item IM
eqv_descriptors': equivocator_descriptors IM
Hpr_x_item: equivocators_transition_item_project
  eqv_descriptors x =
Some (Some xX, eqv_descriptors')
Hpr_tr: equivocators_trace_project eqv_descriptors'
  tr = Some (trX', ieqv_descriptors)
Heq: preX ++ [itemX] ++ [] = trX' ++ [xX]
∃ (pre
   suf : list
           (composite_transition_item
              (equivocator_IM IM))) (item : composite_transition_item
                                              (equivocator_IM
                                                 IM)) 
  (item_descriptors
   pre_descriptors : equivocator_descriptors IM),
  equivocators_trace_project eqv_descriptors suf =
  Some ([], item_descriptors)
  ∧ equivocators_transition_item_project
      item_descriptors item =
    Some (Some itemX, pre_descriptors)
    ∧ equivocators_trace_project pre_descriptors pre =
      Some (preX, ieqv_descriptors)
      ∧ tr ++ [x] = pre ++ [item] ++ suf rewrite app_nil_r in Heq.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
x: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
ieqv_descriptors: equivocator_descriptors IM
preX: list (composite_transition_item IM)
itemX: composite_transition_item IM
IHtr: ∀ (eqv_descriptors : equivocator_descriptors IM) 
  (sufX : list (composite_transition_item IM)),
  equivocators_trace_project eqv_descriptors tr =
  Some
    (preX ++ [itemX] ++ sufX, ieqv_descriptors)
  → ∃ (pre
       suf : list
               (composite_transition_item
                  (equivocator_IM IM))) 
      (item : composite_transition_item
                (equivocator_IM IM)) 
      (item_descriptors
       pre_descriptors : 
       equivocator_descriptors IM),
      equivocators_trace_project
        eqv_descriptors suf =
      Some (sufX, item_descriptors)
      ∧ equivocators_transition_item_project
          item_descriptors item =
        Some (Some itemX, pre_descriptors)
        ∧ equivocators_trace_project
            pre_descriptors pre =
          Some (preX, ieqv_descriptors)
          ∧ tr = pre ++ [item] ++ suf
eqv_descriptors: equivocator_descriptors IM
trX': list (composite_transition_item IM)
xX: composite_transition_item IM
eqv_descriptors': equivocator_descriptors IM
Hpr_x_item: equivocators_transition_item_project
  eqv_descriptors x =
Some (Some xX, eqv_descriptors')
Hpr_tr: equivocators_trace_project eqv_descriptors'
  tr = Some (trX', ieqv_descriptors)
Heq: preX ++ [itemX] = trX' ++ [xX]
∃ (pre
   suf : list
           (composite_transition_item
              (equivocator_IM IM))) (item : composite_transition_item
                                              (equivocator_IM
                                                 IM)) 
  (item_descriptors
   pre_descriptors : equivocator_descriptors IM),
  equivocators_trace_project eqv_descriptors suf =
  Some ([], item_descriptors)
  ∧ equivocators_transition_item_project
      item_descriptors item =
    Some (Some itemX, pre_descriptors)
    ∧ equivocators_trace_project pre_descriptors pre =
      Some (preX, ieqv_descriptors)
      ∧ tr ++ [x] = pre ++ [item] ++ suf apply app_inj_tail in Heq.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
x: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
ieqv_descriptors: equivocator_descriptors IM
preX: list (composite_transition_item IM)
itemX: composite_transition_item IM
IHtr: ∀ (eqv_descriptors : equivocator_descriptors IM) 
  (sufX : list (composite_transition_item IM)),
  equivocators_trace_project eqv_descriptors tr =
  Some
    (preX ++ [itemX] ++ sufX, ieqv_descriptors)
  → ∃ (pre
       suf : list
               (composite_transition_item
                  (equivocator_IM IM))) 
      (item : composite_transition_item
                (equivocator_IM IM)) 
      (item_descriptors
       pre_descriptors : 
       equivocator_descriptors IM),
      equivocators_trace_project
        eqv_descriptors suf =
      Some (sufX, item_descriptors)
      ∧ equivocators_transition_item_project
          item_descriptors item =
        Some (Some itemX, pre_descriptors)
        ∧ equivocators_trace_project
            pre_descriptors pre =
          Some (preX, ieqv_descriptors)
          ∧ tr = pre ++ [item] ++ suf
eqv_descriptors: equivocator_descriptors IM
trX': list (composite_transition_item IM)
xX: composite_transition_item IM
eqv_descriptors': equivocator_descriptors IM
Hpr_x_item: equivocators_transition_item_project
  eqv_descriptors x =
Some (Some xX, eqv_descriptors')
Hpr_tr: equivocators_trace_project eqv_descriptors'
  tr = Some (trX', ieqv_descriptors)
Heq: preX = trX' ∧ itemX = xX
∃ (pre
   suf : list
           (composite_transition_item
              (equivocator_IM IM))) (item : composite_transition_item
                                              (equivocator_IM
                                                 IM)) 
  (item_descriptors
   pre_descriptors : equivocator_descriptors IM),
  equivocators_trace_project eqv_descriptors suf =
  Some ([], item_descriptors)
  ∧ equivocators_transition_item_project
      item_descriptors item =
    Some (Some itemX, pre_descriptors)
    ∧ equivocators_trace_project pre_descriptors pre =
      Some (preX, ieqv_descriptors)
      ∧ tr ++ [x] = pre ++ [item] ++ suf
        destruct Heq.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
x: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
ieqv_descriptors: equivocator_descriptors IM
preX: list (composite_transition_item IM)
itemX: composite_transition_item IM
IHtr: ∀ (eqv_descriptors : equivocator_descriptors IM) 
  (sufX : list (composite_transition_item IM)),
  equivocators_trace_project eqv_descriptors tr =
  Some
    (preX ++ [itemX] ++ sufX, ieqv_descriptors)
  → ∃ (pre
       suf : list
               (composite_transition_item
                  (equivocator_IM IM))) 
      (item : composite_transition_item
                (equivocator_IM IM)) 
      (item_descriptors
       pre_descriptors : 
       equivocator_descriptors IM),
      equivocators_trace_project
        eqv_descriptors suf =
      Some (sufX, item_descriptors)
      ∧ equivocators_transition_item_project
          item_descriptors item =
        Some (Some itemX, pre_descriptors)
        ∧ equivocators_trace_project
            pre_descriptors pre =
          Some (preX, ieqv_descriptors)
          ∧ tr = pre ++ [item] ++ suf
eqv_descriptors: equivocator_descriptors IM
trX': list (composite_transition_item IM)
xX: composite_transition_item IM
eqv_descriptors': equivocator_descriptors IM
Hpr_x_item: equivocators_transition_item_project
  eqv_descriptors x =
Some (Some xX, eqv_descriptors')
Hpr_tr: equivocators_trace_project eqv_descriptors'
  tr = Some (trX', ieqv_descriptors)
H1: preX = trX'
H2: itemX = xX
∃ (pre
   suf : list
           (composite_transition_item
              (equivocator_IM IM))) (item : composite_transition_item
                                              (equivocator_IM
                                                 IM)) 
  (item_descriptors
   pre_descriptors : equivocator_descriptors IM),
  equivocators_trace_project eqv_descriptors suf =
  Some ([], item_descriptors)
  ∧ equivocators_transition_item_project
      item_descriptors item =
    Some (Some itemX, pre_descriptors)
    ∧ equivocators_trace_project pre_descriptors pre =
      Some (preX, ieqv_descriptors)
      ∧ tr ++ [x] = pre ++ [item] ++ suf subst.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
x: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
ieqv_descriptors: equivocator_descriptors IM
trX': list (composite_transition_item IM)
xX: composite_transition_item IM
IHtr: ∀ (eqv_descriptors : equivocator_descriptors IM) 
  (sufX : list (composite_transition_item IM)),
  equivocators_trace_project eqv_descriptors tr =
  Some (trX' ++ [xX] ++ sufX, ieqv_descriptors)
  → ∃ (pre
       suf : list
               (composite_transition_item
                  (equivocator_IM IM))) 
      (item : composite_transition_item
                (equivocator_IM IM)) 
      (item_descriptors
       pre_descriptors : 
       equivocator_descriptors IM),
      equivocators_trace_project
        eqv_descriptors suf =
      Some (sufX, item_descriptors)
      ∧ equivocators_transition_item_project
          item_descriptors item =
        Some (Some xX, pre_descriptors)
        ∧ equivocators_trace_project
            pre_descriptors pre =
          Some (trX', ieqv_descriptors)
          ∧ tr = pre ++ [item] ++ suf
eqv_descriptors, eqv_descriptors': equivocator_descriptors IM
Hpr_x_item: equivocators_transition_item_project
  eqv_descriptors x =
Some (Some xX, eqv_descriptors')
Hpr_tr: equivocators_trace_project eqv_descriptors'
  tr = Some (trX', ieqv_descriptors)
∃ (pre
   suf : list
           (composite_transition_item
              (equivocator_IM IM))) (item : composite_transition_item
                                              (equivocator_IM
                                                 IM)) 
  (item_descriptors
   pre_descriptors : equivocator_descriptors IM),
  equivocators_trace_project eqv_descriptors suf =
  Some ([], item_descriptors)
  ∧ equivocators_transition_item_project
      item_descriptors item =
    Some (Some xX, pre_descriptors)
    ∧ equivocators_trace_project pre_descriptors pre =
      Some (trX', ieqv_descriptors)
      ∧ tr ++ [x] = pre ++ [item] ++ suf
        exists tr, [], x, eqv_descriptors, eqv_descriptors'.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
x: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
ieqv_descriptors: equivocator_descriptors IM
trX': list (composite_transition_item IM)
xX: composite_transition_item IM
IHtr: ∀ (eqv_descriptors : equivocator_descriptors IM) 
  (sufX : list (composite_transition_item IM)),
  equivocators_trace_project eqv_descriptors tr =
  Some (trX' ++ [xX] ++ sufX, ieqv_descriptors)
  → ∃ (pre
       suf : list
               (composite_transition_item
                  (equivocator_IM IM))) 
      (item : composite_transition_item
                (equivocator_IM IM)) 
      (item_descriptors
       pre_descriptors : 
       equivocator_descriptors IM),
      equivocators_trace_project
        eqv_descriptors suf =
      Some (sufX, item_descriptors)
      ∧ equivocators_transition_item_project
          item_descriptors item =
        Some (Some xX, pre_descriptors)
        ∧ equivocators_trace_project
            pre_descriptors pre =
          Some (trX', ieqv_descriptors)
          ∧ tr = pre ++ [item] ++ suf
eqv_descriptors, eqv_descriptors': equivocator_descriptors IM
Hpr_x_item: equivocators_transition_item_project
  eqv_descriptors x =
Some (Some xX, eqv_descriptors')
Hpr_tr: equivocators_trace_project eqv_descriptors'
  tr = Some (trX', ieqv_descriptors)
equivocators_trace_project eqv_descriptors [] =
Some ([], eqv_descriptors)
∧ equivocators_transition_item_project eqv_descriptors
    x = Some (Some xX, eqv_descriptors')
  ∧ equivocators_trace_project eqv_descriptors' tr =
    Some (trX', ieqv_descriptors)
    ∧ tr ++ [x] = tr ++ [x] ++ []
        by rewrite app_nil_r.
      *message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
x: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
ieqv_descriptors: equivocator_descriptors IM
preX: list (composite_transition_item IM)
itemX: composite_transition_item IM
IHtr: ∀ (eqv_descriptors : equivocator_descriptors IM) 
  (sufX : list (composite_transition_item IM)),
  equivocators_trace_project eqv_descriptors tr =
  Some
    (preX ++ [itemX] ++ sufX, ieqv_descriptors)
  → ∃ (pre
       suf : list
               (composite_transition_item
                  (equivocator_IM IM))) 
      (item : composite_transition_item
                (equivocator_IM IM)) 
      (item_descriptors
       pre_descriptors : 
       equivocator_descriptors IM),
      equivocators_trace_project
        eqv_descriptors suf =
      Some (sufX, item_descriptors)
      ∧ equivocators_transition_item_project
          item_descriptors item =
        Some (Some itemX, pre_descriptors)
        ∧ equivocators_trace_project
            pre_descriptors pre =
          Some (preX, ieqv_descriptors)
          ∧ tr = pre ++ [item] ++ suf
eqv_descriptors: equivocator_descriptors IM
sufX, trX': list (composite_transition_item IM)
xX: composite_transition_item IM
eqv_descriptors': equivocator_descriptors IM
Hpr_x_item: equivocators_transition_item_project
  eqv_descriptors x =
Some (Some xX, eqv_descriptors')
Hpr_tr: equivocators_trace_project eqv_descriptors'
  tr = Some (trX', ieqv_descriptors)
sufX': list (composite_transition_item IM)
_xX: composite_transition_item IM
Heq: preX ++ [itemX] ++ sufX' ++ [_xX] = trX' ++ [xX]
Heq_sufX: sufX = sufX' ++ [_xX]
∃ (pre
   suf : list
           (composite_transition_item
              (equivocator_IM IM))) (item : composite_transition_item
                                              (equivocator_IM
                                                 IM)) 
  (item_descriptors
   pre_descriptors : equivocator_descriptors IM),
  equivocators_trace_project eqv_descriptors suf =
  Some (sufX' ++ [_xX], item_descriptors)
  ∧ equivocators_transition_item_project
      item_descriptors item =
    Some (Some itemX, pre_descriptors)
    ∧ equivocators_trace_project pre_descriptors pre =
      Some (preX, ieqv_descriptors)
      ∧ tr ++ [x] = pre ++ [item] ++ suf subst.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
x: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
ieqv_descriptors: equivocator_descriptors IM
preX: list (composite_transition_item IM)
itemX: composite_transition_item IM
IHtr: ∀ (eqv_descriptors : equivocator_descriptors IM) 
  (sufX : list (composite_transition_item IM)),
  equivocators_trace_project eqv_descriptors tr =
  Some
    (preX ++ [itemX] ++ sufX, ieqv_descriptors)
  → ∃ (pre
       suf : list
               (composite_transition_item
                  (equivocator_IM IM))) 
      (item : composite_transition_item
                (equivocator_IM IM)) 
      (item_descriptors
       pre_descriptors : 
       equivocator_descriptors IM),
      equivocators_trace_project
        eqv_descriptors suf =
      Some (sufX, item_descriptors)
      ∧ equivocators_transition_item_project
          item_descriptors item =
        Some (Some itemX, pre_descriptors)
        ∧ equivocators_trace_project
            pre_descriptors pre =
          Some (preX, ieqv_descriptors)
          ∧ tr = pre ++ [item] ++ suf
eqv_descriptors: equivocator_descriptors IM
trX': list (composite_transition_item IM)
xX: composite_transition_item IM
eqv_descriptors': equivocator_descriptors IM
Hpr_x_item: equivocators_transition_item_project
  eqv_descriptors x =
Some (Some xX, eqv_descriptors')
Hpr_tr: equivocators_trace_project eqv_descriptors'
  tr = Some (trX', ieqv_descriptors)
sufX': list (composite_transition_item IM)
_xX: composite_transition_item IM
Heq: preX ++ [itemX] ++ sufX' ++ [_xX] = trX' ++ [xX]
∃ (pre
   suf : list
           (composite_transition_item
              (equivocator_IM IM))) (item : composite_transition_item
                                              (equivocator_IM
                                                 IM)) 
  (item_descriptors
   pre_descriptors : equivocator_descriptors IM),
  equivocators_trace_project eqv_descriptors suf =
  Some (sufX' ++ [_xX], item_descriptors)
  ∧ equivocators_transition_item_project
      item_descriptors item =
    Some (Some itemX, pre_descriptors)
    ∧ equivocators_trace_project pre_descriptors pre =
      Some (preX, ieqv_descriptors)
      ∧ tr ++ [x] = pre ++ [item] ++ suf rewrite! app_assoc in Heq.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
x: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
ieqv_descriptors: equivocator_descriptors IM
preX: list (composite_transition_item IM)
itemX: composite_transition_item IM
IHtr: ∀ (eqv_descriptors : equivocator_descriptors IM) 
  (sufX : list (composite_transition_item IM)),
  equivocators_trace_project eqv_descriptors tr =
  Some
    (preX ++ [itemX] ++ sufX, ieqv_descriptors)
  → ∃ (pre
       suf : list
               (composite_transition_item
                  (equivocator_IM IM))) 
      (item : composite_transition_item
                (equivocator_IM IM)) 
      (item_descriptors
       pre_descriptors : 
       equivocator_descriptors IM),
      equivocators_trace_project
        eqv_descriptors suf =
      Some (sufX, item_descriptors)
      ∧ equivocators_transition_item_project
          item_descriptors item =
        Some (Some itemX, pre_descriptors)
        ∧ equivocators_trace_project
            pre_descriptors pre =
          Some (preX, ieqv_descriptors)
          ∧ tr = pre ++ [item] ++ suf
eqv_descriptors: equivocator_descriptors IM
trX': list (composite_transition_item IM)
xX: composite_transition_item IM
eqv_descriptors': equivocator_descriptors IM
Hpr_x_item: equivocators_transition_item_project
  eqv_descriptors x =
Some (Some xX, eqv_descriptors')
Hpr_tr: equivocators_trace_project eqv_descriptors'
  tr = Some (trX', ieqv_descriptors)
sufX': list (composite_transition_item IM)
_xX: composite_transition_item IM
Heq: ((preX ++ [itemX]) ++ sufX') ++ [_xX] =
trX' ++ [xX]
∃ (pre
   suf : list
           (composite_transition_item
              (equivocator_IM IM))) (item : composite_transition_item
                                              (equivocator_IM
                                                 IM)) 
  (item_descriptors
   pre_descriptors : equivocator_descriptors IM),
  equivocators_trace_project eqv_descriptors suf =
  Some (sufX' ++ [_xX], item_descriptors)
  ∧ equivocators_transition_item_project
      item_descriptors item =
    Some (Some itemX, pre_descriptors)
    ∧ equivocators_trace_project pre_descriptors pre =
      Some (preX, ieqv_descriptors)
      ∧ tr ++ [x] = pre ++ [item] ++ suf apply app_inj_tail in Heq.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
x: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
ieqv_descriptors: equivocator_descriptors IM
preX: list (composite_transition_item IM)
itemX: composite_transition_item IM
IHtr: ∀ (eqv_descriptors : equivocator_descriptors IM) 
  (sufX : list (composite_transition_item IM)),
  equivocators_trace_project eqv_descriptors tr =
  Some
    (preX ++ [itemX] ++ sufX, ieqv_descriptors)
  → ∃ (pre
       suf : list
               (composite_transition_item
                  (equivocator_IM IM))) 
      (item : composite_transition_item
                (equivocator_IM IM)) 
      (item_descriptors
       pre_descriptors : 
       equivocator_descriptors IM),
      equivocators_trace_project
        eqv_descriptors suf =
      Some (sufX, item_descriptors)
      ∧ equivocators_transition_item_project
          item_descriptors item =
        Some (Some itemX, pre_descriptors)
        ∧ equivocators_trace_project
            pre_descriptors pre =
          Some (preX, ieqv_descriptors)
          ∧ tr = pre ++ [item] ++ suf
eqv_descriptors: equivocator_descriptors IM
trX': list (composite_transition_item IM)
xX: composite_transition_item IM
eqv_descriptors': equivocator_descriptors IM
Hpr_x_item: equivocators_transition_item_project
  eqv_descriptors x =
Some (Some xX, eqv_descriptors')
Hpr_tr: equivocators_trace_project eqv_descriptors'
  tr = Some (trX', ieqv_descriptors)
sufX': list (composite_transition_item IM)
_xX: composite_transition_item IM
Heq: (preX ++ [itemX]) ++ sufX' = trX' ∧ _xX = xX
∃ (pre
   suf : list
           (composite_transition_item
              (equivocator_IM IM))) (item : composite_transition_item
                                              (equivocator_IM
                                                 IM)) 
  (item_descriptors
   pre_descriptors : equivocator_descriptors IM),
  equivocators_trace_project eqv_descriptors suf =
  Some (sufX' ++ [_xX], item_descriptors)
  ∧ equivocators_transition_item_project
      item_descriptors item =
    Some (Some itemX, pre_descriptors)
    ∧ equivocators_trace_project pre_descriptors pre =
      Some (preX, ieqv_descriptors)
      ∧ tr ++ [x] = pre ++ [item] ++ suf
        rewrite <- app_assoc in Heq.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
x: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
ieqv_descriptors: equivocator_descriptors IM
preX: list (composite_transition_item IM)
itemX: composite_transition_item IM
IHtr: ∀ (eqv_descriptors : equivocator_descriptors IM) 
  (sufX : list (composite_transition_item IM)),
  equivocators_trace_project eqv_descriptors tr =
  Some
    (preX ++ [itemX] ++ sufX, ieqv_descriptors)
  → ∃ (pre
       suf : list
               (composite_transition_item
                  (equivocator_IM IM))) 
      (item : composite_transition_item
                (equivocator_IM IM)) 
      (item_descriptors
       pre_descriptors : 
       equivocator_descriptors IM),
      equivocators_trace_project
        eqv_descriptors suf =
      Some (sufX, item_descriptors)
      ∧ equivocators_transition_item_project
          item_descriptors item =
        Some (Some itemX, pre_descriptors)
        ∧ equivocators_trace_project
            pre_descriptors pre =
          Some (preX, ieqv_descriptors)
          ∧ tr = pre ++ [item] ++ suf
eqv_descriptors: equivocator_descriptors IM
trX': list (composite_transition_item IM)
xX: composite_transition_item IM
eqv_descriptors': equivocator_descriptors IM
Hpr_x_item: equivocators_transition_item_project
  eqv_descriptors x =
Some (Some xX, eqv_descriptors')
Hpr_tr: equivocators_trace_project eqv_descriptors'
  tr = Some (trX', ieqv_descriptors)
sufX': list (composite_transition_item IM)
_xX: composite_transition_item IM
Heq: preX ++ [itemX] ++ sufX' = trX' ∧ _xX = xX
∃ (pre
   suf : list
           (composite_transition_item
              (equivocator_IM IM))) (item : composite_transition_item
                                              (equivocator_IM
                                                 IM)) 
  (item_descriptors
   pre_descriptors : equivocator_descriptors IM),
  equivocators_trace_project eqv_descriptors suf =
  Some (sufX' ++ [_xX], item_descriptors)
  ∧ equivocators_transition_item_project
      item_descriptors item =
    Some (Some itemX, pre_descriptors)
    ∧ equivocators_trace_project pre_descriptors pre =
      Some (preX, ieqv_descriptors)
      ∧ tr ++ [x] = pre ++ [item] ++ suf destruct Heq.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
x: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
ieqv_descriptors: equivocator_descriptors IM
preX: list (composite_transition_item IM)
itemX: composite_transition_item IM
IHtr: ∀ (eqv_descriptors : equivocator_descriptors IM) 
  (sufX : list (composite_transition_item IM)),
  equivocators_trace_project eqv_descriptors tr =
  Some
    (preX ++ [itemX] ++ sufX, ieqv_descriptors)
  → ∃ (pre
       suf : list
               (composite_transition_item
                  (equivocator_IM IM))) 
      (item : composite_transition_item
                (equivocator_IM IM)) 
      (item_descriptors
       pre_descriptors : 
       equivocator_descriptors IM),
      equivocators_trace_project
        eqv_descriptors suf =
      Some (sufX, item_descriptors)
      ∧ equivocators_transition_item_project
          item_descriptors item =
        Some (Some itemX, pre_descriptors)
        ∧ equivocators_trace_project
            pre_descriptors pre =
          Some (preX, ieqv_descriptors)
          ∧ tr = pre ++ [item] ++ suf
eqv_descriptors: equivocator_descriptors IM
trX': list (composite_transition_item IM)
xX: composite_transition_item IM
eqv_descriptors': equivocator_descriptors IM
Hpr_x_item: equivocators_transition_item_project
  eqv_descriptors x =
Some (Some xX, eqv_descriptors')
Hpr_tr: equivocators_trace_project eqv_descriptors'
  tr = Some (trX', ieqv_descriptors)
sufX': list (composite_transition_item IM)
_xX: composite_transition_item IM
H1: preX ++ [itemX] ++ sufX' = trX'
H2: _xX = xX
∃ (pre
   suf : list
           (composite_transition_item
              (equivocator_IM IM))) (item : composite_transition_item
                                              (equivocator_IM
                                                 IM)) 
  (item_descriptors
   pre_descriptors : equivocator_descriptors IM),
  equivocators_trace_project eqv_descriptors suf =
  Some (sufX' ++ [_xX], item_descriptors)
  ∧ equivocators_transition_item_project
      item_descriptors item =
    Some (Some itemX, pre_descriptors)
    ∧ equivocators_trace_project pre_descriptors pre =
      Some (preX, ieqv_descriptors)
      ∧ tr ++ [x] = pre ++ [item] ++ suf subst.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
x: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
ieqv_descriptors: equivocator_descriptors IM
preX: list (composite_transition_item IM)
itemX: composite_transition_item IM
IHtr: ∀ (eqv_descriptors : equivocator_descriptors IM) 
  (sufX : list (composite_transition_item IM)),
  equivocators_trace_project eqv_descriptors tr =
  Some
    (preX ++ [itemX] ++ sufX, ieqv_descriptors)
  → ∃ (pre
       suf : list
               (composite_transition_item
                  (equivocator_IM IM))) 
      (item : composite_transition_item
                (equivocator_IM IM)) 
      (item_descriptors
       pre_descriptors : 
       equivocator_descriptors IM),
      equivocators_trace_project
        eqv_descriptors suf =
      Some (sufX, item_descriptors)
      ∧ equivocators_transition_item_project
          item_descriptors item =
        Some (Some itemX, pre_descriptors)
        ∧ equivocators_trace_project
            pre_descriptors pre =
          Some (preX, ieqv_descriptors)
          ∧ tr = pre ++ [item] ++ suf
eqv_descriptors: equivocator_descriptors IM
xX: composite_transition_item IM
eqv_descriptors': equivocator_descriptors IM
Hpr_x_item: equivocators_transition_item_project
  eqv_descriptors x =
Some (Some xX, eqv_descriptors')
sufX': list (composite_transition_item IM)
Hpr_tr: equivocators_trace_project eqv_descriptors'
  tr =
Some
  (preX ++ [itemX] ++ sufX', ieqv_descriptors)
∃ (pre
   suf : list
           (composite_transition_item
              (equivocator_IM IM))) (item : composite_transition_item
                                              (equivocator_IM
                                                 IM)) 
  (item_descriptors
   pre_descriptors : equivocator_descriptors IM),
  equivocators_trace_project eqv_descriptors suf =
  Some (sufX' ++ [xX], item_descriptors)
  ∧ equivocators_transition_item_project
      item_descriptors item =
    Some (Some itemX, pre_descriptors)
    ∧ equivocators_trace_project pre_descriptors pre =
      Some (preX, ieqv_descriptors)
      ∧ tr ++ [x] = pre ++ [item] ++ suf
        specialize (IHtr eqv_descriptors' sufX' Hpr_tr).message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
x: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
ieqv_descriptors: equivocator_descriptors IM
preX: list (composite_transition_item IM)
itemX: composite_transition_item IM
eqv_descriptors': equivocator_descriptors IM
sufX': list (composite_transition_item IM)
IHtr: ∃ (pre
   suf : list
           (composite_transition_item
              (equivocator_IM IM))) 
  (item : composite_transition_item
            (equivocator_IM IM)) 
  (item_descriptors
   pre_descriptors : equivocator_descriptors IM),
  equivocators_trace_project eqv_descriptors'
    suf = Some (sufX', item_descriptors)
  ∧ equivocators_transition_item_project
      item_descriptors item =
    Some (Some itemX, pre_descriptors)
    ∧ equivocators_trace_project
        pre_descriptors pre =
      Some (preX, ieqv_descriptors)
      ∧ tr = pre ++ [item] ++ suf
eqv_descriptors: equivocator_descriptors IM
xX: composite_transition_item IM
Hpr_x_item: equivocators_transition_item_project
  eqv_descriptors x =
Some (Some xX, eqv_descriptors')
Hpr_tr: equivocators_trace_project eqv_descriptors'
  tr =
Some
  (preX ++ [itemX] ++ sufX', ieqv_descriptors)
∃ (pre
   suf : list
           (composite_transition_item
              (equivocator_IM IM))) (item : composite_transition_item
                                              (equivocator_IM
                                                 IM)) 
  (item_descriptors
   pre_descriptors : equivocator_descriptors IM),
  equivocators_trace_project eqv_descriptors suf =
  Some (sufX' ++ [xX], item_descriptors)
  ∧ equivocators_transition_item_project
      item_descriptors item =
    Some (Some itemX, pre_descriptors)
    ∧ equivocators_trace_project pre_descriptors pre =
      Some (preX, ieqv_descriptors)
      ∧ tr ++ [x] = pre ++ [item] ++ suf
        destruct IHtr as [pre [suf [item [item_descriptors [pre_descriptors [Hpr_suf [Hpr_item
          [Hpr_pre Heqtr]]]]]]]].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
x: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
ieqv_descriptors: equivocator_descriptors IM
preX: list (composite_transition_item IM)
itemX: composite_transition_item IM
eqv_descriptors': equivocator_descriptors IM
sufX': list (composite_transition_item IM)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
item: composite_transition_item (equivocator_IM IM)
item_descriptors, pre_descriptors: equivocator_descriptors IM
Hpr_suf: equivocators_trace_project eqv_descriptors'
  suf = Some (sufX', item_descriptors)
Hpr_item: equivocators_transition_item_project
  item_descriptors item =
Some (Some itemX, pre_descriptors)
Hpr_pre: equivocators_trace_project pre_descriptors
  pre = Some (preX, ieqv_descriptors)
Heqtr: tr = pre ++ [item] ++ suf
eqv_descriptors: equivocator_descriptors IM
xX: composite_transition_item IM
Hpr_x_item: equivocators_transition_item_project
  eqv_descriptors x =
Some (Some xX, eqv_descriptors')
Hpr_tr: equivocators_trace_project eqv_descriptors'
  tr =
Some
  (preX ++ [itemX] ++ sufX', ieqv_descriptors)
∃ (pre
   suf : list
           (composite_transition_item
              (equivocator_IM IM))) (item : composite_transition_item
                                              (equivocator_IM
                                                 IM)) 
  (item_descriptors
   pre_descriptors : equivocator_descriptors IM),
  equivocators_trace_project eqv_descriptors suf =
  Some (sufX' ++ [xX], item_descriptors)
  ∧ equivocators_transition_item_project
      item_descriptors item =
    Some (Some itemX, pre_descriptors)
    ∧ equivocators_trace_project pre_descriptors pre =
      Some (preX, ieqv_descriptors)
      ∧ tr ++ [x] = pre ++ [item] ++ suf
        exists pre, (suf ++ [x]), item, item_descriptors, pre_descriptors.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
x: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
ieqv_descriptors: equivocator_descriptors IM
preX: list (composite_transition_item IM)
itemX: composite_transition_item IM
eqv_descriptors': equivocator_descriptors IM
sufX': list (composite_transition_item IM)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
item: composite_transition_item (equivocator_IM IM)
item_descriptors, pre_descriptors: equivocator_descriptors IM
Hpr_suf: equivocators_trace_project eqv_descriptors'
  suf = Some (sufX', item_descriptors)
Hpr_item: equivocators_transition_item_project
  item_descriptors item =
Some (Some itemX, pre_descriptors)
Hpr_pre: equivocators_trace_project pre_descriptors
  pre = Some (preX, ieqv_descriptors)
Heqtr: tr = pre ++ [item] ++ suf
eqv_descriptors: equivocator_descriptors IM
xX: composite_transition_item IM
Hpr_x_item: equivocators_transition_item_project
  eqv_descriptors x =
Some (Some xX, eqv_descriptors')
Hpr_tr: equivocators_trace_project eqv_descriptors'
  tr =
Some
  (preX ++ [itemX] ++ sufX', ieqv_descriptors)
equivocators_trace_project eqv_descriptors
  (suf ++ [x]) =
Some (sufX' ++ [xX], item_descriptors)
∧ equivocators_transition_item_project
    item_descriptors item =
  Some (Some itemX, pre_descriptors)
  ∧ equivocators_trace_project pre_descriptors pre =
    Some (preX, ieqv_descriptors)
    ∧ tr ++ [x] = pre ++ [item] ++ suf ++ [x]
        subst tr.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
x: composite_transition_item (equivocator_IM IM)
ieqv_descriptors: equivocator_descriptors IM
preX: list (composite_transition_item IM)
itemX: composite_transition_item IM
eqv_descriptors': equivocator_descriptors IM
sufX': list (composite_transition_item IM)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
item: composite_transition_item (equivocator_IM IM)
item_descriptors, pre_descriptors: equivocator_descriptors IM
Hpr_suf: equivocators_trace_project eqv_descriptors'
  suf = Some (sufX', item_descriptors)
Hpr_item: equivocators_transition_item_project
  item_descriptors item =
Some (Some itemX, pre_descriptors)
Hpr_pre: equivocators_trace_project pre_descriptors
  pre = Some (preX, ieqv_descriptors)
eqv_descriptors: equivocator_descriptors IM
xX: composite_transition_item IM
Hpr_x_item: equivocators_transition_item_project
  eqv_descriptors x =
Some (Some xX, eqv_descriptors')
Hpr_tr: equivocators_trace_project eqv_descriptors'
  (pre ++ [item] ++ suf) =
Some
  (preX ++ [itemX] ++ sufX', ieqv_descriptors)
equivocators_trace_project eqv_descriptors
  (suf ++ [x]) =
Some (sufX' ++ [xX], item_descriptors)
∧ equivocators_transition_item_project
    item_descriptors item =
  Some (Some itemX, pre_descriptors)
  ∧ equivocators_trace_project pre_descriptors pre =
    Some (preX, ieqv_descriptors)
    ∧ (pre ++ [item] ++ suf) ++ [x] =
      pre ++ [item] ++ suf ++ [x] rewrite !app_assoc.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
x: composite_transition_item (equivocator_IM IM)
ieqv_descriptors: equivocator_descriptors IM
preX: list (composite_transition_item IM)
itemX: composite_transition_item IM
eqv_descriptors': equivocator_descriptors IM
sufX': list (composite_transition_item IM)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
item: composite_transition_item (equivocator_IM IM)
item_descriptors, pre_descriptors: equivocator_descriptors IM
Hpr_suf: equivocators_trace_project eqv_descriptors'
  suf = Some (sufX', item_descriptors)
Hpr_item: equivocators_transition_item_project
  item_descriptors item =
Some (Some itemX, pre_descriptors)
Hpr_pre: equivocators_trace_project pre_descriptors
  pre = Some (preX, ieqv_descriptors)
eqv_descriptors: equivocator_descriptors IM
xX: composite_transition_item IM
Hpr_x_item: equivocators_transition_item_project
  eqv_descriptors x =
Some (Some xX, eqv_descriptors')
Hpr_tr: equivocators_trace_project eqv_descriptors'
  (pre ++ [item] ++ suf) =
Some
  (preX ++ [itemX] ++ sufX', ieqv_descriptors)
equivocators_trace_project eqv_descriptors
  (suf ++ [x]) =
Some (sufX' ++ [xX], item_descriptors)
∧ equivocators_transition_item_project
    item_descriptors item =
  Some (Some itemX, pre_descriptors)
  ∧ equivocators_trace_project pre_descriptors pre =
    Some (preX, ieqv_descriptors)
    ∧ ((pre ++ [item]) ++ suf) ++ [x] =
      ((pre ++ [item]) ++ suf) ++ [x]
        repeat split; [| done | done].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
x: composite_transition_item (equivocator_IM IM)
ieqv_descriptors: equivocator_descriptors IM
preX: list (composite_transition_item IM)
itemX: composite_transition_item IM
eqv_descriptors': equivocator_descriptors IM
sufX': list (composite_transition_item IM)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
item: composite_transition_item (equivocator_IM IM)
item_descriptors, pre_descriptors: equivocator_descriptors IM
Hpr_suf: equivocators_trace_project eqv_descriptors'
  suf = Some (sufX', item_descriptors)
Hpr_item: equivocators_transition_item_project
  item_descriptors item =
Some (Some itemX, pre_descriptors)
Hpr_pre: equivocators_trace_project pre_descriptors
  pre = Some (preX, ieqv_descriptors)
eqv_descriptors: equivocator_descriptors IM
xX: composite_transition_item IM
Hpr_x_item: equivocators_transition_item_project
  eqv_descriptors x =
Some (Some xX, eqv_descriptors')
Hpr_tr: equivocators_trace_project eqv_descriptors'
  (pre ++ [item] ++ suf) =
Some
  (preX ++ [itemX] ++ sufX', ieqv_descriptors)
equivocators_trace_project eqv_descriptors
  (suf ++ [x]) =
Some (sufX' ++ [xX], item_descriptors)
        apply equivocators_trace_project_app_iff.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
x: composite_transition_item (equivocator_IM IM)
ieqv_descriptors: equivocator_descriptors IM
preX: list (composite_transition_item IM)
itemX: composite_transition_item IM
eqv_descriptors': equivocator_descriptors IM
sufX': list (composite_transition_item IM)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
item: composite_transition_item (equivocator_IM IM)
item_descriptors, pre_descriptors: equivocator_descriptors IM
Hpr_suf: equivocators_trace_project eqv_descriptors'
  suf = Some (sufX', item_descriptors)
Hpr_item: equivocators_transition_item_project
  item_descriptors item =
Some (Some itemX, pre_descriptors)
Hpr_pre: equivocators_trace_project pre_descriptors
  pre = Some (preX, ieqv_descriptors)
eqv_descriptors: equivocator_descriptors IM
xX: composite_transition_item IM
Hpr_x_item: equivocators_transition_item_project
  eqv_descriptors x =
Some (Some xX, eqv_descriptors')
Hpr_tr: equivocators_trace_project eqv_descriptors'
  (pre ++ [item] ++ suf) =
Some
  (preX ++ [itemX] ++ sufX', ieqv_descriptors)
∃ (preX sufX : list (composite_transition_item IM)) 
  (eqv_descriptors' : equivocator_descriptors IM),
  equivocators_trace_project eqv_descriptors [x] =
  Some (sufX, eqv_descriptors')
  ∧ equivocators_trace_project eqv_descriptors' suf =
    Some (preX, item_descriptors)
    ∧ sufX' ++ [xX] = preX ++ sufX
        exists sufX', [xX], eqv_descriptors'.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
x: composite_transition_item (equivocator_IM IM)
ieqv_descriptors: equivocator_descriptors IM
preX: list (composite_transition_item IM)
itemX: composite_transition_item IM
eqv_descriptors': equivocator_descriptors IM
sufX': list (composite_transition_item IM)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
item: composite_transition_item (equivocator_IM IM)
item_descriptors, pre_descriptors: equivocator_descriptors IM
Hpr_suf: equivocators_trace_project eqv_descriptors'
  suf = Some (sufX', item_descriptors)
Hpr_item: equivocators_transition_item_project
  item_descriptors item =
Some (Some itemX, pre_descriptors)
Hpr_pre: equivocators_trace_project pre_descriptors
  pre = Some (preX, ieqv_descriptors)
eqv_descriptors: equivocator_descriptors IM
xX: composite_transition_item IM
Hpr_x_item: equivocators_transition_item_project
  eqv_descriptors x =
Some (Some xX, eqv_descriptors')
Hpr_tr: equivocators_trace_project eqv_descriptors'
  (pre ++ [item] ++ suf) =
Some
  (preX ++ [itemX] ++ sufX', ieqv_descriptors)
equivocators_trace_project eqv_descriptors [x] =
Some ([xX], eqv_descriptors')
∧ equivocators_trace_project eqv_descriptors' suf =
  Some (sufX', item_descriptors)
  ∧ sufX' ++ [xX] = sufX' ++ [xX]
        repeat split; [| done].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
x: composite_transition_item (equivocator_IM IM)
ieqv_descriptors: equivocator_descriptors IM
preX: list (composite_transition_item IM)
itemX: composite_transition_item IM
eqv_descriptors': equivocator_descriptors IM
sufX': list (composite_transition_item IM)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
item: composite_transition_item (equivocator_IM IM)
item_descriptors, pre_descriptors: equivocator_descriptors IM
Hpr_suf: equivocators_trace_project eqv_descriptors'
  suf = Some (sufX', item_descriptors)
Hpr_item: equivocators_transition_item_project
  item_descriptors item =
Some (Some itemX, pre_descriptors)
Hpr_pre: equivocators_trace_project pre_descriptors
  pre = Some (preX, ieqv_descriptors)
eqv_descriptors: equivocator_descriptors IM
xX: composite_transition_item IM
Hpr_x_item: equivocators_transition_item_project
  eqv_descriptors x =
Some (Some xX, eqv_descriptors')
Hpr_tr: equivocators_trace_project eqv_descriptors'
  (pre ++ [item] ++ suf) =
Some
  (preX ++ [itemX] ++ sufX', ieqv_descriptors)
equivocators_trace_project eqv_descriptors [x] =
Some ([xX], eqv_descriptors')
        by simpl; rewrite Hpr_x_item.
Qed.

A corollary of the above, reflecting a split in the projection to the original trace.

Lemma equivocators_trace_project_app_inv
  (tr : list (composite_transition_item (equivocator_IM IM)))
  (ieqv_descriptors eqv_descriptors : equivocator_descriptors IM)
  (preX sufX : list (composite_transition_item IM))
  : equivocators_trace_project eqv_descriptors tr
    = Some (preX ++ sufX, ieqv_descriptors) ->
  exists
    (pre suf : list (composite_transition_item (equivocator_IM IM)))
    (eqv_descriptors' : equivocator_descriptors IM),
    equivocators_trace_project eqv_descriptors suf = Some (sufX, eqv_descriptors') /\
    equivocators_trace_project eqv_descriptors' pre = Some (preX, ieqv_descriptors) /\
    tr = pre ++ suf.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
tr: list
  (composite_transition_item (equivocator_IM IM))
ieqv_descriptors, eqv_descriptors: equivocator_descriptors IM
preX, sufX: list (composite_transition_item IM)
equivocators_trace_project eqv_descriptors tr =
Some (preX ++ sufX, ieqv_descriptors)
→ ∃ (pre
     suf : list
             (composite_transition_item
                (equivocator_IM IM))) (eqv_descriptors' : 
                                       equivocator_descriptors
                                         IM),
    equivocators_trace_project eqv_descriptors suf =
    Some (sufX, eqv_descriptors')
    ∧ equivocators_trace_project eqv_descriptors' pre =
      Some (preX, ieqv_descriptors) ∧ tr = pre ++ suf
Proof.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
tr: list
  (composite_transition_item (equivocator_IM IM))
ieqv_descriptors, eqv_descriptors: equivocator_descriptors IM
preX, sufX: list (composite_transition_item IM)
equivocators_trace_project eqv_descriptors tr =
Some (preX ++ sufX, ieqv_descriptors)
→ ∃ (pre
     suf : list
             (composite_transition_item
                (equivocator_IM IM))) (eqv_descriptors' : 
                                       equivocator_descriptors
                                         IM),
    equivocators_trace_project eqv_descriptors suf =
    Some (sufX, eqv_descriptors')
    ∧ equivocators_trace_project eqv_descriptors' pre =
      Some (preX, ieqv_descriptors) ∧ tr = pre ++ suf
  intro Hpr_tr.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
tr: list
  (composite_transition_item (equivocator_IM IM))
ieqv_descriptors, eqv_descriptors: equivocator_descriptors IM
preX, sufX: list (composite_transition_item IM)
Hpr_tr: equivocators_trace_project eqv_descriptors tr =
Some (preX ++ sufX, ieqv_descriptors)
∃ (pre
   suf : list
           (composite_transition_item
              (equivocator_IM IM))) (eqv_descriptors' : 
                                     equivocator_descriptors
                                       IM),
  equivocators_trace_project eqv_descriptors suf =
  Some (sufX, eqv_descriptors')
  ∧ equivocators_trace_project eqv_descriptors' pre =
    Some (preX, ieqv_descriptors) ∧ tr = pre ++ suf
  destruct sufX as [| itemX sufX].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
tr: list
  (composite_transition_item (equivocator_IM IM))
ieqv_descriptors, eqv_descriptors: equivocator_descriptors IM
preX: list (composite_transition_item IM)
Hpr_tr: equivocators_trace_project eqv_descriptors tr =
Some (preX ++ [], ieqv_descriptors)
∃ (pre
   suf : list
           (composite_transition_item
              (equivocator_IM IM))) (eqv_descriptors' : 
                                     equivocator_descriptors
                                       IM),
  equivocators_trace_project eqv_descriptors suf =
  Some ([], eqv_descriptors')
  ∧ equivocators_trace_project eqv_descriptors' pre =
    Some (preX, ieqv_descriptors) ∧ tr = pre ++ suf
message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
tr: list
  (composite_transition_item (equivocator_IM IM))
ieqv_descriptors, eqv_descriptors: equivocator_descriptors IM
preX: list (composite_transition_item IM)
itemX: composite_transition_item IM
sufX: list (composite_transition_item IM)
Hpr_tr: equivocators_trace_project eqv_descriptors tr =
Some
  (preX ++ itemX :: sufX, ieqv_descriptors)
∃ (pre
   suf : list
           (composite_transition_item
              (equivocator_IM IM))) 
  (eqv_descriptors' : equivocator_descriptors IM),
  equivocators_trace_project eqv_descriptors suf =
  Some (itemX :: sufX, eqv_descriptors')
  ∧ equivocators_trace_project eqv_descriptors' pre =
    Some (preX, ieqv_descriptors) ∧ 
    tr = pre ++ suf
  -message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
tr: list
  (composite_transition_item (equivocator_IM IM))
ieqv_descriptors, eqv_descriptors: equivocator_descriptors IM
preX: list (composite_transition_item IM)
Hpr_tr: equivocators_trace_project eqv_descriptors tr =
Some (preX ++ [], ieqv_descriptors)
∃ (pre
   suf : list
           (composite_transition_item
              (equivocator_IM IM))) (eqv_descriptors' : 
                                     equivocator_descriptors
                                       IM),
  equivocators_trace_project eqv_descriptors suf =
  Some ([], eqv_descriptors')
  ∧ equivocators_trace_project eqv_descriptors' pre =
    Some (preX, ieqv_descriptors) ∧ tr = pre ++ suf rewrite app_nil_r in Hpr_tr.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
tr: list
  (composite_transition_item (equivocator_IM IM))
ieqv_descriptors, eqv_descriptors: equivocator_descriptors IM
preX: list (composite_transition_item IM)
Hpr_tr: equivocators_trace_project eqv_descriptors tr =
Some (preX, ieqv_descriptors)
∃ (pre
   suf : list
           (composite_transition_item
              (equivocator_IM IM))) (eqv_descriptors' : 
                                     equivocator_descriptors
                                       IM),
  equivocators_trace_project eqv_descriptors suf =
  Some ([], eqv_descriptors')
  ∧ equivocators_trace_project eqv_descriptors' pre =
    Some (preX, ieqv_descriptors) ∧ tr = pre ++ suf
    exists tr, [], eqv_descriptors.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
tr: list
  (composite_transition_item (equivocator_IM IM))
ieqv_descriptors, eqv_descriptors: equivocator_descriptors IM
preX: list (composite_transition_item IM)
Hpr_tr: equivocators_trace_project eqv_descriptors tr =
Some (preX, ieqv_descriptors)
equivocators_trace_project eqv_descriptors [] =
Some ([], eqv_descriptors)
∧ equivocators_trace_project eqv_descriptors tr =
  Some (preX, ieqv_descriptors) ∧ tr = tr ++ []
    by rewrite app_nil_r.
  -message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
tr: list
  (composite_transition_item (equivocator_IM IM))
ieqv_descriptors, eqv_descriptors: equivocator_descriptors IM
preX: list (composite_transition_item IM)
itemX: composite_transition_item IM
sufX: list (composite_transition_item IM)
Hpr_tr: equivocators_trace_project eqv_descriptors tr =
Some
  (preX ++ itemX :: sufX, ieqv_descriptors)
∃ (pre
   suf : list
           (composite_transition_item
              (equivocator_IM IM))) (eqv_descriptors' : 
                                     equivocator_descriptors
                                       IM),
  equivocators_trace_project eqv_descriptors suf =
  Some (itemX :: sufX, eqv_descriptors')
  ∧ equivocators_trace_project eqv_descriptors' pre =
    Some (preX, ieqv_descriptors) ∧ tr = pre ++ suf change (itemX :: sufX) with ([itemX] ++ sufX) in Hpr_tr.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
tr: list
  (composite_transition_item (equivocator_IM IM))
ieqv_descriptors, eqv_descriptors: equivocator_descriptors IM
preX: list (composite_transition_item IM)
itemX: composite_transition_item IM
sufX: list (composite_transition_item IM)
Hpr_tr: equivocators_trace_project eqv_descriptors tr =
Some
  (preX ++ [itemX] ++ sufX, ieqv_descriptors)
∃ (pre
   suf : list
           (composite_transition_item
              (equivocator_IM IM))) (eqv_descriptors' : 
                                     equivocator_descriptors
                                       IM),
  equivocators_trace_project eqv_descriptors suf =
  Some (itemX :: sufX, eqv_descriptors')
  ∧ equivocators_trace_project eqv_descriptors' pre =
    Some (preX, ieqv_descriptors) ∧ tr = pre ++ suf
    apply equivocators_trace_project_app_inv_item in Hpr_tr.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
tr: list
  (composite_transition_item (equivocator_IM IM))
ieqv_descriptors, eqv_descriptors: equivocator_descriptors IM
preX: list (composite_transition_item IM)
itemX: composite_transition_item IM
sufX: list (composite_transition_item IM)
Hpr_tr: ∃ (pre
   suf : list
           (composite_transition_item
              (equivocator_IM IM))) 
  (item : composite_transition_item
            (equivocator_IM IM)) 
  (item_descriptors
   pre_descriptors : equivocator_descriptors
                      IM),
  equivocators_trace_project eqv_descriptors
    suf = Some (sufX, item_descriptors)
  ∧ equivocators_transition_item_project
      item_descriptors item =
    Some (Some itemX, pre_descriptors)
    ∧ equivocators_trace_project
        pre_descriptors pre =
      Some (preX, ieqv_descriptors)
      ∧ tr = pre ++ [item] ++ suf
∃ (pre
   suf : list
           (composite_transition_item
              (equivocator_IM IM))) (eqv_descriptors' : 
                                     equivocator_descriptors
                                       IM),
  equivocators_trace_project eqv_descriptors suf =
  Some (itemX :: sufX, eqv_descriptors')
  ∧ equivocators_trace_project eqv_descriptors' pre =
    Some (preX, ieqv_descriptors) ∧ tr = pre ++ suf
    destruct Hpr_tr as [pre [suf [item [item_descriptors [pre_descriptors [Hpr_suf [Hpr_item
      [Hpr_pre Heqtr]]]]]]]].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
tr: list
  (composite_transition_item (equivocator_IM IM))
ieqv_descriptors, eqv_descriptors: equivocator_descriptors IM
preX: list (composite_transition_item IM)
itemX: composite_transition_item IM
sufX: list (composite_transition_item IM)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
item: composite_transition_item (equivocator_IM IM)
item_descriptors, pre_descriptors: equivocator_descriptors IM
Hpr_suf: equivocators_trace_project eqv_descriptors
  suf = Some (sufX, item_descriptors)
Hpr_item: equivocators_transition_item_project
  item_descriptors item =
Some (Some itemX, pre_descriptors)
Hpr_pre: equivocators_trace_project pre_descriptors
  pre = Some (preX, ieqv_descriptors)
Heqtr: tr = pre ++ [item] ++ suf
∃ (pre
   suf : list
           (composite_transition_item
              (equivocator_IM IM))) (eqv_descriptors' : 
                                     equivocator_descriptors
                                       IM),
  equivocators_trace_project eqv_descriptors suf =
  Some (itemX :: sufX, eqv_descriptors')
  ∧ equivocators_trace_project eqv_descriptors' pre =
    Some (preX, ieqv_descriptors) ∧ tr = pre ++ suf
    exists pre, ([item] ++ suf), pre_descriptors.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
tr: list
  (composite_transition_item (equivocator_IM IM))
ieqv_descriptors, eqv_descriptors: equivocator_descriptors IM
preX: list (composite_transition_item IM)
itemX: composite_transition_item IM
sufX: list (composite_transition_item IM)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
item: composite_transition_item (equivocator_IM IM)
item_descriptors, pre_descriptors: equivocator_descriptors IM
Hpr_suf: equivocators_trace_project eqv_descriptors
  suf = Some (sufX, item_descriptors)
Hpr_item: equivocators_transition_item_project
  item_descriptors item =
Some (Some itemX, pre_descriptors)
Hpr_pre: equivocators_trace_project pre_descriptors
  pre = Some (preX, ieqv_descriptors)
Heqtr: tr = pre ++ [item] ++ suf
equivocators_trace_project eqv_descriptors
  ([item] ++ suf) =
Some (itemX :: sufX, pre_descriptors)
∧ equivocators_trace_project pre_descriptors pre =
  Some (preX, ieqv_descriptors)
  ∧ tr = pre ++ [item] ++ suf
    subst.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
ieqv_descriptors, eqv_descriptors: equivocator_descriptors IM
preX: list (composite_transition_item IM)
itemX: composite_transition_item IM
sufX: list (composite_transition_item IM)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
item: composite_transition_item (equivocator_IM IM)
item_descriptors, pre_descriptors: equivocator_descriptors IM
Hpr_suf: equivocators_trace_project eqv_descriptors
  suf = Some (sufX, item_descriptors)
Hpr_item: equivocators_transition_item_project
  item_descriptors item =
Some (Some itemX, pre_descriptors)
Hpr_pre: equivocators_trace_project pre_descriptors
  pre = Some (preX, ieqv_descriptors)
equivocators_trace_project eqv_descriptors
  ([item] ++ suf) =
Some (itemX :: sufX, pre_descriptors)
∧ equivocators_trace_project pre_descriptors pre =
  Some (preX, ieqv_descriptors)
  ∧ pre ++ [item] ++ suf = pre ++ [item] ++ suf repeat split; [| done].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
ieqv_descriptors, eqv_descriptors: equivocator_descriptors IM
preX: list (composite_transition_item IM)
itemX: composite_transition_item IM
sufX: list (composite_transition_item IM)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
item: composite_transition_item (equivocator_IM IM)
item_descriptors, pre_descriptors: equivocator_descriptors IM
Hpr_suf: equivocators_trace_project eqv_descriptors
  suf = Some (sufX, item_descriptors)
Hpr_item: equivocators_transition_item_project
  item_descriptors item =
Some (Some itemX, pre_descriptors)
Hpr_pre: equivocators_trace_project pre_descriptors
  pre = Some (preX, ieqv_descriptors)
equivocators_trace_project eqv_descriptors
  ([item] ++ suf) =
Some (itemX :: sufX, pre_descriptors)
    apply equivocators_trace_project_app_iff.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
ieqv_descriptors, eqv_descriptors: equivocator_descriptors IM
preX: list (composite_transition_item IM)
itemX: composite_transition_item IM
sufX: list (composite_transition_item IM)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
item: composite_transition_item (equivocator_IM IM)
item_descriptors, pre_descriptors: equivocator_descriptors IM
Hpr_suf: equivocators_trace_project eqv_descriptors
  suf = Some (sufX, item_descriptors)
Hpr_item: equivocators_transition_item_project
  item_descriptors item =
Some (Some itemX, pre_descriptors)
Hpr_pre: equivocators_trace_project pre_descriptors
  pre = Some (preX, ieqv_descriptors)
∃ (preX sufX0 : list (composite_transition_item IM)) 
  (eqv_descriptors' : equivocator_descriptors IM),
  equivocators_trace_project eqv_descriptors suf =
  Some (sufX0, eqv_descriptors')
  ∧ equivocators_trace_project eqv_descriptors' [item] =
    Some (preX, pre_descriptors)
    ∧ itemX :: sufX = preX ++ sufX0
    exists [itemX], sufX, item_descriptors.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
ieqv_descriptors, eqv_descriptors: equivocator_descriptors IM
preX: list (composite_transition_item IM)
itemX: composite_transition_item IM
sufX: list (composite_transition_item IM)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
item: composite_transition_item (equivocator_IM IM)
item_descriptors, pre_descriptors: equivocator_descriptors IM
Hpr_suf: equivocators_trace_project eqv_descriptors
  suf = Some (sufX, item_descriptors)
Hpr_item: equivocators_transition_item_project
  item_descriptors item =
Some (Some itemX, pre_descriptors)
Hpr_pre: equivocators_trace_project pre_descriptors
  pre = Some (preX, ieqv_descriptors)
equivocators_trace_project eqv_descriptors suf =
Some (sufX, item_descriptors)
∧ equivocators_trace_project item_descriptors [item] =
  Some ([itemX], pre_descriptors)
  ∧ itemX :: sufX = [itemX] ++ sufX
    repeat split; [done |].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
ieqv_descriptors, eqv_descriptors: equivocator_descriptors IM
preX: list (composite_transition_item IM)
itemX: composite_transition_item IM
sufX: list (composite_transition_item IM)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
item: composite_transition_item (equivocator_IM IM)
item_descriptors, pre_descriptors: equivocator_descriptors IM
Hpr_suf: equivocators_trace_project eqv_descriptors
  suf = Some (sufX, item_descriptors)
Hpr_item: equivocators_transition_item_project
  item_descriptors item =
Some (Some itemX, pre_descriptors)
Hpr_pre: equivocators_trace_project pre_descriptors
  pre = Some (preX, ieqv_descriptors)
equivocators_trace_project item_descriptors [item] =
Some ([itemX], pre_descriptors)
    by simpl; rewrite Hpr_item.
Qed.

Lemma equivocators_trace_project_preserves_equivocating_indices
  (descriptors idescriptors : equivocator_descriptors IM)
  (tr : list (composite_transition_item (equivocator_IM IM)))
  (trX : list (composite_transition_item IM))
  (is s : composite_state (equivocator_IM IM))
  (Htr : finite_constrained_trace_from_to (free_composite_vlsm (equivocator_IM IM)) is s tr)
  (Hdescriptors : proper_equivocator_descriptors IM descriptors s)
  (Hproject_tr : equivocators_trace_project descriptors tr = Some (trX, idescriptors))
  :
    set_union
      (equivocating_indices IM (enum index) is)
      (newmachine_descriptors_list IM (enum index) idescriptors)
    ⊆
    set_union
      (equivocating_indices IM (enum index) s)
      (newmachine_descriptors_list IM (enum index) descriptors).message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
descriptors, idescriptors: equivocator_descriptors IM
tr: list
  (composite_transition_item (equivocator_IM IM))
trX: list (composite_transition_item IM)
is, s: composite_state (equivocator_IM IM)
Htr: finite_constrained_trace_from_to
  (free_composite_vlsm (equivocator_IM IM)) is s
  tr
Hdescriptors: proper_equivocator_descriptors IM
  descriptors s
Hproject_tr: equivocators_trace_project descriptors
  tr = Some (trX, idescriptors)
set_union (equivocating_indices IM (enum index) is)
  (newmachine_descriptors_list IM (enum index)
     idescriptors)
⊆ set_union (equivocating_indices IM (enum index) s)
    (newmachine_descriptors_list IM (enum index)
       descriptors)
Proof.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
descriptors, idescriptors: equivocator_descriptors IM
tr: list
  (composite_transition_item (equivocator_IM IM))
trX: list (composite_transition_item IM)
is, s: composite_state (equivocator_IM IM)
Htr: finite_constrained_trace_from_to
  (free_composite_vlsm (equivocator_IM IM)) is s
  tr
Hdescriptors: proper_equivocator_descriptors IM
  descriptors s
Hproject_tr: equivocators_trace_project descriptors
  tr = Some (trX, idescriptors)
set_union (equivocating_indices IM (enum index) is)
  (newmachine_descriptors_list IM (enum index)
     idescriptors)
⊆ set_union (equivocating_indices IM (enum index) s)
    (newmachine_descriptors_list IM (enum index)
       descriptors)
  generalize dependent trX.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
descriptors, idescriptors: equivocator_descriptors IM
tr: list
  (composite_transition_item (equivocator_IM IM))
is, s: composite_state (equivocator_IM IM)
Htr: finite_constrained_trace_from_to
  (free_composite_vlsm (equivocator_IM IM)) is s
  tr
Hdescriptors: proper_equivocator_descriptors IM
  descriptors s
∀ trX : list (composite_transition_item IM),
  equivocators_trace_project descriptors tr =
  Some (trX, idescriptors)
  → set_union
      (equivocating_indices IM (enum index) is)
      (newmachine_descriptors_list IM (enum index)
         idescriptors)
    ⊆ set_union
        (equivocating_indices IM (enum index) s)
        (newmachine_descriptors_list IM (enum index)
           descriptors) generalize dependent descriptors.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
idescriptors: equivocator_descriptors IM
tr: list
  (composite_transition_item (equivocator_IM IM))
is, s: composite_state (equivocator_IM IM)
Htr: finite_constrained_trace_from_to
  (free_composite_vlsm (equivocator_IM IM)) is s
  tr
∀ descriptors : equivocator_descriptors IM,
  proper_equivocator_descriptors IM descriptors s
  → ∀ trX : list (composite_transition_item IM),
      equivocators_trace_project descriptors tr =
      Some (trX, idescriptors)
      → set_union
          (equivocating_indices IM (enum index) is)
          (newmachine_descriptors_list IM (enum index)
             idescriptors)
        ⊆ set_union
            (equivocating_indices IM (enum index) s)
            (newmachine_descriptors_list IM
               (enum index) descriptors)
  induction Htr using finite_valid_trace_from_to_rev_ind; [by inversion 2 |].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
idescriptors: equivocator_descriptors IM
si, s: state
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
tr: list transition_item
Htr: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
  si s tr
sf: state
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
iom, oom: option message
l: label
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
Ht: input_valid_transition
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM))) l
  (s, iom) (sf, oom)
IHHtr: ∀ descriptors : equivocator_descriptors IM,
  proper_equivocator_descriptors IM
    descriptors s
  → ∀ trX : list
              (composite_transition_item IM),
      equivocators_trace_project descriptors
        tr = Some (trX, idescriptors)
      → set_union
          (equivocating_indices IM
             (enum index) si)
          (newmachine_descriptors_list IM
             (enum index) idescriptors)
        ⊆ set_union
            (equivocating_indices IM
               (enum index) s)
            (newmachine_descriptors_list IM
               (enum index) descriptors)
∀ descriptors : equivocator_descriptors IM,
  proper_equivocator_descriptors IM descriptors sf
  → ∀ trX : list (composite_transition_item IM),
      equivocators_trace_project descriptors
        (tr ++
         [{|
            l := l;
            input := iom;
            destination := sf;
            output := oom
          |}]) = Some (trX, idescriptors)
      → set_union
          (equivocating_indices IM (enum index) si)
          (newmachine_descriptors_list IM (enum index)
             idescriptors)
        ⊆ set_union
            (equivocating_indices IM (enum index) sf)
            (newmachine_descriptors_list IM
               (enum index) descriptors)
  set (x := {| l := l |}).message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
idescriptors: equivocator_descriptors IM
si, s: state
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
tr: list transition_item
Htr: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
  si s tr
sf: state
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
iom, oom: option message
l: label
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
Ht: input_valid_transition
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM))) l
  (s, iom) (sf, oom)
IHHtr: ∀ descriptors : equivocator_descriptors IM,
  proper_equivocator_descriptors IM
    descriptors s
  → ∀ trX : list
              (composite_transition_item IM),
      equivocators_trace_project descriptors
        tr = Some (trX, idescriptors)
      → set_union
          (equivocating_indices IM
             (enum index) si)
          (newmachine_descriptors_list IM
             (enum index) idescriptors)
        ⊆ set_union
            (equivocating_indices IM
               (enum index) s)
            (newmachine_descriptors_list IM
               (enum index) descriptors)
x:= {|
  l := l;
  input := iom;
  destination := sf;
  output := oom
|}: transition_item
∀ descriptors : equivocator_descriptors IM,
  proper_equivocator_descriptors IM descriptors sf
  → ∀ trX : list (composite_transition_item IM),
      equivocators_trace_project descriptors
        (tr ++ [x]) = Some (trX, idescriptors)
      → set_union
          (equivocating_indices IM (enum index) si)
          (newmachine_descriptors_list IM (enum index)
             idescriptors)
        ⊆ set_union
            (equivocating_indices IM (enum index) sf)
            (newmachine_descriptors_list IM
               (enum index) descriptors)
  intros.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
idescriptors: equivocator_descriptors IM
si, s: state
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
tr: list transition_item
Htr: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
  si s tr
sf: state
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
iom, oom: option message
l: label
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
Ht: input_valid_transition
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM))) l
  (s, iom) (sf, oom)
IHHtr: ∀ descriptors : equivocator_descriptors IM,
  proper_equivocator_descriptors IM
    descriptors s
  → ∀ trX : list
              (composite_transition_item IM),
      equivocators_trace_project descriptors
        tr = Some (trX, idescriptors)
      → set_union
          (equivocating_indices IM
             (enum index) si)
          (newmachine_descriptors_list IM
             (enum index) idescriptors)
        ⊆ set_union
            (equivocating_indices IM
               (enum index) s)
            (newmachine_descriptors_list IM
               (enum index) descriptors)
x:= {|
  l := l;
  input := iom;
  destination := sf;
  output := oom
|}: transition_item
descriptors: equivocator_descriptors IM
Hdescriptors: proper_equivocator_descriptors IM
  descriptors sf
trX: list (composite_transition_item IM)
Hproject_tr: equivocators_trace_project descriptors
  (tr ++ [x]) = Some (trX, idescriptors)
set_union (equivocating_indices IM (enum index) si)
  (newmachine_descriptors_list IM (enum index)
     idescriptors)
⊆ set_union (equivocating_indices IM (enum index) sf)
    (newmachine_descriptors_list IM (enum index)
       descriptors)
  apply equivocators_trace_project_app_iff in Hproject_tr.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
idescriptors: equivocator_descriptors IM
si, s: state
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
tr: list transition_item
Htr: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
  si s tr
sf: state
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
iom, oom: option message
l: label
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
Ht: input_valid_transition
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM))) l
  (s, iom) (sf, oom)
IHHtr: ∀ descriptors : equivocator_descriptors IM,
  proper_equivocator_descriptors IM
    descriptors s
  → ∀ trX : list
              (composite_transition_item IM),
      equivocators_trace_project descriptors
        tr = Some (trX, idescriptors)
      → set_union
          (equivocating_indices IM
             (enum index) si)
          (newmachine_descriptors_list IM
             (enum index) idescriptors)
        ⊆ set_union
            (equivocating_indices IM
               (enum index) s)
            (newmachine_descriptors_list IM
               (enum index) descriptors)
x:= {|
  l := l;
  input := iom;
  destination := sf;
  output := oom
|}: transition_item
descriptors: equivocator_descriptors IM
Hdescriptors: proper_equivocator_descriptors IM
  descriptors sf
trX: list (composite_transition_item IM)
Hproject_tr: ∃ (preX
   sufX : list
            (composite_transition_item
               IM)) (eqv_descriptors' : 
                     equivocator_descriptors
                      IM),
  equivocators_trace_project descriptors
    [x] = Some (sufX, eqv_descriptors')
  ∧ equivocators_trace_project
      eqv_descriptors' tr =
    Some (preX, idescriptors)
    ∧ trX = preX ++ sufX
set_union (equivocating_indices IM (enum index) si)
  (newmachine_descriptors_list IM (enum index)
     idescriptors)
⊆ set_union (equivocating_indices IM (enum index) sf)
    (newmachine_descriptors_list IM (enum index)
       descriptors)
  destruct Hproject_tr as [preX [sufX [descriptors' [Hproject_x [Hproject_tr _]]]]].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
idescriptors: equivocator_descriptors IM
si, s: state
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
tr: list transition_item
Htr: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
  si s tr
sf: state
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
iom, oom: option message
l: label
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
Ht: input_valid_transition
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM))) l
  (s, iom) (sf, oom)
IHHtr: ∀ descriptors : equivocator_descriptors IM,
  proper_equivocator_descriptors IM
    descriptors s
  → ∀ trX : list
              (composite_transition_item IM),
      equivocators_trace_project descriptors
        tr = Some (trX, idescriptors)
      → set_union
          (equivocating_indices IM
             (enum index) si)
          (newmachine_descriptors_list IM
             (enum index) idescriptors)
        ⊆ set_union
            (equivocating_indices IM
               (enum index) s)
            (newmachine_descriptors_list IM
               (enum index) descriptors)
x:= {|
  l := l;
  input := iom;
  destination := sf;
  output := oom
|}: transition_item
descriptors: equivocator_descriptors IM
Hdescriptors: proper_equivocator_descriptors IM
  descriptors sf
trX, preX, sufX: list (composite_transition_item IM)
descriptors': equivocator_descriptors IM
Hproject_x: equivocators_trace_project descriptors
  [x] = Some (sufX, descriptors')
Hproject_tr: equivocators_trace_project descriptors'
  tr = Some (preX, idescriptors)
set_union (equivocating_indices IM (enum index) si)
  (newmachine_descriptors_list IM (enum index)
     idescriptors)
⊆ set_union (equivocating_indices IM (enum index) sf)
    (newmachine_descriptors_list IM (enum index)
       descriptors)
  simpl in Hproject_x.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
idescriptors: equivocator_descriptors IM
si, s: state
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
tr: list transition_item
Htr: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
  si s tr
sf: state
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
iom, oom: option message
l: label
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
Ht: input_valid_transition
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM))) l
  (s, iom) (sf, oom)
IHHtr: ∀ descriptors : equivocator_descriptors IM,
  proper_equivocator_descriptors IM
    descriptors s
  → ∀ trX : list
              (composite_transition_item IM),
      equivocators_trace_project descriptors
        tr = Some (trX, idescriptors)
      → set_union
          (equivocating_indices IM
             (enum index) si)
          (newmachine_descriptors_list IM
             (enum index) idescriptors)
        ⊆ set_union
            (equivocating_indices IM
               (enum index) s)
            (newmachine_descriptors_list IM
               (enum index) descriptors)
x:= {|
  l := l;
  input := iom;
  destination := sf;
  output := oom
|}: transition_item
descriptors: equivocator_descriptors IM
Hdescriptors: proper_equivocator_descriptors IM
  descriptors sf
trX, preX, sufX: list (composite_transition_item IM)
descriptors': equivocator_descriptors IM
Hproject_x: match
  equivocators_transition_item_project
    descriptors x
with
| Some (Some item', odescriptor) =>
    Some ([item'], odescriptor)
| Some (None, odescriptor) =>
    Some ([], odescriptor)
| None => None
end = Some (sufX, descriptors')
Hproject_tr: equivocators_trace_project descriptors'
  tr = Some (preX, idescriptors)
set_union (equivocating_indices IM (enum index) si)
  (newmachine_descriptors_list IM (enum index)
     idescriptors)
⊆ set_union (equivocating_indices IM (enum index) sf)
    (newmachine_descriptors_list IM (enum index)
       descriptors)
  destruct
    (equivocators_transition_item_project descriptors x)
    as [(oitemx, _descriptors') |] eqn: Hpr_x ; [| by congruence].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
idescriptors: equivocator_descriptors IM
si, s: state
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
tr: list transition_item
Htr: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
  si s tr
sf: state
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
iom, oom: option message
l: label
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
Ht: input_valid_transition
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM))) l
  (s, iom) (sf, oom)
IHHtr: ∀ descriptors : equivocator_descriptors IM,
  proper_equivocator_descriptors IM
    descriptors s
  → ∀ trX : list
              (composite_transition_item IM),
      equivocators_trace_project descriptors
        tr = Some (trX, idescriptors)
      → set_union
          (equivocating_indices IM
             (enum index) si)
          (newmachine_descriptors_list IM
             (enum index) idescriptors)
        ⊆ set_union
            (equivocating_indices IM
               (enum index) s)
            (newmachine_descriptors_list IM
               (enum index) descriptors)
x:= {|
  l := l;
  input := iom;
  destination := sf;
  output := oom
|}: transition_item
descriptors: equivocator_descriptors IM
Hdescriptors: proper_equivocator_descriptors IM
  descriptors sf
trX, preX, sufX: list (composite_transition_item IM)
descriptors': equivocator_descriptors IM
oitemx: option (composite_transition_item IM)
_descriptors': equivocator_descriptors IM
Hpr_x: equivocators_transition_item_project
  descriptors x = Some (oitemx, _descriptors')
Hproject_x: match oitemx with
| Some item' =>
    Some ([item'], _descriptors')
| None => Some ([], _descriptors')
end = Some (sufX, descriptors')
Hproject_tr: equivocators_trace_project descriptors'
  tr = Some (preX, idescriptors)
set_union (equivocating_indices IM (enum index) si)
  (newmachine_descriptors_list IM (enum index)
     idescriptors)
⊆ set_union (equivocating_indices IM (enum index) sf)
    (newmachine_descriptors_list IM (enum index)
       descriptors)
  assert (_descriptors' = descriptors') as -> by (destruct oitemx; congruence).message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
idescriptors: equivocator_descriptors IM
si, s: state
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
tr: list transition_item
Htr: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
  si s tr
sf: state
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
iom, oom: option message
l: label
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
Ht: input_valid_transition
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM))) l
  (s, iom) (sf, oom)
IHHtr: ∀ descriptors : equivocator_descriptors IM,
  proper_equivocator_descriptors IM
    descriptors s
  → ∀ trX : list
              (composite_transition_item IM),
      equivocators_trace_project descriptors
        tr = Some (trX, idescriptors)
      → set_union
          (equivocating_indices IM
             (enum index) si)
          (newmachine_descriptors_list IM
             (enum index) idescriptors)
        ⊆ set_union
            (equivocating_indices IM
               (enum index) s)
            (newmachine_descriptors_list IM
               (enum index) descriptors)
x:= {|
  l := l;
  input := iom;
  destination := sf;
  output := oom
|}: transition_item
descriptors: equivocator_descriptors IM
Hdescriptors: proper_equivocator_descriptors IM
  descriptors sf
trX, preX, sufX: list (composite_transition_item IM)
descriptors': equivocator_descriptors IM
oitemx: option (composite_transition_item IM)
Hproject_x: match oitemx with
| Some item' =>
    Some ([item'], descriptors')
| None => Some ([], descriptors')
end = Some (sufX, descriptors')
Hpr_x: equivocators_transition_item_project
  descriptors x = Some (oitemx, descriptors')
Hproject_tr: equivocators_trace_project descriptors'
  tr = Some (preX, idescriptors)
set_union (equivocating_indices IM (enum index) si)
  (newmachine_descriptors_list IM (enum index)
     idescriptors)
⊆ set_union (equivocating_indices IM (enum index) sf)
    (newmachine_descriptors_list IM (enum index)
       descriptors)
  clear Hproject_x trX sufX.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
idescriptors: equivocator_descriptors IM
si, s: state
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
tr: list transition_item
Htr: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
  si s tr
sf: state
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
iom, oom: option message
l: label
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
Ht: input_valid_transition
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM))) l
  (s, iom) (sf, oom)
IHHtr: ∀ descriptors : equivocator_descriptors IM,
  proper_equivocator_descriptors IM
    descriptors s
  → ∀ trX : list
              (composite_transition_item IM),
      equivocators_trace_project descriptors
        tr = Some (trX, idescriptors)
      → set_union
          (equivocating_indices IM
             (enum index) si)
          (newmachine_descriptors_list IM
             (enum index) idescriptors)
        ⊆ set_union
            (equivocating_indices IM
               (enum index) s)
            (newmachine_descriptors_list IM
               (enum index) descriptors)
x:= {|
  l := l;
  input := iom;
  destination := sf;
  output := oom
|}: transition_item
descriptors: equivocator_descriptors IM
Hdescriptors: proper_equivocator_descriptors IM
  descriptors sf
preX: list (composite_transition_item IM)
descriptors': equivocator_descriptors IM
oitemx: option (composite_transition_item IM)
Hpr_x: equivocators_transition_item_project
  descriptors x = Some (oitemx, descriptors')
Hproject_tr: equivocators_trace_project descriptors'
  tr = Some (preX, idescriptors)
set_union (equivocating_indices IM (enum index) si)
  (newmachine_descriptors_list IM (enum index)
     idescriptors)
⊆ set_union (equivocating_indices IM (enum index) sf)
    (newmachine_descriptors_list IM (enum index)
       descriptors)
  destruct Ht as [[_ [_ Hv]] Ht].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
idescriptors: equivocator_descriptors IM
si, s: state
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
tr: list transition_item
Htr: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
  si s tr
sf: state
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
iom, oom: option message
l: label
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
Hv: valid l (s, iom)
Ht: transition l (s, iom) = (sf, oom)
IHHtr: ∀ descriptors : equivocator_descriptors IM,
  proper_equivocator_descriptors IM
    descriptors s
  → ∀ trX : list
              (composite_transition_item IM),
      equivocators_trace_project descriptors
        tr = Some (trX, idescriptors)
      → set_union
          (equivocating_indices IM
             (enum index) si)
          (newmachine_descriptors_list IM
             (enum index) idescriptors)
        ⊆ set_union
            (equivocating_indices IM
               (enum index) s)
            (newmachine_descriptors_list IM
               (enum index) descriptors)
x:= {|
  l := l;
  input := iom;
  destination := sf;
  output := oom
|}: transition_item
descriptors: equivocator_descriptors IM
Hdescriptors: proper_equivocator_descriptors IM
  descriptors sf
preX: list (composite_transition_item IM)
descriptors': equivocator_descriptors IM
oitemx: option (composite_transition_item IM)
Hpr_x: equivocators_transition_item_project
  descriptors x = Some (oitemx, descriptors')
Hproject_tr: equivocators_trace_project descriptors'
  tr = Some (preX, idescriptors)
set_union (equivocating_indices IM (enum index) si)
  (newmachine_descriptors_list IM (enum index)
     idescriptors)
⊆ set_union (equivocating_indices IM (enum index) sf)
    (newmachine_descriptors_list IM (enum index)
       descriptors)
  specialize
    (equivocators_transition_item_project_preserves_equivocating_indices descriptors x
       oitemx descriptors' s Hdescriptors Ht Hv Hpr_x) as Hx_preserves.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
idescriptors: equivocator_descriptors IM
si, s: state
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
tr: list transition_item
Htr: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
  si s tr
sf: state
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
iom, oom: option message
l: label
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
Hv: valid l (s, iom)
Ht: transition l (s, iom) = (sf, oom)
IHHtr: ∀ descriptors : equivocator_descriptors IM,
  proper_equivocator_descriptors IM
    descriptors s
  → ∀ trX : list
              (composite_transition_item IM),
      equivocators_trace_project descriptors
        tr = Some (trX, idescriptors)
      → set_union
          (equivocating_indices IM
             (enum index) si)
          (newmachine_descriptors_list IM
             (enum index) idescriptors)
        ⊆ set_union
            (equivocating_indices IM
               (enum index) s)
            (newmachine_descriptors_list IM
               (enum index) descriptors)
x:= {|
  l := l;
  input := iom;
  destination := sf;
  output := oom
|}: transition_item
descriptors: equivocator_descriptors IM
Hdescriptors: proper_equivocator_descriptors IM
  descriptors sf
preX: list (composite_transition_item IM)
descriptors': equivocator_descriptors IM
oitemx: option (composite_transition_item IM)
Hpr_x: equivocators_transition_item_project
  descriptors x = Some (oitemx, descriptors')
Hproject_tr: equivocators_trace_project descriptors'
  tr = Some (preX, idescriptors)
Hx_preserves: set_union
  (equivocating_indices IM 
     (enum index) s)
  (newmachine_descriptors_list IM
     (enum index) descriptors')
⊆ set_union
    (equivocating_indices IM
       (enum index) (destination x))
    (newmachine_descriptors_list IM
       (enum index) descriptors)
set_union (equivocating_indices IM (enum index) si)
  (newmachine_descriptors_list IM (enum index)
     idescriptors)
⊆ set_union (equivocating_indices IM (enum index) sf)
    (newmachine_descriptors_list IM (enum index)
       descriptors)
  specialize
    (equivocators_transition_item_project_proper_characterization descriptors x Hdescriptors)
    as Hpr_x_char.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
idescriptors: equivocator_descriptors IM
si, s: state
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
tr: list transition_item
Htr: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
  si s tr
sf: state
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
iom, oom: option message
l: label
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
Hv: valid l (s, iom)
Ht: transition l (s, iom) = (sf, oom)
IHHtr: ∀ descriptors : equivocator_descriptors IM,
  proper_equivocator_descriptors IM
    descriptors s
  → ∀ trX : list
              (composite_transition_item IM),
      equivocators_trace_project descriptors
        tr = Some (trX, idescriptors)
      → set_union
          (equivocating_indices IM
             (enum index) si)
          (newmachine_descriptors_list IM
             (enum index) idescriptors)
        ⊆ set_union
            (equivocating_indices IM
               (enum index) s)
            (newmachine_descriptors_list IM
               (enum index) descriptors)
x:= {|
  l := l;
  input := iom;
  destination := sf;
  output := oom
|}: transition_item
descriptors: equivocator_descriptors IM
Hdescriptors: proper_equivocator_descriptors IM
  descriptors sf
preX: list (composite_transition_item IM)
descriptors': equivocator_descriptors IM
oitemx: option (composite_transition_item IM)
Hpr_x: equivocators_transition_item_project
  descriptors x = Some (oitemx, descriptors')
Hproject_tr: equivocators_trace_project descriptors'
  tr = Some (preX, idescriptors)
Hx_preserves: set_union
  (equivocating_indices IM 
     (enum index) s)
  (newmachine_descriptors_list IM
     (enum index) descriptors')
⊆ set_union
    (equivocating_indices IM
       (enum index) (destination x))
    (newmachine_descriptors_list IM
       (enum index) descriptors)
Hpr_x_char: ∃ (oitem : option
             (composite_transition_item
                IM)) (eqv_descriptors' : 
                      equivocator_descriptors
                      IM),
  equivocators_transition_item_project
    descriptors x =
  Some (oitem, eqv_descriptors')
  ∧ match oitem with
    | Some itemx =>
        (∃ Hex : existing_equivocator_label
                   (IM
                      (projT1 (VLSM.l x)))
                   (projT2 (VLSM.l x)),
           existT (projT1 (VLSM.l x))
             (existing_equivocator_label_extract
                (IM (projT1 (VLSM.l x)))
                (projT2 (VLSM.l x)) Hex) =
           VLSM.l itemx)
        ∧ input x = input itemx
          ∧ output x = output itemx
            ∧ equivocators_state_project
                IM descriptors
                (destination x) =
              destination itemx
              ∧ eqv_descriptors'
                  (projT1 (VLSM.l x)) =
                equivocator_label_descriptor
                  (projT2 (VLSM.l x))
    | None => True
    end
    ∧ (∀ s : composite_state
               (equivocator_IM IM),
         composite_valid
           (equivocator_IM IM) 
           (VLSM.l x) (
           s, input x)
         → composite_transition
             (equivocator_IM IM)
             (VLSM.l x) (
             s, input x) =
           (destination x, output x)
           → proper_equivocator_descriptors
               IM eqv_descriptors' s
             ∧ eqv_descriptors' =
               equivocator_descriptors_update
                 IM descriptors
                 (projT1 (VLSM.l x))
                 (eqv_descriptors'
                    (projT1 (VLSM.l x)))
               ∧ s =
                 state_update
                   (equivocator_IM IM)
                   (destination x)
                   (projT1 (VLSM.l x))
                   (s (projT1 (VLSM.l x)))
                 ∧ previous_state_descriptor_prop
                     (IM
                      (projT1 (VLSM.l x)))
                     (descriptors
                      (projT1 (VLSM.l x)))
                     (s
                      (projT1 (VLSM.l x)))
                     (eqv_descriptors'
                      (projT1 (VLSM.l x)))
                   ∧ match oitem with
                     | Some itemx =>
                      ∀ sx : 
                      composite_state IM,
                      sx =
                      equivocators_state_project
                      IM eqv_descriptors'
                      s
                      → 
                      composite_valid IM
                      (VLSM.l itemx)
                      (sx, input itemx)
                      ∧ 
                      composite_transition
                      IM 
                      (VLSM.l itemx)
                      (sx, input itemx) =
                      (
                      destination itemx,
                      output itemx)
                     | None =>
                      equivocators_state_project
                      IM descriptors
                      (destination x) =
                      equivocators_state_project
                      IM eqv_descriptors'
                      s
                     end)
set_union (equivocating_indices IM (enum index) si)
  (newmachine_descriptors_list IM (enum index)
     idescriptors)
⊆ set_union (equivocating_indices IM (enum index) sf)
    (newmachine_descriptors_list IM (enum index)
       descriptors)
  rewrite Hpr_x in Hpr_x_char.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
idescriptors: equivocator_descriptors IM
si, s: state
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
tr: list transition_item
Htr: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
  si s tr
sf: state
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
iom, oom: option message
l: label
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
Hv: valid l (s, iom)
Ht: transition l (s, iom) = (sf, oom)
IHHtr: ∀ descriptors : equivocator_descriptors IM,
  proper_equivocator_descriptors IM
    descriptors s
  → ∀ trX : list
              (composite_transition_item IM),
      equivocators_trace_project descriptors
        tr = Some (trX, idescriptors)
      → set_union
          (equivocating_indices IM
             (enum index) si)
          (newmachine_descriptors_list IM
             (enum index) idescriptors)
        ⊆ set_union
            (equivocating_indices IM
               (enum index) s)
            (newmachine_descriptors_list IM
               (enum index) descriptors)
x:= {|
  l := l;
  input := iom;
  destination := sf;
  output := oom
|}: transition_item
descriptors: equivocator_descriptors IM
Hdescriptors: proper_equivocator_descriptors IM
  descriptors sf
preX: list (composite_transition_item IM)
descriptors': equivocator_descriptors IM
oitemx: option (composite_transition_item IM)
Hpr_x: equivocators_transition_item_project
  descriptors x = Some (oitemx, descriptors')
Hproject_tr: equivocators_trace_project descriptors'
  tr = Some (preX, idescriptors)
Hx_preserves: set_union
  (equivocating_indices IM 
     (enum index) s)
  (newmachine_descriptors_list IM
     (enum index) descriptors')
⊆ set_union
    (equivocating_indices IM
       (enum index) (destination x))
    (newmachine_descriptors_list IM
       (enum index) descriptors)
Hpr_x_char: ∃ (oitem : option
             (composite_transition_item
                IM)) (eqv_descriptors' : 
                      equivocator_descriptors
                      IM),
  Some (oitemx, descriptors') =
  Some (oitem, eqv_descriptors')
  ∧ match oitem with
    | Some itemx =>
        (∃ Hex : existing_equivocator_label
                   (IM
                      (projT1 (VLSM.l x)))
                   (projT2 (VLSM.l x)),
           existT (projT1 (VLSM.l x))
             (existing_equivocator_label_extract
                (IM (projT1 (VLSM.l x)))
                (projT2 (VLSM.l x)) Hex) =
           VLSM.l itemx)
        ∧ input x = input itemx
          ∧ output x = output itemx
            ∧ equivocators_state_project
                IM descriptors
                (destination x) =
              destination itemx
              ∧ eqv_descriptors'
                  (projT1 (VLSM.l x)) =
                equivocator_label_descriptor
                  (projT2 (VLSM.l x))
    | None => True
    end
    ∧ (∀ s : composite_state
               (equivocator_IM IM),
         composite_valid
           (equivocator_IM IM) 
           (VLSM.l x) (
           s, input x)
         → composite_transition
             (equivocator_IM IM)
             (VLSM.l x) (
             s, input x) =
           (destination x, output x)
           → proper_equivocator_descriptors
               IM eqv_descriptors' s
             ∧ eqv_descriptors' =
               equivocator_descriptors_update
                 IM descriptors
                 (projT1 (VLSM.l x))
                 (eqv_descriptors'
                    (projT1 (VLSM.l x)))
               ∧ s =
                 state_update
                   (equivocator_IM IM)
                   (destination x)
                   (projT1 (VLSM.l x))
                   (s (projT1 (VLSM.l x)))
                 ∧ previous_state_descriptor_prop
                     (IM
                      (projT1 (VLSM.l x)))
                     (descriptors
                      (projT1 (VLSM.l x)))
                     (s
                      (projT1 (VLSM.l x)))
                     (eqv_descriptors'
                      (projT1 (VLSM.l x)))
                   ∧ match oitem with
                     | Some itemx =>
                      ∀ sx : 
                      composite_state IM,
                      sx =
                      equivocators_state_project
                      IM eqv_descriptors'
                      s
                      → 
                      composite_valid IM
                      (VLSM.l itemx)
                      (sx, input itemx)
                      ∧ 
                      composite_transition
                      IM 
                      (VLSM.l itemx)
                      (sx, input itemx) =
                      (
                      destination itemx,
                      output itemx)
                     | None =>
                      equivocators_state_project
                      IM descriptors
                      (destination x) =
                      equivocators_state_project
                      IM eqv_descriptors'
                      s
                     end)
set_union (equivocating_indices IM (enum index) si)
  (newmachine_descriptors_list IM (enum index)
     idescriptors)
⊆ set_union (equivocating_indices IM (enum index) sf)
    (newmachine_descriptors_list IM (enum index)
       descriptors)
  destruct Hpr_x_char as [_ [_ [[= <- <-] [_ Hchar2]]]].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
idescriptors: equivocator_descriptors IM
si, s: state
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
tr: list transition_item
Htr: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
  si s tr
sf: state
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
iom, oom: option message
l: label
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
Hv: valid l (s, iom)
Ht: transition l (s, iom) = (sf, oom)
IHHtr: ∀ descriptors : equivocator_descriptors IM,
  proper_equivocator_descriptors IM
    descriptors s
  → ∀ trX : list
              (composite_transition_item IM),
      equivocators_trace_project descriptors
        tr = Some (trX, idescriptors)
      → set_union
          (equivocating_indices IM
             (enum index) si)
          (newmachine_descriptors_list IM
             (enum index) idescriptors)
        ⊆ set_union
            (equivocating_indices IM
               (enum index) s)
            (newmachine_descriptors_list IM
               (enum index) descriptors)
x:= {|
  l := l;
  input := iom;
  destination := sf;
  output := oom
|}: transition_item
descriptors: equivocator_descriptors IM
Hdescriptors: proper_equivocator_descriptors IM
  descriptors sf
preX: list (composite_transition_item IM)
descriptors': equivocator_descriptors IM
oitemx: option (composite_transition_item IM)
Hpr_x: equivocators_transition_item_project
  descriptors x = Some (oitemx, descriptors')
Hproject_tr: equivocators_trace_project descriptors'
  tr = Some (preX, idescriptors)
Hx_preserves: set_union
  (equivocating_indices IM 
     (enum index) s)
  (newmachine_descriptors_list IM
     (enum index) descriptors')
⊆ set_union
    (equivocating_indices IM
       (enum index) (destination x))
    (newmachine_descriptors_list IM
       (enum index) descriptors)
Hchar2: ∀ s : composite_state (equivocator_IM IM),
  composite_valid (equivocator_IM IM)
    (VLSM.l x) (s, input x)
  → composite_transition 
      (equivocator_IM IM) 
      (VLSM.l x) (s, input x) =
    (destination x, output x)
    → proper_equivocator_descriptors IM
        descriptors' s
      ∧ descriptors' =
        equivocator_descriptors_update IM
          descriptors (projT1 (VLSM.l x))
          (descriptors' (projT1 (VLSM.l x)))
        ∧ s =
          state_update 
            (equivocator_IM IM)
            (destination x)
            (projT1 (VLSM.l x))
            (s (projT1 (VLSM.l x)))
          ∧ previous_state_descriptor_prop
              (IM (projT1 (VLSM.l x)))
              (descriptors
                 (projT1 (VLSM.l x)))
              (s (projT1 (VLSM.l x)))
              (descriptors'
                 (projT1 (VLSM.l x)))
            ∧ match oitemx with
              | Some itemx =>
                  ∀ sx : composite_state IM,
                    sx =
                    equivocators_state_project
                      IM descriptors' s
                    → composite_valid IM
                      (VLSM.l itemx)
                      (sx, input itemx)
                      ∧ 
                      composite_transition IM
                      (VLSM.l itemx)
                      (sx, input itemx) =
                      (
                      destination itemx,
                      output itemx)
              | None =>
                  equivocators_state_project
                    IM descriptors
                    (destination x) =
                  equivocators_state_project
                    IM descriptors' s
              end
set_union (equivocating_indices IM (enum index) si)
  (newmachine_descriptors_list IM (enum index)
     idescriptors)
⊆ set_union (equivocating_indices IM (enum index) sf)
    (newmachine_descriptors_list IM (enum index)
       descriptors)
  specialize (Hchar2 s Hv Ht) as [Hdescriptors' _].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
idescriptors: equivocator_descriptors IM
si, s: state
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
tr: list transition_item
Htr: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
  si s tr
sf: state
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
iom, oom: option message
l: label
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
Hv: valid l (s, iom)
Ht: transition l (s, iom) = (sf, oom)
IHHtr: ∀ descriptors : equivocator_descriptors IM,
  proper_equivocator_descriptors IM
    descriptors s
  → ∀ trX : list
              (composite_transition_item IM),
      equivocators_trace_project descriptors
        tr = Some (trX, idescriptors)
      → set_union
          (equivocating_indices IM
             (enum index) si)
          (newmachine_descriptors_list IM
             (enum index) idescriptors)
        ⊆ set_union
            (equivocating_indices IM
               (enum index) s)
            (newmachine_descriptors_list IM
               (enum index) descriptors)
x:= {|
  l := l;
  input := iom;
  destination := sf;
  output := oom
|}: transition_item
descriptors: equivocator_descriptors IM
Hdescriptors: proper_equivocator_descriptors IM
  descriptors sf
preX: list (composite_transition_item IM)
descriptors': equivocator_descriptors IM
oitemx: option (composite_transition_item IM)
Hpr_x: equivocators_transition_item_project
  descriptors x = Some (oitemx, descriptors')
Hproject_tr: equivocators_trace_project descriptors'
  tr = Some (preX, idescriptors)
Hx_preserves: set_union
  (equivocating_indices IM 
     (enum index) s)
  (newmachine_descriptors_list IM
     (enum index) descriptors')
⊆ set_union
    (equivocating_indices IM
       (enum index) (destination x))
    (newmachine_descriptors_list IM
       (enum index) descriptors)
Hdescriptors': proper_equivocator_descriptors IM
  descriptors' s
set_union (equivocating_indices IM (enum index) si)
  (newmachine_descriptors_list IM (enum index)
     idescriptors)
⊆ set_union (equivocating_indices IM (enum index) sf)
    (newmachine_descriptors_list IM (enum index)
       descriptors)
  specialize (IHHtr _ Hdescriptors' _ Hproject_tr).message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
idescriptors: equivocator_descriptors IM
si, s: state
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
tr: list transition_item
Htr: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
  si s tr
sf: state
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
iom, oom: option message
l: label
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
Hv: valid l (s, iom)
Ht: transition l (s, iom) = (sf, oom)
descriptors': equivocator_descriptors IM
IHHtr: set_union
  (equivocating_indices IM (enum index) si)
  (newmachine_descriptors_list IM 
     (enum index) idescriptors)
⊆ set_union
    (equivocating_indices IM (enum index) s)
    (newmachine_descriptors_list IM
       (enum index) descriptors')
x:= {|
  l := l;
  input := iom;
  destination := sf;
  output := oom
|}: transition_item
descriptors: equivocator_descriptors IM
Hdescriptors: proper_equivocator_descriptors IM
  descriptors sf
preX: list (composite_transition_item IM)
oitemx: option (composite_transition_item IM)
Hpr_x: equivocators_transition_item_project
  descriptors x = Some (oitemx, descriptors')
Hproject_tr: equivocators_trace_project descriptors'
  tr = Some (preX, idescriptors)
Hx_preserves: set_union
  (equivocating_indices IM 
     (enum index) s)
  (newmachine_descriptors_list IM
     (enum index) descriptors')
⊆ set_union
    (equivocating_indices IM
       (enum index) (destination x))
    (newmachine_descriptors_list IM
       (enum index) descriptors)
Hdescriptors': proper_equivocator_descriptors IM
  descriptors' s
set_union (equivocating_indices IM (enum index) si)
  (newmachine_descriptors_list IM (enum index)
     idescriptors)
⊆ set_union (equivocating_indices IM (enum index) sf)
    (newmachine_descriptors_list IM (enum index)
       descriptors)
  by etransitivity.
Qed.

The state and descriptors obtained after applying equivocators_trace_project on a preloaded valid trace satisfy the previous_state_descriptor_property.

Lemma equivocators_trace_project_from_state_descriptors
  (descriptors idescriptors : equivocator_descriptors IM)
  (tr : list (composite_transition_item (equivocator_IM IM)))
  (trX : list (composite_transition_item IM))
  (is s : composite_state (equivocator_IM IM))
  (Htr : finite_constrained_trace_from_to (free_composite_vlsm (equivocator_IM IM)) is s tr)
  (Hdescriptors : proper_equivocator_descriptors IM descriptors s)
  (Hproject_tr : equivocators_trace_project descriptors tr = Some (trX, idescriptors))
  : forall eqv, previous_state_descriptor_prop (IM eqv) (descriptors eqv) (is eqv) (idescriptors eqv).message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
descriptors, idescriptors: equivocator_descriptors IM
tr: list
  (composite_transition_item (equivocator_IM IM))
trX: list (composite_transition_item IM)
is, s: composite_state (equivocator_IM IM)
Htr: finite_constrained_trace_from_to
  (free_composite_vlsm (equivocator_IM IM)) is s
  tr
Hdescriptors: proper_equivocator_descriptors IM
  descriptors s
Hproject_tr: equivocators_trace_project descriptors
  tr = Some (trX, idescriptors)
∀ eqv : index,
  previous_state_descriptor_prop (IM eqv)
    (descriptors eqv) (is eqv) (idescriptors eqv)
Proof.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
descriptors, idescriptors: equivocator_descriptors IM
tr: list
  (composite_transition_item (equivocator_IM IM))
trX: list (composite_transition_item IM)
is, s: composite_state (equivocator_IM IM)
Htr: finite_constrained_trace_from_to
  (free_composite_vlsm (equivocator_IM IM)) is s
  tr
Hdescriptors: proper_equivocator_descriptors IM
  descriptors s
Hproject_tr: equivocators_trace_project descriptors
  tr = Some (trX, idescriptors)
∀ eqv : index,
  previous_state_descriptor_prop (IM eqv)
    (descriptors eqv) (is eqv) (idescriptors eqv)
  generalize dependent trX.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
descriptors, idescriptors: equivocator_descriptors IM
tr: list
  (composite_transition_item (equivocator_IM IM))
is, s: composite_state (equivocator_IM IM)
Htr: finite_constrained_trace_from_to
  (free_composite_vlsm (equivocator_IM IM)) is s
  tr
Hdescriptors: proper_equivocator_descriptors IM
  descriptors s
∀ trX : list (composite_transition_item IM),
  equivocators_trace_project descriptors tr =
  Some (trX, idescriptors)
  → ∀ eqv : index,
      previous_state_descriptor_prop (IM eqv)
        (descriptors eqv) (is eqv) (idescriptors eqv)
  generalize dependent descriptors.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
idescriptors: equivocator_descriptors IM
tr: list
  (composite_transition_item (equivocator_IM IM))
is, s: composite_state (equivocator_IM IM)
Htr: finite_constrained_trace_from_to
  (free_composite_vlsm (equivocator_IM IM)) is s
  tr
∀ descriptors : equivocator_descriptors IM,
  proper_equivocator_descriptors IM descriptors s
  → ∀ trX : list (composite_transition_item IM),
      equivocators_trace_project descriptors tr =
      Some (trX, idescriptors)
      → ∀ eqv : index,
          previous_state_descriptor_prop (IM eqv)
            (descriptors eqv) (is eqv)
            (idescriptors eqv)
  generalize dependent s.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
idescriptors: equivocator_descriptors IM
tr: list
  (composite_transition_item (equivocator_IM IM))
is: composite_state (equivocator_IM IM)
∀ s : composite_state (equivocator_IM IM),
  finite_constrained_trace_from_to
    (free_composite_vlsm (equivocator_IM IM)) is s tr
  → ∀ descriptors : equivocator_descriptors IM,
      proper_equivocator_descriptors IM descriptors s
      → ∀ trX : list (composite_transition_item IM),
          equivocators_trace_project descriptors tr =
          Some (trX, idescriptors)
          → ∀ eqv : index,
              previous_state_descriptor_prop (IM eqv)
                (descriptors eqv) (is eqv)
                (idescriptors eqv)
  induction tr using rev_ind; intros.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
idescriptors: equivocator_descriptors IM
is, s: composite_state (equivocator_IM IM)
Htr: finite_constrained_trace_from_to
  (free_composite_vlsm (equivocator_IM IM)) is s
  []
descriptors: equivocator_descriptors IM
Hdescriptors: proper_equivocator_descriptors IM
  descriptors s
trX: list (composite_transition_item IM)
Hproject_tr: equivocators_trace_project descriptors
  [] = Some (trX, idescriptors)
eqv: index
previous_state_descriptor_prop (IM eqv)
  (descriptors eqv) (is eqv) (idescriptors eqv)
message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
idescriptors: equivocator_descriptors IM
x: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
is: composite_state (equivocator_IM IM)
IHtr: ∀ s : composite_state (equivocator_IM IM),
  finite_constrained_trace_from_to
    (free_composite_vlsm (equivocator_IM IM))
    is s tr
  → ∀ descriptors : equivocator_descriptors IM,
      proper_equivocator_descriptors IM
        descriptors s
      → ∀ trX : list
                  (composite_transition_item IM),
          equivocators_trace_project
            descriptors tr =
          Some (trX, idescriptors)
          → ∀ eqv : index,
              previous_state_descriptor_prop
                (IM eqv) 
                (descriptors eqv) 
                (is eqv) 
                (idescriptors eqv)
s: composite_state (equivocator_IM IM)
Htr: finite_constrained_trace_from_to
  (free_composite_vlsm (equivocator_IM IM)) is s
  (tr ++ [x])
descriptors: equivocator_descriptors IM
Hdescriptors: proper_equivocator_descriptors IM
  descriptors s
trX: list (composite_transition_item IM)
Hproject_tr: equivocators_trace_project descriptors
  (tr ++ [x]) = Some (trX, idescriptors)
eqv: index
previous_state_descriptor_prop 
  (IM eqv) (descriptors eqv) 
  (is eqv) (idescriptors eqv)
  -message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
idescriptors: equivocator_descriptors IM
is, s: composite_state (equivocator_IM IM)
Htr: finite_constrained_trace_from_to
  (free_composite_vlsm (equivocator_IM IM)) is s
  []
descriptors: equivocator_descriptors IM
Hdescriptors: proper_equivocator_descriptors IM
  descriptors s
trX: list (composite_transition_item IM)
Hproject_tr: equivocators_trace_project descriptors
  [] = Some (trX, idescriptors)
eqv: index
previous_state_descriptor_prop (IM eqv)
  (descriptors eqv) (is eqv) (idescriptors eqv) by inversion Hproject_tr; subst; destruct (idescriptors eqv); simpl; [| lia].
  -message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
idescriptors: equivocator_descriptors IM
x: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
is: composite_state (equivocator_IM IM)
IHtr: ∀ s : composite_state (equivocator_IM IM),
  finite_constrained_trace_from_to
    (free_composite_vlsm (equivocator_IM IM))
    is s tr
  → ∀ descriptors : equivocator_descriptors IM,
      proper_equivocator_descriptors IM
        descriptors s
      → ∀ trX : list
                  (composite_transition_item IM),
          equivocators_trace_project
            descriptors tr =
          Some (trX, idescriptors)
          → ∀ eqv : index,
              previous_state_descriptor_prop
                (IM eqv) 
                (descriptors eqv) 
                (is eqv) 
                (idescriptors eqv)
s: composite_state (equivocator_IM IM)
Htr: finite_constrained_trace_from_to
  (free_composite_vlsm (equivocator_IM IM)) is s
  (tr ++ [x])
descriptors: equivocator_descriptors IM
Hdescriptors: proper_equivocator_descriptors IM
  descriptors s
trX: list (composite_transition_item IM)
Hproject_tr: equivocators_trace_project descriptors
  (tr ++ [x]) = Some (trX, idescriptors)
eqv: index
previous_state_descriptor_prop (IM eqv)
  (descriptors eqv) (is eqv) (idescriptors eqv) apply finite_valid_trace_from_to_last in Htr as Heq_s.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
idescriptors: equivocator_descriptors IM
x: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
is: composite_state (equivocator_IM IM)
IHtr: ∀ s : composite_state (equivocator_IM IM),
  finite_constrained_trace_from_to
    (free_composite_vlsm (equivocator_IM IM))
    is s tr
  → ∀ descriptors : equivocator_descriptors IM,
      proper_equivocator_descriptors IM
        descriptors s
      → ∀ trX : list
                  (composite_transition_item IM),
          equivocators_trace_project
            descriptors tr =
          Some (trX, idescriptors)
          → ∀ eqv : index,
              previous_state_descriptor_prop
                (IM eqv) 
                (descriptors eqv) 
                (is eqv) 
                (idescriptors eqv)
s: composite_state (equivocator_IM IM)
Htr: finite_constrained_trace_from_to
  (free_composite_vlsm (equivocator_IM IM)) is s
  (tr ++ [x])
descriptors: equivocator_descriptors IM
Hdescriptors: proper_equivocator_descriptors IM
  descriptors s
trX: list (composite_transition_item IM)
Hproject_tr: equivocators_trace_project descriptors
  (tr ++ [x]) = Some (trX, idescriptors)
eqv: index
Heq_s: finite_trace_last is (tr ++ [x]) = s
previous_state_descriptor_prop (IM eqv)
  (descriptors eqv) (is eqv) (idescriptors eqv)
    rewrite finite_trace_last_is_last in Heq_s.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
idescriptors: equivocator_descriptors IM
x: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
is: composite_state (equivocator_IM IM)
IHtr: ∀ s : composite_state (equivocator_IM IM),
  finite_constrained_trace_from_to
    (free_composite_vlsm (equivocator_IM IM))
    is s tr
  → ∀ descriptors : equivocator_descriptors IM,
      proper_equivocator_descriptors IM
        descriptors s
      → ∀ trX : list
                  (composite_transition_item IM),
          equivocators_trace_project
            descriptors tr =
          Some (trX, idescriptors)
          → ∀ eqv : index,
              previous_state_descriptor_prop
                (IM eqv) 
                (descriptors eqv) 
                (is eqv) 
                (idescriptors eqv)
s: composite_state (equivocator_IM IM)
Htr: finite_constrained_trace_from_to
  (free_composite_vlsm (equivocator_IM IM)) is s
  (tr ++ [x])
descriptors: equivocator_descriptors IM
Hdescriptors: proper_equivocator_descriptors IM
  descriptors s
trX: list (composite_transition_item IM)
Hproject_tr: equivocators_trace_project descriptors
  (tr ++ [x]) = Some (trX, idescriptors)
eqv: index
Heq_s: destination x = s
previous_state_descriptor_prop (IM eqv)
  (descriptors eqv) (is eqv) (idescriptors eqv) subst s.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
idescriptors: equivocator_descriptors IM
x: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
is: composite_state (equivocator_IM IM)
IHtr: ∀ s : composite_state (equivocator_IM IM),
  finite_constrained_trace_from_to
    (free_composite_vlsm (equivocator_IM IM))
    is s tr
  → ∀ descriptors : equivocator_descriptors IM,
      proper_equivocator_descriptors IM
        descriptors s
      → ∀ trX : list
                  (composite_transition_item IM),
          equivocators_trace_project
            descriptors tr =
          Some (trX, idescriptors)
          → ∀ eqv : index,
              previous_state_descriptor_prop
                (IM eqv) 
                (descriptors eqv) 
                (is eqv) 
                (idescriptors eqv)
Htr: finite_constrained_trace_from_to
  (free_composite_vlsm (equivocator_IM IM)) is
  (destination x) (tr ++ [x])
descriptors: equivocator_descriptors IM
Hdescriptors: proper_equivocator_descriptors IM
  descriptors (destination x)
trX: list (composite_transition_item IM)
Hproject_tr: equivocators_trace_project descriptors
  (tr ++ [x]) = Some (trX, idescriptors)
eqv: index
previous_state_descriptor_prop (IM eqv)
  (descriptors eqv) (is eqv) (idescriptors eqv)
    apply finite_valid_trace_from_to_app_split in Htr.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
idescriptors: equivocator_descriptors IM
x: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
is: composite_state (equivocator_IM IM)
IHtr: ∀ s : composite_state (equivocator_IM IM),
  finite_constrained_trace_from_to
    (free_composite_vlsm (equivocator_IM IM))
    is s tr
  → ∀ descriptors : equivocator_descriptors IM,
      proper_equivocator_descriptors IM
        descriptors s
      → ∀ trX : list
                  (composite_transition_item IM),
          equivocators_trace_project
            descriptors tr =
          Some (trX, idescriptors)
          → ∀ eqv : index,
              previous_state_descriptor_prop
                (IM eqv) 
                (descriptors eqv) 
                (is eqv) 
                (idescriptors eqv)
Htr: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
  is (finite_trace_last is tr) tr
∧ finite_valid_trace_from_to
    (preloaded_with_all_messages_vlsm
       (free_composite_vlsm (equivocator_IM IM)))
    (finite_trace_last is tr) 
    (destination x) [x]
descriptors: equivocator_descriptors IM
Hdescriptors: proper_equivocator_descriptors IM
  descriptors (destination x)
trX: list (composite_transition_item IM)
Hproject_tr: equivocators_trace_project descriptors
  (tr ++ [x]) = Some (trX, idescriptors)
eqv: index
previous_state_descriptor_prop (IM eqv)
  (descriptors eqv) (is eqv) (idescriptors eqv)
    destruct Htr as [Htr Hx].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
idescriptors: equivocator_descriptors IM
x: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
is: composite_state (equivocator_IM IM)
IHtr: ∀ s : composite_state (equivocator_IM IM),
  finite_constrained_trace_from_to
    (free_composite_vlsm (equivocator_IM IM))
    is s tr
  → ∀ descriptors : equivocator_descriptors IM,
      proper_equivocator_descriptors IM
        descriptors s
      → ∀ trX : list
                  (composite_transition_item IM),
          equivocators_trace_project
            descriptors tr =
          Some (trX, idescriptors)
          → ∀ eqv : index,
              previous_state_descriptor_prop
                (IM eqv) 
                (descriptors eqv) 
                (is eqv) 
                (idescriptors eqv)
Htr: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
  is (finite_trace_last is tr) tr
Hx: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
  (finite_trace_last is tr) 
  (destination x) [x]
descriptors: equivocator_descriptors IM
Hdescriptors: proper_equivocator_descriptors IM
  descriptors (destination x)
trX: list (composite_transition_item IM)
Hproject_tr: equivocators_trace_project descriptors
  (tr ++ [x]) = Some (trX, idescriptors)
eqv: index
previous_state_descriptor_prop (IM eqv)
  (descriptors eqv) (is eqv) (idescriptors eqv)
    specialize (equivocators_pre_trace_cannot_decrease_state_size IM _ _ _ Htr) as His_tr.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
idescriptors: equivocator_descriptors IM
x: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
is: composite_state (equivocator_IM IM)
IHtr: ∀ s : composite_state (equivocator_IM IM),
  finite_constrained_trace_from_to
    (free_composite_vlsm (equivocator_IM IM))
    is s tr
  → ∀ descriptors : equivocator_descriptors IM,
      proper_equivocator_descriptors IM
        descriptors s
      → ∀ trX : list
                  (composite_transition_item IM),
          equivocators_trace_project
            descriptors tr =
          Some (trX, idescriptors)
          → ∀ eqv : index,
              previous_state_descriptor_prop
                (IM eqv) 
                (descriptors eqv) 
                (is eqv) 
                (idescriptors eqv)
Htr: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
  is (finite_trace_last is tr) tr
Hx: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
  (finite_trace_last is tr) 
  (destination x) [x]
descriptors: equivocator_descriptors IM
Hdescriptors: proper_equivocator_descriptors IM
  descriptors (destination x)
trX: list (composite_transition_item IM)
Hproject_tr: equivocators_trace_project descriptors
  (tr ++ [x]) = Some (trX, idescriptors)
eqv: index
His_tr: ∀ eqv : index,
  equivocator_state_n (is eqv)
  ≤ equivocator_state_n
      (finite_trace_last is tr eqv)
previous_state_descriptor_prop (IM eqv)
  (descriptors eqv) (is eqv) (idescriptors eqv)
    specialize (equivocators_pre_trace_cannot_decrease_state_size IM _ _ _ Hx) as Htr_x.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
idescriptors: equivocator_descriptors IM
x: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
is: composite_state (equivocator_IM IM)
IHtr: ∀ s : composite_state (equivocator_IM IM),
  finite_constrained_trace_from_to
    (free_composite_vlsm (equivocator_IM IM))
    is s tr
  → ∀ descriptors : equivocator_descriptors IM,
      proper_equivocator_descriptors IM
        descriptors s
      → ∀ trX : list
                  (composite_transition_item IM),
          equivocators_trace_project
            descriptors tr =
          Some (trX, idescriptors)
          → ∀ eqv : index,
              previous_state_descriptor_prop
                (IM eqv) 
                (descriptors eqv) 
                (is eqv) 
                (idescriptors eqv)
Htr: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
  is (finite_trace_last is tr) tr
Hx: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
  (finite_trace_last is tr) 
  (destination x) [x]
descriptors: equivocator_descriptors IM
Hdescriptors: proper_equivocator_descriptors IM
  descriptors (destination x)
trX: list (composite_transition_item IM)
Hproject_tr: equivocators_trace_project descriptors
  (tr ++ [x]) = Some (trX, idescriptors)
eqv: index
His_tr: ∀ eqv : index,
  equivocator_state_n (is eqv)
  ≤ equivocator_state_n
      (finite_trace_last is tr eqv)
Htr_x: ∀ eqv : index,
  equivocator_state_n
    (finite_trace_last is tr eqv)
  ≤ equivocator_state_n (destination x eqv)
previous_state_descriptor_prop (IM eqv)
  (descriptors eqv) (is eqv) (idescriptors eqv)
    specialize (IHtr _ Htr).message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
idescriptors: equivocator_descriptors IM
x: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
is: composite_state (equivocator_IM IM)
IHtr: ∀ descriptors : equivocator_descriptors IM,
  proper_equivocator_descriptors IM descriptors
    (finite_trace_last is tr)
  → ∀ trX : list (composite_transition_item IM),
      equivocators_trace_project descriptors tr =
      Some (trX, idescriptors)
      → ∀ eqv : index,
          previous_state_descriptor_prop
            (IM eqv) (descriptors eqv) 
            (is eqv) (idescriptors eqv)
Htr: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
  is (finite_trace_last is tr) tr
Hx: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
  (finite_trace_last is tr) 
  (destination x) [x]
descriptors: equivocator_descriptors IM
Hdescriptors: proper_equivocator_descriptors IM
  descriptors (destination x)
trX: list (composite_transition_item IM)
Hproject_tr: equivocators_trace_project descriptors
  (tr ++ [x]) = Some (trX, idescriptors)
eqv: index
His_tr: ∀ eqv : index,
  equivocator_state_n (is eqv)
  ≤ equivocator_state_n
      (finite_trace_last is tr eqv)
Htr_x: ∀ eqv : index,
  equivocator_state_n
    (finite_trace_last is tr eqv)
  ≤ equivocator_state_n (destination x eqv)
previous_state_descriptor_prop (IM eqv)
  (descriptors eqv) (is eqv) (idescriptors eqv)
    specialize (equivocators_transition_item_project_proper_characterization descriptors x)
      as Hproperx.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
idescriptors: equivocator_descriptors IM
x: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
is: composite_state (equivocator_IM IM)
IHtr: ∀ descriptors : equivocator_descriptors IM,
  proper_equivocator_descriptors IM descriptors
    (finite_trace_last is tr)
  → ∀ trX : list (composite_transition_item IM),
      equivocators_trace_project descriptors tr =
      Some (trX, idescriptors)
      → ∀ eqv : index,
          previous_state_descriptor_prop
            (IM eqv) (descriptors eqv) 
            (is eqv) (idescriptors eqv)
Htr: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
  is (finite_trace_last is tr) tr
Hx: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
  (finite_trace_last is tr) 
  (destination x) [x]
descriptors: equivocator_descriptors IM
Hdescriptors: proper_equivocator_descriptors IM
  descriptors (destination x)
trX: list (composite_transition_item IM)
Hproject_tr: equivocators_trace_project descriptors
  (tr ++ [x]) = Some (trX, idescriptors)
eqv: index
His_tr: ∀ eqv : index,
  equivocator_state_n (is eqv)
  ≤ equivocator_state_n
      (finite_trace_last is tr eqv)
Htr_x: ∀ eqv : index,
  equivocator_state_n
    (finite_trace_last is tr eqv)
  ≤ equivocator_state_n (destination x eqv)
Hproperx: proper_equivocator_descriptors IM
  descriptors (destination x)
→ ∃ (oitem : option
               (composite_transition_item
                  IM)) 
    (eqv_descriptors' : 
     equivocator_descriptors IM),
    equivocators_transition_item_project
      descriptors x =
    Some (oitem, eqv_descriptors')
    ∧ match oitem with
      | Some itemx =>
          (∃ Hex : existing_equivocator_label
                     (IM (projT1 (l x)))
                     (projT2 (l x)),
             existT (projT1 (l x))
               (existing_equivocator_label_extract
                  (IM (projT1 (l x)))
                  (projT2 (l x)) Hex) =
             l itemx)
          ∧ input x = input itemx
            ∧ output x = output itemx
              ∧ equivocators_state_project
                  IM descriptors
                  (destination x) =
                destination itemx
                ∧ eqv_descriptors'
                    (projT1 (l x)) =
                  equivocator_label_descriptor
                    (projT2 (l x))
      | None => True
      end
      ∧ (∀ s : composite_state
                 (equivocator_IM IM),
           composite_valid
             (equivocator_IM IM) 
             (l x) (s, input x)
           → composite_transition
               (equivocator_IM IM) 
               (l x) (s, input x) =
             (destination x, output x)
             → proper_equivocator_descriptors
                 IM eqv_descriptors' s
               ∧ eqv_descriptors' =
                 equivocator_descriptors_update
                   IM descriptors
                   (projT1 (l x))
                   (eqv_descriptors'
                      (projT1 (l x)))
                 ∧ s =
                   state_update
                     (equivocator_IM IM)
                     (destination x)
                     (projT1 (l x))
                     (s (projT1 (l x)))
                   ∧ previous_state_descriptor_prop
                      (IM (projT1 (l x)))
                      (descriptors
                      (projT1 (l x)))
                      (s (projT1 (l x)))
                      (eqv_descriptors'
                      (projT1 (l x)))
                     ∧ 
                     match oitem with
                     | Some itemx =>
                      ∀ sx : 
                      composite_state IM,
                      sx =
                      equivocators_state_project
                      IM eqv_descriptors' s
                      → 
                      composite_valid IM
                      (l itemx)
                      (sx, input itemx)
                      ∧ 
                      composite_transition
                      IM 
                      (l itemx)
                      (sx, input itemx) =
                      (
                      destination itemx,
                      output itemx)
                     | None =>
                      equivocators_state_project
                      IM descriptors
                      (destination x) =
                      equivocators_state_project
                      IM eqv_descriptors' s
                     end)
previous_state_descriptor_prop (IM eqv)
  (descriptors eqv) (is eqv) (idescriptors eqv)
    specialize (Hproperx Hdescriptors).message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
idescriptors: equivocator_descriptors IM
x: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
is: composite_state (equivocator_IM IM)
IHtr: ∀ descriptors : equivocator_descriptors IM,
  proper_equivocator_descriptors IM descriptors
    (finite_trace_last is tr)
  → ∀ trX : list (composite_transition_item IM),
      equivocators_trace_project descriptors tr =
      Some (trX, idescriptors)
      → ∀ eqv : index,
          previous_state_descriptor_prop
            (IM eqv) (descriptors eqv) 
            (is eqv) (idescriptors eqv)
Htr: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
  is (finite_trace_last is tr) tr
Hx: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
  (finite_trace_last is tr) 
  (destination x) [x]
descriptors: equivocator_descriptors IM
Hdescriptors: proper_equivocator_descriptors IM
  descriptors (destination x)
trX: list (composite_transition_item IM)
Hproject_tr: equivocators_trace_project descriptors
  (tr ++ [x]) = Some (trX, idescriptors)
eqv: index
His_tr: ∀ eqv : index,
  equivocator_state_n (is eqv)
  ≤ equivocator_state_n
      (finite_trace_last is tr eqv)
Htr_x: ∀ eqv : index,
  equivocator_state_n
    (finite_trace_last is tr eqv)
  ≤ equivocator_state_n (destination x eqv)
Hproperx: ∃ (oitem : option
             (composite_transition_item IM)) 
  (eqv_descriptors' : equivocator_descriptors
                      IM),
  equivocators_transition_item_project
    descriptors x =
  Some (oitem, eqv_descriptors')
  ∧ match oitem with
    | Some itemx =>
        (∃ Hex : existing_equivocator_label
                   (IM (projT1 (l x)))
                   (projT2 (l x)),
           existT (projT1 (l x))
             (existing_equivocator_label_extract
                (IM (projT1 (l x)))
                (projT2 (l x)) Hex) =
           l itemx)
        ∧ input x = input itemx
          ∧ output x = output itemx
            ∧ equivocators_state_project IM
                descriptors 
                (destination x) =
              destination itemx
              ∧ eqv_descriptors'
                  (projT1 (l x)) =
                equivocator_label_descriptor
                  (projT2 (l x))
    | None => True
    end
    ∧ (∀ s : composite_state
               (equivocator_IM IM),
         composite_valid
           (equivocator_IM IM) 
           (l x) (s, input x)
         → composite_transition
             (equivocator_IM IM) 
             (l x) (s, input x) =
           (destination x, output x)
           → proper_equivocator_descriptors
               IM eqv_descriptors' s
             ∧ eqv_descriptors' =
               equivocator_descriptors_update
                 IM descriptors
                 (projT1 (l x))
                 (eqv_descriptors'
                    (projT1 (l x)))
               ∧ s =
                 state_update
                   (equivocator_IM IM)
                   (destination x)
                   (projT1 (l x))
                   (s (projT1 (l x)))
                 ∧ previous_state_descriptor_prop
                     (IM (projT1 (l x)))
                     (descriptors
                      (projT1 (l x)))
                     (s (projT1 (l x)))
                     (eqv_descriptors'
                      (projT1 (l x)))
                   ∧ match oitem with
                     | Some itemx =>
                      ∀ sx : 
                      composite_state IM,
                      sx =
                      equivocators_state_project
                      IM eqv_descriptors' s
                      → 
                      composite_valid IM
                      (l itemx)
                      (sx, input itemx)
                      ∧ 
                      composite_transition
                      IM 
                      (l itemx)
                      (sx, input itemx) =
                      (
                      destination itemx,
                      output itemx)
                     | None =>
                      equivocators_state_project
                      IM descriptors
                      (destination x) =
                      equivocators_state_project
                      IM eqv_descriptors' s
                     end)
previous_state_descriptor_prop (IM eqv)
  (descriptors eqv) (is eqv) (idescriptors eqv)
    destruct Hproperx as [oitem [final_descriptors' [Hprojectx [Hitemx Hproperx]]]].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
idescriptors: equivocator_descriptors IM
x: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
is: composite_state (equivocator_IM IM)
IHtr: ∀ descriptors : equivocator_descriptors IM,
  proper_equivocator_descriptors IM descriptors
    (finite_trace_last is tr)
  → ∀ trX : list (composite_transition_item IM),
      equivocators_trace_project descriptors tr =
      Some (trX, idescriptors)
      → ∀ eqv : index,
          previous_state_descriptor_prop
            (IM eqv) (descriptors eqv) 
            (is eqv) (idescriptors eqv)
Htr: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
  is (finite_trace_last is tr) tr
Hx: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
  (finite_trace_last is tr) 
  (destination x) [x]
descriptors: equivocator_descriptors IM
Hdescriptors: proper_equivocator_descriptors IM
  descriptors (destination x)
trX: list (composite_transition_item IM)
Hproject_tr: equivocators_trace_project descriptors
  (tr ++ [x]) = Some (trX, idescriptors)
eqv: index
His_tr: ∀ eqv : index,
  equivocator_state_n (is eqv)
  ≤ equivocator_state_n
      (finite_trace_last is tr eqv)
Htr_x: ∀ eqv : index,
  equivocator_state_n
    (finite_trace_last is tr eqv)
  ≤ equivocator_state_n (destination x eqv)
oitem: option (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hprojectx: equivocators_transition_item_project
  descriptors x =
Some (oitem, final_descriptors')
Hitemx: match oitem with
| Some itemx =>
    (∃ Hex : existing_equivocator_label
               (IM (projT1 (l x)))
               (projT2 (l x)),
       existT (projT1 (l x))
         (existing_equivocator_label_extract
            (IM (projT1 (l x)))
            (projT2 (l x)) Hex) = 
       l itemx)
    ∧ input x = input itemx
      ∧ output x = output itemx
        ∧ equivocators_state_project IM
            descriptors 
            (destination x) =
          destination itemx
          ∧ final_descriptors' (projT1 (l x)) =
            equivocator_label_descriptor
              (projT2 (l x))
| None => True
end
Hproperx: ∀ s : composite_state (equivocator_IM IM),
  composite_valid (equivocator_IM IM) 
    (l x) (s, input x)
  → composite_transition
      (equivocator_IM IM) 
      (l x) (s, input x) =
    (destination x, output x)
    → proper_equivocator_descriptors IM
        final_descriptors' s
      ∧ final_descriptors' =
        equivocator_descriptors_update IM
          descriptors (projT1 (l x))
          (final_descriptors'
             (projT1 (l x)))
        ∧ s =
          state_update 
            (equivocator_IM IM)
            (destination x) 
            (projT1 (l x))
            (s (projT1 (l x)))
          ∧ previous_state_descriptor_prop
              (IM (projT1 (l x)))
              (descriptors (projT1 (l x)))
              (s (projT1 (l x)))
              (final_descriptors'
                 (projT1 (l x)))
            ∧ match oitem with
              | Some itemx =>
                  ∀ sx : composite_state IM,
                    sx =
                    equivocators_state_project
                      IM final_descriptors'
                      s
                    → composite_valid IM
                      (l itemx)
                      (sx, input itemx)
                      ∧ 
                      composite_transition
                      IM 
                      (l itemx)
                      (sx, input itemx) =
                      (
                      destination itemx,
                      output itemx)
              | None =>
                  equivocators_state_project
                    IM descriptors
                    (destination x) =
                  equivocators_state_project
                    IM final_descriptors' s
              end
previous_state_descriptor_prop (IM eqv)
  (descriptors eqv) (is eqv) (idescriptors eqv)
    specialize (Hproperx (finite_trace_last is tr)).message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
idescriptors: equivocator_descriptors IM
x: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
is: composite_state (equivocator_IM IM)
IHtr: ∀ descriptors : equivocator_descriptors IM,
  proper_equivocator_descriptors IM descriptors
    (finite_trace_last is tr)
  → ∀ trX : list (composite_transition_item IM),
      equivocators_trace_project descriptors tr =
      Some (trX, idescriptors)
      → ∀ eqv : index,
          previous_state_descriptor_prop
            (IM eqv) (descriptors eqv) 
            (is eqv) (idescriptors eqv)
Htr: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
  is (finite_trace_last is tr) tr
Hx: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
  (finite_trace_last is tr) 
  (destination x) [x]
descriptors: equivocator_descriptors IM
Hdescriptors: proper_equivocator_descriptors IM
  descriptors (destination x)
trX: list (composite_transition_item IM)
Hproject_tr: equivocators_trace_project descriptors
  (tr ++ [x]) = Some (trX, idescriptors)
eqv: index
His_tr: ∀ eqv : index,
  equivocator_state_n (is eqv)
  ≤ equivocator_state_n
      (finite_trace_last is tr eqv)
Htr_x: ∀ eqv : index,
  equivocator_state_n
    (finite_trace_last is tr eqv)
  ≤ equivocator_state_n (destination x eqv)
oitem: option (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hprojectx: equivocators_transition_item_project
  descriptors x =
Some (oitem, final_descriptors')
Hitemx: match oitem with
| Some itemx =>
    (∃ Hex : existing_equivocator_label
               (IM (projT1 (l x)))
               (projT2 (l x)),
       existT (projT1 (l x))
         (existing_equivocator_label_extract
            (IM (projT1 (l x)))
            (projT2 (l x)) Hex) = 
       l itemx)
    ∧ input x = input itemx
      ∧ output x = output itemx
        ∧ equivocators_state_project IM
            descriptors 
            (destination x) =
          destination itemx
          ∧ final_descriptors' (projT1 (l x)) =
            equivocator_label_descriptor
              (projT2 (l x))
| None => True
end
Hproperx: composite_valid (equivocator_IM IM) 
  (l x) (finite_trace_last is tr, input x)
→ composite_transition 
    (equivocator_IM IM) 
    (l x)
    (finite_trace_last is tr, input x) =
  (destination x, output x)
  → proper_equivocator_descriptors IM
      final_descriptors'
      (finite_trace_last is tr)
    ∧ final_descriptors' =
      equivocator_descriptors_update IM
        descriptors (projT1 (l x))
        (final_descriptors' (projT1 (l x)))
      ∧ finite_trace_last is tr =
        state_update (equivocator_IM IM)
          (destination x) 
          (projT1 (l x))
          (finite_trace_last is tr
             (projT1 (l x)))
        ∧ previous_state_descriptor_prop
            (IM (projT1 (l x)))
            (descriptors (projT1 (l x)))
            (finite_trace_last is tr
               (projT1 (l x)))
            (final_descriptors'
               (projT1 (l x)))
          ∧ match oitem with
            | Some itemx =>
                ∀ sx : composite_state IM,
                  sx =
                  equivocators_state_project
                    IM final_descriptors'
                    (finite_trace_last is
                      tr)
                  → composite_valid IM
                      (l itemx)
                      (
                      sx, input itemx)
                    ∧ composite_transition
                      IM 
                      (l itemx)
                      (sx, input itemx) =
                      (
                      destination itemx,
                      output itemx)
            | None =>
                equivocators_state_project
                  IM descriptors
                  (destination x) =
                equivocators_state_project
                  IM final_descriptors'
                  (finite_trace_last is tr)
            end
previous_state_descriptor_prop (IM eqv)
  (descriptors eqv) (is eqv) (idescriptors eqv)
    rewrite equivocators_trace_project_app_iff in Hproject_tr.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
idescriptors: equivocator_descriptors IM
x: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
is: composite_state (equivocator_IM IM)
IHtr: ∀ descriptors : equivocator_descriptors IM,
  proper_equivocator_descriptors IM descriptors
    (finite_trace_last is tr)
  → ∀ trX : list (composite_transition_item IM),
      equivocators_trace_project descriptors tr =
      Some (trX, idescriptors)
      → ∀ eqv : index,
          previous_state_descriptor_prop
            (IM eqv) (descriptors eqv) 
            (is eqv) (idescriptors eqv)
Htr: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
  is (finite_trace_last is tr) tr
Hx: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
  (finite_trace_last is tr) 
  (destination x) [x]
descriptors: equivocator_descriptors IM
Hdescriptors: proper_equivocator_descriptors IM
  descriptors (destination x)
trX: list (composite_transition_item IM)
Hproject_tr: ∃ (preX
   sufX : list
            (composite_transition_item
               IM)) (eqv_descriptors' : 
                     equivocator_descriptors
                      IM),
  equivocators_trace_project descriptors
    [x] = Some (sufX, eqv_descriptors')
  ∧ equivocators_trace_project
      eqv_descriptors' tr =
    Some (preX, idescriptors)
    ∧ trX = preX ++ sufX
eqv: index
His_tr: ∀ eqv : index,
  equivocator_state_n (is eqv)
  ≤ equivocator_state_n
      (finite_trace_last is tr eqv)
Htr_x: ∀ eqv : index,
  equivocator_state_n
    (finite_trace_last is tr eqv)
  ≤ equivocator_state_n (destination x eqv)
oitem: option (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hprojectx: equivocators_transition_item_project
  descriptors x =
Some (oitem, final_descriptors')
Hitemx: match oitem with
| Some itemx =>
    (∃ Hex : existing_equivocator_label
               (IM (projT1 (l x)))
               (projT2 (l x)),
       existT (projT1 (l x))
         (existing_equivocator_label_extract
            (IM (projT1 (l x)))
            (projT2 (l x)) Hex) = 
       l itemx)
    ∧ input x = input itemx
      ∧ output x = output itemx
        ∧ equivocators_state_project IM
            descriptors 
            (destination x) =
          destination itemx
          ∧ final_descriptors' (projT1 (l x)) =
            equivocator_label_descriptor
              (projT2 (l x))
| None => True
end
Hproperx: composite_valid (equivocator_IM IM) 
  (l x) (finite_trace_last is tr, input x)
→ composite_transition 
    (equivocator_IM IM) 
    (l x)
    (finite_trace_last is tr, input x) =
  (destination x, output x)
  → proper_equivocator_descriptors IM
      final_descriptors'
      (finite_trace_last is tr)
    ∧ final_descriptors' =
      equivocator_descriptors_update IM
        descriptors (projT1 (l x))
        (final_descriptors' (projT1 (l x)))
      ∧ finite_trace_last is tr =
        state_update (equivocator_IM IM)
          (destination x) 
          (projT1 (l x))
          (finite_trace_last is tr
             (projT1 (l x)))
        ∧ previous_state_descriptor_prop
            (IM (projT1 (l x)))
            (descriptors (projT1 (l x)))
            (finite_trace_last is tr
               (projT1 (l x)))
            (final_descriptors'
               (projT1 (l x)))
          ∧ match oitem with
            | Some itemx =>
                ∀ sx : composite_state IM,
                  sx =
                  equivocators_state_project
                    IM final_descriptors'
                    (finite_trace_last is
                      tr)
                  → composite_valid IM
                      (l itemx)
                      (
                      sx, input itemx)
                    ∧ composite_transition
                      IM 
                      (l itemx)
                      (sx, input itemx) =
                      (
                      destination itemx,
                      output itemx)
            | None =>
                equivocators_state_project
                  IM descriptors
                  (destination x) =
                equivocators_state_project
                  IM final_descriptors'
                  (finite_trace_last is tr)
            end
previous_state_descriptor_prop (IM eqv)
  (descriptors eqv) (is eqv) (idescriptors eqv)
    simpl in *.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
idescriptors: equivocator_descriptors IM
x: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
is: composite_state (equivocator_IM IM)
IHtr: ∀ descriptors : equivocator_descriptors IM,
  proper_equivocator_descriptors IM descriptors
    (finite_trace_last is tr)
  → ∀ trX : list (composite_transition_item IM),
      equivocators_trace_project descriptors tr =
      Some (trX, idescriptors)
      → ∀ eqv : index,
          previous_state_descriptor_prop
            (IM eqv) (descriptors eqv) 
            (is eqv) (idescriptors eqv)
Htr: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
  is (finite_trace_last is tr) tr
Hx: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
  (finite_trace_last is tr) 
  (destination x) [x]
descriptors: equivocator_descriptors IM
Hdescriptors: proper_equivocator_descriptors IM
  descriptors (destination x)
trX: list (composite_transition_item IM)
Hproject_tr: ∃ (preX
   sufX : list
            (composite_transition_item
               IM)) (eqv_descriptors' : 
                     equivocator_descriptors
                      IM),
  match
    equivocators_transition_item_project
      descriptors x
  with
  | Some (Some item', odescriptor) =>
      Some ([item'], odescriptor)
  | Some (None, odescriptor) =>
      Some ([], odescriptor)
  | None => None
  end = Some (sufX, eqv_descriptors')
  ∧ equivocators_trace_project
      eqv_descriptors' tr =
    Some (preX, idescriptors)
    ∧ trX = preX ++ sufX
eqv: index
His_tr: ∀ eqv : index,
  equivocator_state_n (is eqv)
  ≤ equivocator_state_n
      (finite_trace_last is tr eqv)
Htr_x: ∀ eqv : index,
  equivocator_state_n
    (finite_trace_last is tr eqv)
  ≤ equivocator_state_n (destination x eqv)
oitem: option (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hprojectx: equivocators_transition_item_project
  descriptors x =
Some (oitem, final_descriptors')
Hitemx: match oitem with
| Some itemx =>
    (∃ Hex : existing_equivocator_label
               (IM (projT1 (l x)))
               (projT2 (l x)),
       existT (projT1 (l x))
         (existing_equivocator_label_extract
            (IM (projT1 (l x)))
            (projT2 (l x)) Hex) = 
       l itemx)
    ∧ input x = input itemx
      ∧ output x = output itemx
        ∧ equivocators_state_project IM
            descriptors 
            (destination x) =
          destination itemx
          ∧ final_descriptors' (projT1 (l x)) =
            equivocator_label_descriptor
              (projT2 (l x))
| None => True
end
Hproperx: (let (i, li) := l x in
 equivocator_valid (IM i) li
   (finite_trace_last is tr i, input x))
→ (let (i, li) := l x in
   let (si', om') :=
     equivocator_transition 
       (IM i) li
       (finite_trace_last is tr i, input x) in
   (state_update (equivocator_IM IM)
      (finite_trace_last is tr) i si', om')) =
  (destination x, output x)
  → proper_equivocator_descriptors IM
      final_descriptors'
      (finite_trace_last is tr)
    ∧ final_descriptors' =
      equivocator_descriptors_update IM
        descriptors (projT1 (l x))
        (final_descriptors' (projT1 (l x)))
      ∧ finite_trace_last is tr =
        state_update (equivocator_IM IM)
          (destination x) 
          (projT1 (l x))
          (finite_trace_last is tr
             (projT1 (l x)))
        ∧ previous_state_descriptor_prop
            (IM (projT1 (l x)))
            (descriptors (projT1 (l x)))
            (finite_trace_last is tr
               (projT1 (l x)))
            (final_descriptors'
               (projT1 (l x)))
          ∧ match oitem with
            | Some itemx =>
                ∀ sx : composite_state IM,
                  sx =
                  equivocators_state_project
                    IM final_descriptors'
                    (finite_trace_last is
                      tr)
                  → (let 
                      (i, li) := 
                      l itemx in
                     valid li
                      (sx i, input itemx))
                    ∧ (let 
                      (i, li) := 
                      l itemx in
                      let 
                      (si', om') :=
                      transition li
                      (sx i, input itemx) in
                      (
                      state_update IM sx i
                      si', om')) =
                      (
                      destination itemx,
                      output itemx)
            | None =>
                equivocators_state_project
                  IM descriptors
                  (destination x) =
                equivocators_state_project
                  IM final_descriptors'
                  (finite_trace_last is tr)
            end
previous_state_descriptor_prop (IM eqv)
  (descriptors eqv) (is eqv) (idescriptors eqv)
    rewrite Hprojectx in Hproject_tr.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
idescriptors: equivocator_descriptors IM
x: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
is: composite_state (equivocator_IM IM)
IHtr: ∀ descriptors : equivocator_descriptors IM,
  proper_equivocator_descriptors IM descriptors
    (finite_trace_last is tr)
  → ∀ trX : list (composite_transition_item IM),
      equivocators_trace_project descriptors tr =
      Some (trX, idescriptors)
      → ∀ eqv : index,
          previous_state_descriptor_prop
            (IM eqv) (descriptors eqv) 
            (is eqv) (idescriptors eqv)
Htr: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
  is (finite_trace_last is tr) tr
Hx: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
  (finite_trace_last is tr) 
  (destination x) [x]
descriptors: equivocator_descriptors IM
Hdescriptors: proper_equivocator_descriptors IM
  descriptors (destination x)
trX: list (composite_transition_item IM)
oitem: option (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hproject_tr: ∃ (preX
   sufX : list
            (composite_transition_item
               IM)) (eqv_descriptors' : 
                     equivocator_descriptors
                      IM),
  match oitem with
  | Some item' =>
      Some ([item'], final_descriptors')
  | None =>
      Some ([], final_descriptors')
  end = Some (sufX, eqv_descriptors')
  ∧ equivocators_trace_project
      eqv_descriptors' tr =
    Some (preX, idescriptors)
    ∧ trX = preX ++ sufX
eqv: index
His_tr: ∀ eqv : index,
  equivocator_state_n (is eqv)
  ≤ equivocator_state_n
      (finite_trace_last is tr eqv)
Htr_x: ∀ eqv : index,
  equivocator_state_n
    (finite_trace_last is tr eqv)
  ≤ equivocator_state_n (destination x eqv)
Hprojectx: equivocators_transition_item_project
  descriptors x =
Some (oitem, final_descriptors')
Hitemx: match oitem with
| Some itemx =>
    (∃ Hex : existing_equivocator_label
               (IM (projT1 (l x)))
               (projT2 (l x)),
       existT (projT1 (l x))
         (existing_equivocator_label_extract
            (IM (projT1 (l x)))
            (projT2 (l x)) Hex) = 
       l itemx)
    ∧ input x = input itemx
      ∧ output x = output itemx
        ∧ equivocators_state_project IM
            descriptors 
            (destination x) =
          destination itemx
          ∧ final_descriptors' (projT1 (l x)) =
            equivocator_label_descriptor
              (projT2 (l x))
| None => True
end
Hproperx: (let (i, li) := l x in
 equivocator_valid (IM i) li
   (finite_trace_last is tr i, input x))
→ (let (i, li) := l x in
   let (si', om') :=
     equivocator_transition 
       (IM i) li
       (finite_trace_last is tr i, input x) in
   (state_update (equivocator_IM IM)
      (finite_trace_last is tr) i si', om')) =
  (destination x, output x)
  → proper_equivocator_descriptors IM
      final_descriptors'
      (finite_trace_last is tr)
    ∧ final_descriptors' =
      equivocator_descriptors_update IM
        descriptors (projT1 (l x))
        (final_descriptors' (projT1 (l x)))
      ∧ finite_trace_last is tr =
        state_update (equivocator_IM IM)
          (destination x) 
          (projT1 (l x))
          (finite_trace_last is tr
             (projT1 (l x)))
        ∧ previous_state_descriptor_prop
            (IM (projT1 (l x)))
            (descriptors (projT1 (l x)))
            (finite_trace_last is tr
               (projT1 (l x)))
            (final_descriptors'
               (projT1 (l x)))
          ∧ match oitem with
            | Some itemx =>
                ∀ sx : composite_state IM,
                  sx =
                  equivocators_state_project
                    IM final_descriptors'
                    (finite_trace_last is
                      tr)
                  → (let 
                      (i, li) := 
                      l itemx in
                     valid li
                      (sx i, input itemx))
                    ∧ (let 
                      (i, li) := 
                      l itemx in
                      let 
                      (si', om') :=
                      transition li
                      (sx i, input itemx) in
                      (
                      state_update IM sx i
                      si', om')) =
                      (
                      destination itemx,
                      output itemx)
            | None =>
                equivocators_state_project
                  IM descriptors
                  (destination x) =
                equivocators_state_project
                  IM final_descriptors'
                  (finite_trace_last is tr)
            end
previous_state_descriptor_prop (IM eqv)
  (descriptors eqv) (is eqv) (idescriptors eqv)
    inversion Hx.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
idescriptors: equivocator_descriptors IM
x: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
is: composite_state (equivocator_IM IM)
IHtr: ∀ descriptors : equivocator_descriptors IM,
  proper_equivocator_descriptors IM descriptors
    (finite_trace_last is tr)
  → ∀ trX : list (composite_transition_item IM),
      equivocators_trace_project descriptors tr =
      Some (trX, idescriptors)
      → ∀ eqv : index,
          previous_state_descriptor_prop
            (IM eqv) (descriptors eqv) 
            (is eqv) (idescriptors eqv)
Htr: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
  is (finite_trace_last is tr) tr
Hx: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
  (finite_trace_last is tr) 
  (destination x) [x]
descriptors: equivocator_descriptors IM
Hdescriptors: proper_equivocator_descriptors IM
  descriptors (destination x)
trX: list (composite_transition_item IM)
oitem: option (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hproject_tr: ∃ (preX
   sufX : list
            (composite_transition_item
               IM)) (eqv_descriptors' : 
                     equivocator_descriptors
                      IM),
  match oitem with
  | Some item' =>
      Some ([item'], final_descriptors')
  | None =>
      Some ([], final_descriptors')
  end = Some (sufX, eqv_descriptors')
  ∧ equivocators_trace_project
      eqv_descriptors' tr =
    Some (preX, idescriptors)
    ∧ trX = preX ++ sufX
eqv: index
His_tr: ∀ eqv : index,
  equivocator_state_n (is eqv)
  ≤ equivocator_state_n
      (finite_trace_last is tr eqv)
Htr_x: ∀ eqv : index,
  equivocator_state_n
    (finite_trace_last is tr eqv)
  ≤ equivocator_state_n (destination x eqv)
Hprojectx: equivocators_transition_item_project
  descriptors x =
Some (oitem, final_descriptors')
Hitemx: match oitem with
| Some itemx =>
    (∃ Hex : existing_equivocator_label
               (IM (projT1 (VLSM.l x)))
               (projT2 (VLSM.l x)),
       existT (projT1 (VLSM.l x))
         (existing_equivocator_label_extract
            (IM (projT1 (VLSM.l x)))
            (projT2 (VLSM.l x)) Hex) =
       VLSM.l itemx)
    ∧ input x = input itemx
      ∧ output x = output itemx
        ∧ equivocators_state_project IM
            descriptors 
            (destination x) =
          destination itemx
          ∧ final_descriptors'
              (projT1 (VLSM.l x)) =
            equivocator_label_descriptor
              (projT2 (VLSM.l x))
| None => True
end
Hproperx: (let (i, li) := VLSM.l x in
 equivocator_valid (IM i) li
   (finite_trace_last is tr i, input x))
→ (let (i, li) := VLSM.l x in
   let (si', om') :=
     equivocator_transition 
       (IM i) li
       (finite_trace_last is tr i, input x) in
   (state_update (equivocator_IM IM)
      (finite_trace_last is tr) i si', om')) =
  (destination x, output x)
  → proper_equivocator_descriptors IM
      final_descriptors'
      (finite_trace_last is tr)
    ∧ final_descriptors' =
      equivocator_descriptors_update IM
        descriptors (projT1 (VLSM.l x))
        (final_descriptors'
           (projT1 (VLSM.l x)))
      ∧ finite_trace_last is tr =
        state_update (equivocator_IM IM)
          (destination x)
          (projT1 (VLSM.l x))
          (finite_trace_last is tr
             (projT1 (VLSM.l x)))
        ∧ previous_state_descriptor_prop
            (IM (projT1 (VLSM.l x)))
            (descriptors
               (projT1 (VLSM.l x)))
            (finite_trace_last is tr
               (projT1 (VLSM.l x)))
            (final_descriptors'
               (projT1 (VLSM.l x)))
          ∧ match oitem with
            | Some itemx =>
                ∀ sx : composite_state IM,
                  sx =
                  equivocators_state_project
                    IM final_descriptors'
                    (finite_trace_last is
                      tr)
                  → (let 
                      (i, li) :=
                      VLSM.l itemx in
                     valid li
                      (sx i, input itemx))
                    ∧ (let 
                      (i, li) :=
                      VLSM.l itemx in
                      let 
                      (si', om') :=
                      transition li
                      (sx i, input itemx) in
                      (
                      state_update IM sx i
                      si', om')) =
                      (
                      destination itemx,
                      output itemx)
            | None =>
                equivocators_state_project
                  IM descriptors
                  (destination x) =
                equivocators_state_project
                  IM final_descriptors'
                  (finite_trace_last is tr)
            end
s, f: state
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
tl: list transition_item
s': state
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
iom, oom: option message
l: label
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
Htl: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
  s
  (destination
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |}) []
Ht: input_valid_transition
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM))) l
  (finite_trace_last is tr, iom) (
  s, oom)
H1: s' = finite_trace_last is tr
H2: f = destination x
H4: {|
  l := l;
  input := iom;
  destination := s;
  output := oom
|} = x
H5: tl = []
previous_state_descriptor_prop (IM eqv)
  (descriptors eqv) (is eqv) (idescriptors eqv) subst tl s' x f.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
idescriptors: equivocator_descriptors IM
tr: list
  (composite_transition_item (equivocator_IM IM))
is: composite_state (equivocator_IM IM)
IHtr: ∀ descriptors : equivocator_descriptors IM,
  proper_equivocator_descriptors IM descriptors
    (finite_trace_last is tr)
  → ∀ trX : list (composite_transition_item IM),
      equivocators_trace_project descriptors tr =
      Some (trX, idescriptors)
      → ∀ eqv : index,
          previous_state_descriptor_prop
            (IM eqv) (descriptors eqv) 
            (is eqv) (idescriptors eqv)
Htr: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
  is (finite_trace_last is tr) tr
s: state
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
iom, oom: option message
l: label
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
Hx: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
  (finite_trace_last is tr)
  (destination
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |})
  [{|
     l := l;
     input := iom;
     destination := s;
     output := oom
   |}]
descriptors: equivocator_descriptors IM
Hdescriptors: proper_equivocator_descriptors IM
  descriptors
  (destination
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |})
trX: list (composite_transition_item IM)
oitem: option (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hproject_tr: ∃ (preX
   sufX : list
            (composite_transition_item
               IM)) (eqv_descriptors' : 
                     equivocator_descriptors
                      IM),
  match oitem with
  | Some item' =>
      Some ([item'], final_descriptors')
  | None =>
      Some ([], final_descriptors')
  end = Some (sufX, eqv_descriptors')
  ∧ equivocators_trace_project
      eqv_descriptors' tr =
    Some (preX, idescriptors)
    ∧ trX = preX ++ sufX
eqv: index
His_tr: ∀ eqv : index,
  equivocator_state_n (is eqv)
  ≤ equivocator_state_n
      (finite_trace_last is tr eqv)
Hproperx: (let (i, li) :=
   VLSM.l
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |} in
 equivocator_valid (IM i) li
   (finite_trace_last is tr i,
    input
      {|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}))
→ (let (i, li) :=
     VLSM.l
       {|
         l := l;
         input := iom;
         destination := s;
         output := oom
       |} in
   let (si', om') :=
     equivocator_transition 
       (IM i) li
       (finite_trace_last is tr i,
        input
          {|
            l := l;
            input := iom;
            destination := s;
            output := oom
          |}) in
   (state_update (equivocator_IM IM)
      (finite_trace_last is tr) i si', om')) =
  (destination
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |},
   output
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |})
  → proper_equivocator_descriptors IM
      final_descriptors'
      (finite_trace_last is tr)
    ∧ final_descriptors' =
      equivocator_descriptors_update IM
        descriptors
        (projT1
           (VLSM.l
              {|
                l := l;
                input := iom;
                destination := s;
                output := oom
              |}))
        (final_descriptors'
           (projT1
              (VLSM.l
                 {|
                   l := l;
                   input := iom;
                   destination := s;
                   output := oom
                 |})))
      ∧ finite_trace_last is tr =
        state_update (equivocator_IM IM)
          (destination
             {|
               l := l;
               input := iom;
               destination := s;
               output := oom
             |})
          (projT1
             (VLSM.l
                {|
                  l := l;
                  input := iom;
                  destination := s;
                  output := oom
                |}))
          (finite_trace_last is tr
             (projT1
                (VLSM.l
                   {|
                     l := l;
                     input := iom;
                     destination := s;
                     output := oom
                   |})))
        ∧ previous_state_descriptor_prop
            (IM
               (projT1
                  (VLSM.l
                     {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                     |})))
            (descriptors
               (projT1
                  (VLSM.l
                     {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                     |})))
            (finite_trace_last is tr
               (projT1
                  (VLSM.l
                     {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                     |})))
            (final_descriptors'
               (projT1
                  (VLSM.l
                     {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                     |})))
          ∧ match oitem with
            | Some itemx =>
                ∀ sx : composite_state IM,
                  sx =
                  equivocators_state_project
                    IM final_descriptors'
                    (finite_trace_last is
                      tr)
                  → (let 
                      (i, li) :=
                      VLSM.l itemx in
                     valid li
                      (sx i, input itemx))
                    ∧ (let 
                      (i, li) :=
                      VLSM.l itemx in
                      let 
                      (si', om') :=
                      transition li
                      (sx i, input itemx) in
                      (
                      state_update IM sx i
                      si', om')) =
                      (
                      destination itemx,
                      output itemx)
            | None =>
                equivocators_state_project
                  IM descriptors
                  (destination
                     {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                     |}) =
                equivocators_state_project
                  IM final_descriptors'
                  (finite_trace_last is tr)
            end
Hitemx: match oitem with
| Some itemx =>
    (∃ Hex : existing_equivocator_label
               (IM
                  (projT1
                     (VLSM.l
                      {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                      |})))
               (projT2
                  (VLSM.l
                     {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                     |})),
       existT
         (projT1
            (VLSM.l
               {|
                 l := l;
                 input := iom;
                 destination := s;
                 output := oom
               |}))
         (existing_equivocator_label_extract
            (IM
               (projT1
                  (VLSM.l
                     {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                     |})))
            (projT2
               (VLSM.l
                  {|
                    l := l;
                    input := iom;
                    destination := s;
                    output := oom
                  |})) Hex) = 
       VLSM.l itemx)
    ∧ input
        {|
          l := l;
          input := iom;
          destination := s;
          output := oom
        |} = input itemx
      ∧ output
          {|
            l := l;
            input := iom;
            destination := s;
            output := oom
          |} = output itemx
        ∧ equivocators_state_project IM
            descriptors
            (destination
               {|
                 l := l;
                 input := iom;
                 destination := s;
                 output := oom
               |}) = destination itemx
          ∧ final_descriptors'
              (projT1
                 (VLSM.l
                    {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                    |})) =
            equivocator_label_descriptor
              (projT2
                 (VLSM.l
                    {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                    |}))
| None => True
end
Hprojectx: equivocators_transition_item_project
  descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some (oitem, final_descriptors')
Htr_x: ∀ eqv : index,
  equivocator_state_n
    (finite_trace_last is tr eqv)
  ≤ equivocator_state_n
      (destination
         {|
           l := l;
           input := iom;
           destination := s;
           output := oom
         |} eqv)
Htl: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
  s
  (destination
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |}) []
Ht: input_valid_transition
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM))) l
  (finite_trace_last is tr, iom) (
  s, oom)
previous_state_descriptor_prop (IM eqv)
  (descriptors eqv) (is eqv) (idescriptors eqv) clear Hx Htl.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
idescriptors: equivocator_descriptors IM
tr: list
  (composite_transition_item (equivocator_IM IM))
is: composite_state (equivocator_IM IM)
IHtr: ∀ descriptors : equivocator_descriptors IM,
  proper_equivocator_descriptors IM descriptors
    (finite_trace_last is tr)
  → ∀ trX : list (composite_transition_item IM),
      equivocators_trace_project descriptors tr =
      Some (trX, idescriptors)
      → ∀ eqv : index,
          previous_state_descriptor_prop
            (IM eqv) (descriptors eqv) 
            (is eqv) (idescriptors eqv)
Htr: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
  is (finite_trace_last is tr) tr
s: state
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
iom, oom: option message
l: label
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
descriptors: equivocator_descriptors IM
Hdescriptors: proper_equivocator_descriptors IM
  descriptors
  (destination
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |})
trX: list (composite_transition_item IM)
oitem: option (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hproject_tr: ∃ (preX
   sufX : list
            (composite_transition_item
               IM)) (eqv_descriptors' : 
                     equivocator_descriptors
                      IM),
  match oitem with
  | Some item' =>
      Some ([item'], final_descriptors')
  | None =>
      Some ([], final_descriptors')
  end = Some (sufX, eqv_descriptors')
  ∧ equivocators_trace_project
      eqv_descriptors' tr =
    Some (preX, idescriptors)
    ∧ trX = preX ++ sufX
eqv: index
His_tr: ∀ eqv : index,
  equivocator_state_n (is eqv)
  ≤ equivocator_state_n
      (finite_trace_last is tr eqv)
Hproperx: (let (i, li) :=
   VLSM.l
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |} in
 equivocator_valid (IM i) li
   (finite_trace_last is tr i,
    input
      {|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}))
→ (let (i, li) :=
     VLSM.l
       {|
         l := l;
         input := iom;
         destination := s;
         output := oom
       |} in
   let (si', om') :=
     equivocator_transition 
       (IM i) li
       (finite_trace_last is tr i,
        input
          {|
            l := l;
            input := iom;
            destination := s;
            output := oom
          |}) in
   (state_update (equivocator_IM IM)
      (finite_trace_last is tr) i si', om')) =
  (destination
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |},
   output
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |})
  → proper_equivocator_descriptors IM
      final_descriptors'
      (finite_trace_last is tr)
    ∧ final_descriptors' =
      equivocator_descriptors_update IM
        descriptors
        (projT1
           (VLSM.l
              {|
                l := l;
                input := iom;
                destination := s;
                output := oom
              |}))
        (final_descriptors'
           (projT1
              (VLSM.l
                 {|
                   l := l;
                   input := iom;
                   destination := s;
                   output := oom
                 |})))
      ∧ finite_trace_last is tr =
        state_update (equivocator_IM IM)
          (destination
             {|
               l := l;
               input := iom;
               destination := s;
               output := oom
             |})
          (projT1
             (VLSM.l
                {|
                  l := l;
                  input := iom;
                  destination := s;
                  output := oom
                |}))
          (finite_trace_last is tr
             (projT1
                (VLSM.l
                   {|
                     l := l;
                     input := iom;
                     destination := s;
                     output := oom
                   |})))
        ∧ previous_state_descriptor_prop
            (IM
               (projT1
                  (VLSM.l
                     {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                     |})))
            (descriptors
               (projT1
                  (VLSM.l
                     {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                     |})))
            (finite_trace_last is tr
               (projT1
                  (VLSM.l
                     {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                     |})))
            (final_descriptors'
               (projT1
                  (VLSM.l
                     {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                     |})))
          ∧ match oitem with
            | Some itemx =>
                ∀ sx : composite_state IM,
                  sx =
                  equivocators_state_project
                    IM final_descriptors'
                    (finite_trace_last is
                      tr)
                  → (let 
                      (i, li) :=
                      VLSM.l itemx in
                     valid li
                      (sx i, input itemx))
                    ∧ (let 
                      (i, li) :=
                      VLSM.l itemx in
                      let 
                      (si', om') :=
                      transition li
                      (sx i, input itemx) in
                      (
                      state_update IM sx i
                      si', om')) =
                      (
                      destination itemx,
                      output itemx)
            | None =>
                equivocators_state_project
                  IM descriptors
                  (destination
                     {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                     |}) =
                equivocators_state_project
                  IM final_descriptors'
                  (finite_trace_last is tr)
            end
Hitemx: match oitem with
| Some itemx =>
    (∃ Hex : existing_equivocator_label
               (IM
                  (projT1
                     (VLSM.l
                      {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                      |})))
               (projT2
                  (VLSM.l
                     {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                     |})),
       existT
         (projT1
            (VLSM.l
               {|
                 l := l;
                 input := iom;
                 destination := s;
                 output := oom
               |}))
         (existing_equivocator_label_extract
            (IM
               (projT1
                  (VLSM.l
                     {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                     |})))
            (projT2
               (VLSM.l
                  {|
                    l := l;
                    input := iom;
                    destination := s;
                    output := oom
                  |})) Hex) = 
       VLSM.l itemx)
    ∧ input
        {|
          l := l;
          input := iom;
          destination := s;
          output := oom
        |} = input itemx
      ∧ output
          {|
            l := l;
            input := iom;
            destination := s;
            output := oom
          |} = output itemx
        ∧ equivocators_state_project IM
            descriptors
            (destination
               {|
                 l := l;
                 input := iom;
                 destination := s;
                 output := oom
               |}) = destination itemx
          ∧ final_descriptors'
              (projT1
                 (VLSM.l
                    {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                    |})) =
            equivocator_label_descriptor
              (projT2
                 (VLSM.l
                    {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                    |}))
| None => True
end
Hprojectx: equivocators_transition_item_project
  descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some (oitem, final_descriptors')
Htr_x: ∀ eqv : index,
  equivocator_state_n
    (finite_trace_last is tr eqv)
  ≤ equivocator_state_n
      (destination
         {|
           l := l;
           input := iom;
           destination := s;
           output := oom
         |} eqv)
Ht: input_valid_transition
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM))) l
  (finite_trace_last is tr, iom) (
  s, oom)
previous_state_descriptor_prop (IM eqv)
  (descriptors eqv) (is eqv) (idescriptors eqv)
    destruct Ht as [[_ [_ Hv]] Ht].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
idescriptors: equivocator_descriptors IM
tr: list
  (composite_transition_item (equivocator_IM IM))
is: composite_state (equivocator_IM IM)
IHtr: ∀ descriptors : equivocator_descriptors IM,
  proper_equivocator_descriptors IM descriptors
    (finite_trace_last is tr)
  → ∀ trX : list (composite_transition_item IM),
      equivocators_trace_project descriptors tr =
      Some (trX, idescriptors)
      → ∀ eqv : index,
          previous_state_descriptor_prop
            (IM eqv) (descriptors eqv) 
            (is eqv) (idescriptors eqv)
Htr: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
  is (finite_trace_last is tr) tr
s: state
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
iom, oom: option message
l: label
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
descriptors: equivocator_descriptors IM
Hdescriptors: proper_equivocator_descriptors IM
  descriptors
  (destination
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |})
trX: list (composite_transition_item IM)
oitem: option (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hproject_tr: ∃ (preX
   sufX : list
            (composite_transition_item
               IM)) (eqv_descriptors' : 
                     equivocator_descriptors
                      IM),
  match oitem with
  | Some item' =>
      Some ([item'], final_descriptors')
  | None =>
      Some ([], final_descriptors')
  end = Some (sufX, eqv_descriptors')
  ∧ equivocators_trace_project
      eqv_descriptors' tr =
    Some (preX, idescriptors)
    ∧ trX = preX ++ sufX
eqv: index
His_tr: ∀ eqv : index,
  equivocator_state_n (is eqv)
  ≤ equivocator_state_n
      (finite_trace_last is tr eqv)
Hproperx: (let (i, li) :=
   VLSM.l
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |} in
 equivocator_valid (IM i) li
   (finite_trace_last is tr i,
    input
      {|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}))
→ (let (i, li) :=
     VLSM.l
       {|
         l := l;
         input := iom;
         destination := s;
         output := oom
       |} in
   let (si', om') :=
     equivocator_transition 
       (IM i) li
       (finite_trace_last is tr i,
        input
          {|
            l := l;
            input := iom;
            destination := s;
            output := oom
          |}) in
   (state_update (equivocator_IM IM)
      (finite_trace_last is tr) i si', om')) =
  (destination
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |},
   output
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |})
  → proper_equivocator_descriptors IM
      final_descriptors'
      (finite_trace_last is tr)
    ∧ final_descriptors' =
      equivocator_descriptors_update IM
        descriptors
        (projT1
           (VLSM.l
              {|
                l := l;
                input := iom;
                destination := s;
                output := oom
              |}))
        (final_descriptors'
           (projT1
              (VLSM.l
                 {|
                   l := l;
                   input := iom;
                   destination := s;
                   output := oom
                 |})))
      ∧ finite_trace_last is tr =
        state_update (equivocator_IM IM)
          (destination
             {|
               l := l;
               input := iom;
               destination := s;
               output := oom
             |})
          (projT1
             (VLSM.l
                {|
                  l := l;
                  input := iom;
                  destination := s;
                  output := oom
                |}))
          (finite_trace_last is tr
             (projT1
                (VLSM.l
                   {|
                     l := l;
                     input := iom;
                     destination := s;
                     output := oom
                   |})))
        ∧ previous_state_descriptor_prop
            (IM
               (projT1
                  (VLSM.l
                     {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                     |})))
            (descriptors
               (projT1
                  (VLSM.l
                     {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                     |})))
            (finite_trace_last is tr
               (projT1
                  (VLSM.l
                     {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                     |})))
            (final_descriptors'
               (projT1
                  (VLSM.l
                     {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                     |})))
          ∧ match oitem with
            | Some itemx =>
                ∀ sx : composite_state IM,
                  sx =
                  equivocators_state_project
                    IM final_descriptors'
                    (finite_trace_last is
                      tr)
                  → (let 
                      (i, li) :=
                      VLSM.l itemx in
                     valid li
                      (sx i, input itemx))
                    ∧ (let 
                      (i, li) :=
                      VLSM.l itemx in
                      let 
                      (si', om') :=
                      transition li
                      (sx i, input itemx) in
                      (
                      state_update IM sx i
                      si', om')) =
                      (
                      destination itemx,
                      output itemx)
            | None =>
                equivocators_state_project
                  IM descriptors
                  (destination
                     {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                     |}) =
                equivocators_state_project
                  IM final_descriptors'
                  (finite_trace_last is tr)
            end
Hitemx: match oitem with
| Some itemx =>
    (∃ Hex : existing_equivocator_label
               (IM
                  (projT1
                     (VLSM.l
                      {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                      |})))
               (projT2
                  (VLSM.l
                     {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                     |})),
       existT
         (projT1
            (VLSM.l
               {|
                 l := l;
                 input := iom;
                 destination := s;
                 output := oom
               |}))
         (existing_equivocator_label_extract
            (IM
               (projT1
                  (VLSM.l
                     {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                     |})))
            (projT2
               (VLSM.l
                  {|
                    l := l;
                    input := iom;
                    destination := s;
                    output := oom
                  |})) Hex) = 
       VLSM.l itemx)
    ∧ input
        {|
          l := l;
          input := iom;
          destination := s;
          output := oom
        |} = input itemx
      ∧ output
          {|
            l := l;
            input := iom;
            destination := s;
            output := oom
          |} = output itemx
        ∧ equivocators_state_project IM
            descriptors
            (destination
               {|
                 l := l;
                 input := iom;
                 destination := s;
                 output := oom
               |}) = destination itemx
          ∧ final_descriptors'
              (projT1
                 (VLSM.l
                    {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                    |})) =
            equivocator_label_descriptor
              (projT2
                 (VLSM.l
                    {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                    |}))
| None => True
end
Hprojectx: equivocators_transition_item_project
  descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some (oitem, final_descriptors')
Htr_x: ∀ eqv : index,
  equivocator_state_n
    (finite_trace_last is tr eqv)
  ≤ equivocator_state_n
      (destination
         {|
           l := l;
           input := iom;
           destination := s;
           output := oom
         |} eqv)
Hv: valid l (finite_trace_last is tr, iom)
Ht: transition l (finite_trace_last is tr, iom) =
(s, oom)
previous_state_descriptor_prop (IM eqv)
  (descriptors eqv) (is eqv) (idescriptors eqv)
    specialize (Hproperx Hv Ht).message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
idescriptors: equivocator_descriptors IM
tr: list
  (composite_transition_item (equivocator_IM IM))
is: composite_state (equivocator_IM IM)
IHtr: ∀ descriptors : equivocator_descriptors IM,
  proper_equivocator_descriptors IM descriptors
    (finite_trace_last is tr)
  → ∀ trX : list (composite_transition_item IM),
      equivocators_trace_project descriptors tr =
      Some (trX, idescriptors)
      → ∀ eqv : index,
          previous_state_descriptor_prop
            (IM eqv) (descriptors eqv) 
            (is eqv) (idescriptors eqv)
Htr: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
  is (finite_trace_last is tr) tr
s: state
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
iom, oom: option message
l: label
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
descriptors: equivocator_descriptors IM
Hdescriptors: proper_equivocator_descriptors IM
  descriptors
  (destination
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |})
trX: list (composite_transition_item IM)
oitem: option (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hproject_tr: ∃ (preX
   sufX : list
            (composite_transition_item
               IM)) (eqv_descriptors' : 
                     equivocator_descriptors
                      IM),
  match oitem with
  | Some item' =>
      Some ([item'], final_descriptors')
  | None =>
      Some ([], final_descriptors')
  end = Some (sufX, eqv_descriptors')
  ∧ equivocators_trace_project
      eqv_descriptors' tr =
    Some (preX, idescriptors)
    ∧ trX = preX ++ sufX
eqv: index
His_tr: ∀ eqv : index,
  equivocator_state_n (is eqv)
  ≤ equivocator_state_n
      (finite_trace_last is tr eqv)
Hproperx: proper_equivocator_descriptors IM
  final_descriptors'
  (finite_trace_last is tr)
∧ final_descriptors' =
  equivocator_descriptors_update IM
    descriptors
    (projT1
       (VLSM.l
          {|
            l := l;
            input := iom;
            destination := s;
            output := oom
          |}))
    (final_descriptors'
       (projT1
          (VLSM.l
             {|
               l := l;
               input := iom;
               destination := s;
               output := oom
             |})))
  ∧ finite_trace_last is tr =
    state_update (equivocator_IM IM)
      (destination
         {|
           l := l;
           input := iom;
           destination := s;
           output := oom
         |})
      (projT1
         (VLSM.l
            {|
              l := l;
              input := iom;
              destination := s;
              output := oom
            |}))
      (finite_trace_last is tr
         (projT1
            (VLSM.l
               {|
                 l := l;
                 input := iom;
                 destination := s;
                 output := oom
               |})))
    ∧ previous_state_descriptor_prop
        (IM
           (projT1
              (VLSM.l
                 {|
                   l := l;
                   input := iom;
                   destination := s;
                   output := oom
                 |})))
        (descriptors
           (projT1
              (VLSM.l
                 {|
                   l := l;
                   input := iom;
                   destination := s;
                   output := oom
                 |})))
        (finite_trace_last is tr
           (projT1
              (VLSM.l
                 {|
                   l := l;
                   input := iom;
                   destination := s;
                   output := oom
                 |})))
        (final_descriptors'
           (projT1
              (VLSM.l
                 {|
                   l := l;
                   input := iom;
                   destination := s;
                   output := oom
                 |})))
      ∧ match oitem with
        | Some itemx =>
            ∀ sx : composite_state IM,
              sx =
              equivocators_state_project IM
                final_descriptors'
                (finite_trace_last is tr)
              → (let (i, li) :=
                   VLSM.l itemx in
                 valid li
                   (sx i, input itemx))
                ∧ (let 
                     (i, li) :=
                     VLSM.l itemx in
                   let 
                     (si', om') :=
                     transition li
                      (sx i, input itemx) in
                   (state_update IM sx i
                      si', om')) =
                  (destination itemx,
                   output itemx)
        | None =>
            equivocators_state_project IM
              descriptors
              (destination
                 {|
                   l := l;
                   input := iom;
                   destination := s;
                   output := oom
                 |}) =
            equivocators_state_project IM
              final_descriptors'
              (finite_trace_last is tr)
        end
Hitemx: match oitem with
| Some itemx =>
    (∃ Hex : existing_equivocator_label
               (IM
                  (projT1
                     (VLSM.l
                      {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                      |})))
               (projT2
                  (VLSM.l
                     {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                     |})),
       existT
         (projT1
            (VLSM.l
               {|
                 l := l;
                 input := iom;
                 destination := s;
                 output := oom
               |}))
         (existing_equivocator_label_extract
            (IM
               (projT1
                  (VLSM.l
                     {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                     |})))
            (projT2
               (VLSM.l
                  {|
                    l := l;
                    input := iom;
                    destination := s;
                    output := oom
                  |})) Hex) = 
       VLSM.l itemx)
    ∧ input
        {|
          l := l;
          input := iom;
          destination := s;
          output := oom
        |} = input itemx
      ∧ output
          {|
            l := l;
            input := iom;
            destination := s;
            output := oom
          |} = output itemx
        ∧ equivocators_state_project IM
            descriptors
            (destination
               {|
                 l := l;
                 input := iom;
                 destination := s;
                 output := oom
               |}) = destination itemx
          ∧ final_descriptors'
              (projT1
                 (VLSM.l
                    {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                    |})) =
            equivocator_label_descriptor
              (projT2
                 (VLSM.l
                    {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                    |}))
| None => True
end
Hprojectx: equivocators_transition_item_project
  descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some (oitem, final_descriptors')
Htr_x: ∀ eqv : index,
  equivocator_state_n
    (finite_trace_last is tr eqv)
  ≤ equivocator_state_n
      (destination
         {|
           l := l;
           input := iom;
           destination := s;
           output := oom
         |} eqv)
Hv: valid l (finite_trace_last is tr, iom)
Ht: transition l (finite_trace_last is tr, iom) =
(s, oom)
previous_state_descriptor_prop (IM eqv)
  (descriptors eqv) (is eqv) (idescriptors eqv) simpl in Hproperx.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
idescriptors: equivocator_descriptors IM
tr: list
  (composite_transition_item (equivocator_IM IM))
is: composite_state (equivocator_IM IM)
IHtr: ∀ descriptors : equivocator_descriptors IM,
  proper_equivocator_descriptors IM descriptors
    (finite_trace_last is tr)
  → ∀ trX : list (composite_transition_item IM),
      equivocators_trace_project descriptors tr =
      Some (trX, idescriptors)
      → ∀ eqv : index,
          previous_state_descriptor_prop
            (IM eqv) (descriptors eqv) 
            (is eqv) (idescriptors eqv)
Htr: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
  is (finite_trace_last is tr) tr
s: state
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
iom, oom: option message
l: label
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
descriptors: equivocator_descriptors IM
Hdescriptors: proper_equivocator_descriptors IM
  descriptors
  (destination
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |})
trX: list (composite_transition_item IM)
oitem: option (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hproject_tr: ∃ (preX
   sufX : list
            (composite_transition_item
               IM)) (eqv_descriptors' : 
                     equivocator_descriptors
                      IM),
  match oitem with
  | Some item' =>
      Some ([item'], final_descriptors')
  | None =>
      Some ([], final_descriptors')
  end = Some (sufX, eqv_descriptors')
  ∧ equivocators_trace_project
      eqv_descriptors' tr =
    Some (preX, idescriptors)
    ∧ trX = preX ++ sufX
eqv: index
His_tr: ∀ eqv : index,
  equivocator_state_n (is eqv)
  ≤ equivocator_state_n
      (finite_trace_last is tr eqv)
Hproperx: proper_equivocator_descriptors IM
  final_descriptors'
  (finite_trace_last is tr)
∧ final_descriptors' =
  equivocator_descriptors_update IM
    descriptors (projT1 l)
    (final_descriptors' (projT1 l))
  ∧ finite_trace_last is tr =
    state_update (equivocator_IM IM) s
      (projT1 l)
      (finite_trace_last is tr (projT1 l))
    ∧ previous_state_descriptor_prop
        (IM (projT1 l))
        (descriptors (projT1 l))
        (finite_trace_last is tr (projT1 l))
        (final_descriptors' (projT1 l))
      ∧ match oitem with
        | Some itemx =>
            ∀ sx : composite_state IM,
              sx =
              equivocators_state_project IM
                final_descriptors'
                (finite_trace_last is tr)
              → (let (i, li) :=
                   VLSM.l itemx in
                 valid li
                   (sx i, input itemx))
                ∧ (let 
                     (i, li) :=
                     VLSM.l itemx in
                   let 
                     (si', om') :=
                     transition li
                      (sx i, input itemx) in
                   (state_update IM sx i
                      si', om')) =
                  (destination itemx,
                   output itemx)
        | None =>
            equivocators_state_project IM
              descriptors s =
            equivocators_state_project IM
              final_descriptors'
              (finite_trace_last is tr)
        end
Hitemx: match oitem with
| Some itemx =>
    (∃ Hex : existing_equivocator_label
               (IM
                  (projT1
                     (VLSM.l
                      {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                      |})))
               (projT2
                  (VLSM.l
                     {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                     |})),
       existT
         (projT1
            (VLSM.l
               {|
                 l := l;
                 input := iom;
                 destination := s;
                 output := oom
               |}))
         (existing_equivocator_label_extract
            (IM
               (projT1
                  (VLSM.l
                     {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                     |})))
            (projT2
               (VLSM.l
                  {|
                    l := l;
                    input := iom;
                    destination := s;
                    output := oom
                  |})) Hex) = 
       VLSM.l itemx)
    ∧ input
        {|
          l := l;
          input := iom;
          destination := s;
          output := oom
        |} = input itemx
      ∧ output
          {|
            l := l;
            input := iom;
            destination := s;
            output := oom
          |} = output itemx
        ∧ equivocators_state_project IM
            descriptors
            (destination
               {|
                 l := l;
                 input := iom;
                 destination := s;
                 output := oom
               |}) = destination itemx
          ∧ final_descriptors'
              (projT1
                 (VLSM.l
                    {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                    |})) =
            equivocator_label_descriptor
              (projT2
                 (VLSM.l
                    {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                    |}))
| None => True
end
Hprojectx: equivocators_transition_item_project
  descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some (oitem, final_descriptors')
Htr_x: ∀ eqv : index,
  equivocator_state_n
    (finite_trace_last is tr eqv)
  ≤ equivocator_state_n
      (destination
         {|
           l := l;
           input := iom;
           destination := s;
           output := oom
         |} eqv)
Hv: valid l (finite_trace_last is tr, iom)
Ht: transition l (finite_trace_last is tr, iom) =
(s, oom)
previous_state_descriptor_prop (IM eqv)
  (descriptors eqv) (is eqv) (idescriptors eqv)
    destruct Hproperx as [Hproper' [Heq_final_descriptors' [Heq_ltr [Hex_new Hx]]]].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
idescriptors: equivocator_descriptors IM
tr: list
  (composite_transition_item (equivocator_IM IM))
is: composite_state (equivocator_IM IM)
IHtr: ∀ descriptors : equivocator_descriptors IM,
  proper_equivocator_descriptors IM descriptors
    (finite_trace_last is tr)
  → ∀ trX : list (composite_transition_item IM),
      equivocators_trace_project descriptors tr =
      Some (trX, idescriptors)
      → ∀ eqv : index,
          previous_state_descriptor_prop
            (IM eqv) (descriptors eqv) 
            (is eqv) (idescriptors eqv)
Htr: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
  is (finite_trace_last is tr) tr
s: state
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
iom, oom: option message
l: label
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
descriptors: equivocator_descriptors IM
Hdescriptors: proper_equivocator_descriptors IM
  descriptors
  (destination
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |})
trX: list (composite_transition_item IM)
oitem: option (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hproject_tr: ∃ (preX
   sufX : list
            (composite_transition_item
               IM)) (eqv_descriptors' : 
                     equivocator_descriptors
                      IM),
  match oitem with
  | Some item' =>
      Some ([item'], final_descriptors')
  | None =>
      Some ([], final_descriptors')
  end = Some (sufX, eqv_descriptors')
  ∧ equivocators_trace_project
      eqv_descriptors' tr =
    Some (preX, idescriptors)
    ∧ trX = preX ++ sufX
eqv: index
His_tr: ∀ eqv : index,
  equivocator_state_n (is eqv)
  ≤ equivocator_state_n
      (finite_trace_last is tr eqv)
Hproper': proper_equivocator_descriptors IM
  final_descriptors'
  (finite_trace_last is tr)
Heq_final_descriptors': final_descriptors' =
equivocator_descriptors_update
  IM descriptors (projT1 l)
  (final_descriptors'
     (projT1 l))
Heq_ltr: finite_trace_last is tr =
state_update (equivocator_IM IM) s
  (projT1 l)
  (finite_trace_last is tr (projT1 l))
Hex_new: previous_state_descriptor_prop
  (IM (projT1 l)) (descriptors (projT1 l))
  (finite_trace_last is tr (projT1 l))
  (final_descriptors' (projT1 l))
Hx: match oitem with
| Some itemx =>
    ∀ sx : composite_state IM,
      sx =
      equivocators_state_project IM
        final_descriptors'
        (finite_trace_last is tr)
      → (let (i, li) := VLSM.l itemx in
         valid li (sx i, input itemx))
        ∧ (let (i, li) := VLSM.l itemx in
           let (si', om') :=
             transition li (sx i, input itemx) in
           (state_update IM sx i si', om')) =
          (destination itemx, output itemx)
| None =>
    equivocators_state_project IM descriptors s =
    equivocators_state_project IM
      final_descriptors'
      (finite_trace_last is tr)
end
Hitemx: match oitem with
| Some itemx =>
    (∃ Hex : existing_equivocator_label
               (IM
                  (projT1
                     (VLSM.l
                      {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                      |})))
               (projT2
                  (VLSM.l
                     {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                     |})),
       existT
         (projT1
            (VLSM.l
               {|
                 l := l;
                 input := iom;
                 destination := s;
                 output := oom
               |}))
         (existing_equivocator_label_extract
            (IM
               (projT1
                  (VLSM.l
                     {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                     |})))
            (projT2
               (VLSM.l
                  {|
                    l := l;
                    input := iom;
                    destination := s;
                    output := oom
                  |})) Hex) = 
       VLSM.l itemx)
    ∧ input
        {|
          l := l;
          input := iom;
          destination := s;
          output := oom
        |} = input itemx
      ∧ output
          {|
            l := l;
            input := iom;
            destination := s;
            output := oom
          |} = output itemx
        ∧ equivocators_state_project IM
            descriptors
            (destination
               {|
                 l := l;
                 input := iom;
                 destination := s;
                 output := oom
               |}) = destination itemx
          ∧ final_descriptors'
              (projT1
                 (VLSM.l
                    {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                    |})) =
            equivocator_label_descriptor
              (projT2
                 (VLSM.l
                    {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                    |}))
| None => True
end
Hprojectx: equivocators_transition_item_project
  descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some (oitem, final_descriptors')
Htr_x: ∀ eqv : index,
  equivocator_state_n
    (finite_trace_last is tr eqv)
  ≤ equivocator_state_n
      (destination
         {|
           l := l;
           input := iom;
           destination := s;
           output := oom
         |} eqv)
Hv: valid l (finite_trace_last is tr, iom)
Ht: transition l (finite_trace_last is tr, iom) =
(s, oom)
previous_state_descriptor_prop (IM eqv)
  (descriptors eqv) (is eqv) (idescriptors eqv)
    specialize (IHtr _ Hproper').message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
idescriptors: equivocator_descriptors IM
tr: list
  (composite_transition_item (equivocator_IM IM))
is: composite_state (equivocator_IM IM)
final_descriptors': equivocator_descriptors IM
IHtr: ∀ trX : list (composite_transition_item IM),
  equivocators_trace_project final_descriptors'
    tr = Some (trX, idescriptors)
  → ∀ eqv : index,
      previous_state_descriptor_prop 
        (IM eqv) (final_descriptors' eqv)
        (is eqv) (idescriptors eqv)
Htr: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
  is (finite_trace_last is tr) tr
s: state
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
iom, oom: option message
l: label
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
descriptors: equivocator_descriptors IM
Hdescriptors: proper_equivocator_descriptors IM
  descriptors
  (destination
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |})
trX: list (composite_transition_item IM)
oitem: option (composite_transition_item IM)
Hproject_tr: ∃ (preX
   sufX : list
            (composite_transition_item
               IM)) (eqv_descriptors' : 
                     equivocator_descriptors
                      IM),
  match oitem with
  | Some item' =>
      Some ([item'], final_descriptors')
  | None =>
      Some ([], final_descriptors')
  end = Some (sufX, eqv_descriptors')
  ∧ equivocators_trace_project
      eqv_descriptors' tr =
    Some (preX, idescriptors)
    ∧ trX = preX ++ sufX
eqv: index
His_tr: ∀ eqv : index,
  equivocator_state_n (is eqv)
  ≤ equivocator_state_n
      (finite_trace_last is tr eqv)
Hproper': proper_equivocator_descriptors IM
  final_descriptors'
  (finite_trace_last is tr)
Heq_final_descriptors': final_descriptors' =
equivocator_descriptors_update
  IM descriptors (projT1 l)
  (final_descriptors'
     (projT1 l))
Heq_ltr: finite_trace_last is tr =
state_update (equivocator_IM IM) s
  (projT1 l)
  (finite_trace_last is tr (projT1 l))
Hex_new: previous_state_descriptor_prop
  (IM (projT1 l)) (descriptors (projT1 l))
  (finite_trace_last is tr (projT1 l))
  (final_descriptors' (projT1 l))
Hx: match oitem with
| Some itemx =>
    ∀ sx : composite_state IM,
      sx =
      equivocators_state_project IM
        final_descriptors'
        (finite_trace_last is tr)
      → (let (i, li) := VLSM.l itemx in
         valid li (sx i, input itemx))
        ∧ (let (i, li) := VLSM.l itemx in
           let (si', om') :=
             transition li (sx i, input itemx) in
           (state_update IM sx i si', om')) =
          (destination itemx, output itemx)
| None =>
    equivocators_state_project IM descriptors s =
    equivocators_state_project IM
      final_descriptors'
      (finite_trace_last is tr)
end
Hitemx: match oitem with
| Some itemx =>
    (∃ Hex : existing_equivocator_label
               (IM
                  (projT1
                     (VLSM.l
                      {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                      |})))
               (projT2
                  (VLSM.l
                     {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                     |})),
       existT
         (projT1
            (VLSM.l
               {|
                 l := l;
                 input := iom;
                 destination := s;
                 output := oom
               |}))
         (existing_equivocator_label_extract
            (IM
               (projT1
                  (VLSM.l
                     {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                     |})))
            (projT2
               (VLSM.l
                  {|
                    l := l;
                    input := iom;
                    destination := s;
                    output := oom
                  |})) Hex) = 
       VLSM.l itemx)
    ∧ input
        {|
          l := l;
          input := iom;
          destination := s;
          output := oom
        |} = input itemx
      ∧ output
          {|
            l := l;
            input := iom;
            destination := s;
            output := oom
          |} = output itemx
        ∧ equivocators_state_project IM
            descriptors
            (destination
               {|
                 l := l;
                 input := iom;
                 destination := s;
                 output := oom
               |}) = destination itemx
          ∧ final_descriptors'
              (projT1
                 (VLSM.l
                    {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                    |})) =
            equivocator_label_descriptor
              (projT2
                 (VLSM.l
                    {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                    |}))
| None => True
end
Hprojectx: equivocators_transition_item_project
  descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some (oitem, final_descriptors')
Htr_x: ∀ eqv : index,
  equivocator_state_n
    (finite_trace_last is tr eqv)
  ≤ equivocator_state_n
      (destination
         {|
           l := l;
           input := iom;
           destination := s;
           output := oom
         |} eqv)
Hv: valid l (finite_trace_last is tr, iom)
Ht: transition l (finite_trace_last is tr, iom) =
(s, oom)
previous_state_descriptor_prop (IM eqv)
  (descriptors eqv) (is eqv) (idescriptors eqv)
    assert (Hex_new' : previous_state_descriptor_prop (IM eqv) (final_descriptors' eqv) (is eqv)
                        (idescriptors eqv)).message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
idescriptors: equivocator_descriptors IM
tr: list
  (composite_transition_item (equivocator_IM IM))
is: composite_state (equivocator_IM IM)
final_descriptors': equivocator_descriptors IM
IHtr: ∀ trX : list (composite_transition_item IM),
  equivocators_trace_project final_descriptors'
    tr = Some (trX, idescriptors)
  → ∀ eqv : index,
      previous_state_descriptor_prop 
        (IM eqv) (final_descriptors' eqv)
        (is eqv) (idescriptors eqv)
Htr: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
  is (finite_trace_last is tr) tr
s: state
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
iom, oom: option message
l: label
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
descriptors: equivocator_descriptors IM
Hdescriptors: proper_equivocator_descriptors IM
  descriptors
  (destination
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |})
trX: list (composite_transition_item IM)
oitem: option (composite_transition_item IM)
Hproject_tr: ∃ (preX
   sufX : list
            (composite_transition_item
               IM)) (eqv_descriptors' : 
                     equivocator_descriptors
                      IM),
  match oitem with
  | Some item' =>
      Some ([item'], final_descriptors')
  | None =>
      Some ([], final_descriptors')
  end = Some (sufX, eqv_descriptors')
  ∧ equivocators_trace_project
      eqv_descriptors' tr =
    Some (preX, idescriptors)
    ∧ trX = preX ++ sufX
eqv: index
His_tr: ∀ eqv : index,
  equivocator_state_n (is eqv)
  ≤ equivocator_state_n
      (finite_trace_last is tr eqv)
Hproper': proper_equivocator_descriptors IM
  final_descriptors'
  (finite_trace_last is tr)
Heq_final_descriptors': final_descriptors' =
equivocator_descriptors_update
  IM descriptors (projT1 l)
  (final_descriptors'
     (projT1 l))
Heq_ltr: finite_trace_last is tr =
state_update (equivocator_IM IM) s
  (projT1 l)
  (finite_trace_last is tr (projT1 l))
Hex_new: previous_state_descriptor_prop
  (IM (projT1 l)) (descriptors (projT1 l))
  (finite_trace_last is tr (projT1 l))
  (final_descriptors' (projT1 l))
Hx: match oitem with
| Some itemx =>
    ∀ sx : composite_state IM,
      sx =
      equivocators_state_project IM
        final_descriptors'
        (finite_trace_last is tr)
      → (let (i, li) := VLSM.l itemx in
         valid li (sx i, input itemx))
        ∧ (let (i, li) := VLSM.l itemx in
           let (si', om') :=
             transition li (sx i, input itemx) in
           (state_update IM sx i si', om')) =
          (destination itemx, output itemx)
| None =>
    equivocators_state_project IM descriptors s =
    equivocators_state_project IM
      final_descriptors'
      (finite_trace_last is tr)
end
Hitemx: match oitem with
| Some itemx =>
    (∃ Hex : existing_equivocator_label
               (IM
                  (projT1
                     (VLSM.l
                      {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                      |})))
               (projT2
                  (VLSM.l
                     {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                     |})),
       existT
         (projT1
            (VLSM.l
               {|
                 l := l;
                 input := iom;
                 destination := s;
                 output := oom
               |}))
         (existing_equivocator_label_extract
            (IM
               (projT1
                  (VLSM.l
                     {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                     |})))
            (projT2
               (VLSM.l
                  {|
                    l := l;
                    input := iom;
                    destination := s;
                    output := oom
                  |})) Hex) = 
       VLSM.l itemx)
    ∧ input
        {|
          l := l;
          input := iom;
          destination := s;
          output := oom
        |} = input itemx
      ∧ output
          {|
            l := l;
            input := iom;
            destination := s;
            output := oom
          |} = output itemx
        ∧ equivocators_state_project IM
            descriptors
            (destination
               {|
                 l := l;
                 input := iom;
                 destination := s;
                 output := oom
               |}) = destination itemx
          ∧ final_descriptors'
              (projT1
                 (VLSM.l
                    {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                    |})) =
            equivocator_label_descriptor
              (projT2
                 (VLSM.l
                    {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                    |}))
| None => True
end
Hprojectx: equivocators_transition_item_project
  descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some (oitem, final_descriptors')
Htr_x: ∀ eqv : index,
  equivocator_state_n
    (finite_trace_last is tr eqv)
  ≤ equivocator_state_n
      (destination
         {|
           l := l;
           input := iom;
           destination := s;
           output := oom
         |} eqv)
Hv: valid l (finite_trace_last is tr, iom)
Ht: transition l (finite_trace_last is tr, iom) =
(s, oom)
previous_state_descriptor_prop (IM eqv)
  (final_descriptors' eqv) (is eqv) (idescriptors eqv)
message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
idescriptors: equivocator_descriptors IM
tr: list
  (composite_transition_item (equivocator_IM IM))
is: composite_state (equivocator_IM IM)
final_descriptors': equivocator_descriptors IM
IHtr: ∀ trX : list (composite_transition_item IM),
  equivocators_trace_project final_descriptors'
    tr = Some (trX, idescriptors)
  → ∀ eqv : index,
      previous_state_descriptor_prop 
        (IM eqv) (final_descriptors' eqv)
        (is eqv) (idescriptors eqv)
Htr: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
  is (finite_trace_last is tr) tr
s: state
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
iom, oom: option message
l: label
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
descriptors: equivocator_descriptors IM
Hdescriptors: proper_equivocator_descriptors IM
  descriptors
  (destination
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |})
trX: list (composite_transition_item IM)
oitem: option (composite_transition_item IM)
Hproject_tr: ∃ (preX
   sufX : list
            (composite_transition_item
               IM)) (eqv_descriptors' : 
                     equivocator_descriptors
                      IM),
  match oitem with
  | Some item' =>
      Some ([item'], final_descriptors')
  | None =>
      Some ([], final_descriptors')
  end = Some (sufX, eqv_descriptors')
  ∧ equivocators_trace_project
      eqv_descriptors' tr =
    Some (preX, idescriptors)
    ∧ trX = preX ++ sufX
eqv: index
His_tr: ∀ eqv : index,
  equivocator_state_n (is eqv)
  ≤ equivocator_state_n
      (finite_trace_last is tr eqv)
Hproper': proper_equivocator_descriptors IM
  final_descriptors'
  (finite_trace_last is tr)
Heq_final_descriptors': final_descriptors' =
equivocator_descriptors_update
  IM descriptors (projT1 l)
  (final_descriptors'
     (projT1 l))
Heq_ltr: finite_trace_last is tr =
state_update (equivocator_IM IM) s
  (projT1 l)
  (finite_trace_last is tr (projT1 l))
Hex_new: previous_state_descriptor_prop
  (IM (projT1 l)) (descriptors (projT1 l))
  (finite_trace_last is tr (projT1 l))
  (final_descriptors' (projT1 l))
Hx: match oitem with
| Some itemx =>
    ∀ sx : composite_state IM,
      sx =
      equivocators_state_project IM
        final_descriptors'
        (finite_trace_last is tr)
      → (let (i, li) := VLSM.l itemx in
         valid li (sx i, input itemx))
        ∧ (let (i, li) := VLSM.l itemx in
           let (si', om') :=
             transition li (sx i, input itemx) in
           (state_update IM sx i si', om')) =
          (destination itemx, output itemx)
| None =>
    equivocators_state_project IM descriptors s =
    equivocators_state_project IM
      final_descriptors'
      (finite_trace_last is tr)
end
Hitemx: match oitem with
| Some itemx =>
    (∃ Hex : existing_equivocator_label
               (IM
                  (projT1
                     (VLSM.l
                      {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                      |})))
               (projT2
                  (VLSM.l
                     {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                     |})),
       existT
         (projT1
            (VLSM.l
               {|
                 l := l;
                 input := iom;
                 destination := s;
                 output := oom
               |}))
         (existing_equivocator_label_extract
            (IM
               (projT1
                  (VLSM.l
                     {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                     |})))
            (projT2
               (VLSM.l
                  {|
                    l := l;
                    input := iom;
                    destination := s;
                    output := oom
                  |})) Hex) = 
       VLSM.l itemx)
    ∧ input
        {|
          l := l;
          input := iom;
          destination := s;
          output := oom
        |} = input itemx
      ∧ output
          {|
            l := l;
            input := iom;
            destination := s;
            output := oom
          |} = output itemx
        ∧ equivocators_state_project IM
            descriptors
            (destination
               {|
                 l := l;
                 input := iom;
                 destination := s;
                 output := oom
               |}) = destination itemx
          ∧ final_descriptors'
              (projT1
                 (VLSM.l
                    {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                    |})) =
            equivocator_label_descriptor
              (projT2
                 (VLSM.l
                    {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                    |}))
| None => True
end
Hprojectx: equivocators_transition_item_project
  descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some (oitem, final_descriptors')
Htr_x: ∀ eqv : index,
  equivocator_state_n
    (finite_trace_last is tr eqv)
  ≤ equivocator_state_n
      (destination
         {|
           l := l;
           input := iom;
           destination := s;
           output := oom
         |} eqv)
Hv: valid l (finite_trace_last is tr, iom)
Ht: transition l (finite_trace_last is tr, iom) =
(s, oom)
Hex_new': previous_state_descriptor_prop 
  (IM eqv) (final_descriptors' eqv)
  (is eqv) (idescriptors eqv)
previous_state_descriptor_prop 
  (IM eqv) (descriptors eqv) 
  (is eqv) (idescriptors eqv)
    {message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
idescriptors: equivocator_descriptors IM
tr: list
  (composite_transition_item (equivocator_IM IM))
is: composite_state (equivocator_IM IM)
final_descriptors': equivocator_descriptors IM
IHtr: ∀ trX : list (composite_transition_item IM),
  equivocators_trace_project final_descriptors'
    tr = Some (trX, idescriptors)
  → ∀ eqv : index,
      previous_state_descriptor_prop 
        (IM eqv) (final_descriptors' eqv)
        (is eqv) (idescriptors eqv)
Htr: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
  is (finite_trace_last is tr) tr
s: state
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
iom, oom: option message
l: label
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
descriptors: equivocator_descriptors IM
Hdescriptors: proper_equivocator_descriptors IM
  descriptors
  (destination
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |})
trX: list (composite_transition_item IM)
oitem: option (composite_transition_item IM)
Hproject_tr: ∃ (preX
   sufX : list
            (composite_transition_item
               IM)) (eqv_descriptors' : 
                     equivocator_descriptors
                      IM),
  match oitem with
  | Some item' =>
      Some ([item'], final_descriptors')
  | None =>
      Some ([], final_descriptors')
  end = Some (sufX, eqv_descriptors')
  ∧ equivocators_trace_project
      eqv_descriptors' tr =
    Some (preX, idescriptors)
    ∧ trX = preX ++ sufX
eqv: index
His_tr: ∀ eqv : index,
  equivocator_state_n (is eqv)
  ≤ equivocator_state_n
      (finite_trace_last is tr eqv)
Hproper': proper_equivocator_descriptors IM
  final_descriptors'
  (finite_trace_last is tr)
Heq_final_descriptors': final_descriptors' =
equivocator_descriptors_update
  IM descriptors (projT1 l)
  (final_descriptors'
     (projT1 l))
Heq_ltr: finite_trace_last is tr =
state_update (equivocator_IM IM) s
  (projT1 l)
  (finite_trace_last is tr (projT1 l))
Hex_new: previous_state_descriptor_prop
  (IM (projT1 l)) (descriptors (projT1 l))
  (finite_trace_last is tr (projT1 l))
  (final_descriptors' (projT1 l))
Hx: match oitem with
| Some itemx =>
    ∀ sx : composite_state IM,
      sx =
      equivocators_state_project IM
        final_descriptors'
        (finite_trace_last is tr)
      → (let (i, li) := VLSM.l itemx in
         valid li (sx i, input itemx))
        ∧ (let (i, li) := VLSM.l itemx in
           let (si', om') :=
             transition li (sx i, input itemx) in
           (state_update IM sx i si', om')) =
          (destination itemx, output itemx)
| None =>
    equivocators_state_project IM descriptors s =
    equivocators_state_project IM
      final_descriptors'
      (finite_trace_last is tr)
end
Hitemx: match oitem with
| Some itemx =>
    (∃ Hex : existing_equivocator_label
               (IM
                  (projT1
                     (VLSM.l
                      {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                      |})))
               (projT2
                  (VLSM.l
                     {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                     |})),
       existT
         (projT1
            (VLSM.l
               {|
                 l := l;
                 input := iom;
                 destination := s;
                 output := oom
               |}))
         (existing_equivocator_label_extract
            (IM
               (projT1
                  (VLSM.l
                     {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                     |})))
            (projT2
               (VLSM.l
                  {|
                    l := l;
                    input := iom;
                    destination := s;
                    output := oom
                  |})) Hex) = 
       VLSM.l itemx)
    ∧ input
        {|
          l := l;
          input := iom;
          destination := s;
          output := oom
        |} = input itemx
      ∧ output
          {|
            l := l;
            input := iom;
            destination := s;
            output := oom
          |} = output itemx
        ∧ equivocators_state_project IM
            descriptors
            (destination
               {|
                 l := l;
                 input := iom;
                 destination := s;
                 output := oom
               |}) = destination itemx
          ∧ final_descriptors'
              (projT1
                 (VLSM.l
                    {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                    |})) =
            equivocator_label_descriptor
              (projT2
                 (VLSM.l
                    {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                    |}))
| None => True
end
Hprojectx: equivocators_transition_item_project
  descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some (oitem, final_descriptors')
Htr_x: ∀ eqv : index,
  equivocator_state_n
    (finite_trace_last is tr eqv)
  ≤ equivocator_state_n
      (destination
         {|
           l := l;
           input := iom;
           destination := s;
           output := oom
         |} eqv)
Hv: valid l (finite_trace_last is tr, iom)
Ht: transition l (finite_trace_last is tr, iom) =
(s, oom)
previous_state_descriptor_prop (IM eqv)
  (final_descriptors' eqv) (is eqv) (idescriptors eqv)
      destruct Hproject_tr as [preX [sufX [_final_descriptors' [H_final_descriptors'
        [Hproject_tr HtrX]]]]].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
idescriptors: equivocator_descriptors IM
tr: list
  (composite_transition_item (equivocator_IM IM))
is: composite_state (equivocator_IM IM)
final_descriptors': equivocator_descriptors IM
IHtr: ∀ trX : list (composite_transition_item IM),
  equivocators_trace_project final_descriptors'
    tr = Some (trX, idescriptors)
  → ∀ eqv : index,
      previous_state_descriptor_prop 
        (IM eqv) (final_descriptors' eqv)
        (is eqv) (idescriptors eqv)
Htr: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
  is (finite_trace_last is tr) tr
s: state
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
iom, oom: option message
l: label
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
descriptors: equivocator_descriptors IM
Hdescriptors: proper_equivocator_descriptors IM
  descriptors
  (destination
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |})
trX: list (composite_transition_item IM)
oitem: option (composite_transition_item IM)
preX, sufX: list (composite_transition_item IM)
_final_descriptors': equivocator_descriptors IM
H_final_descriptors': match oitem with
| Some item' =>
    Some
      ([item'],
       final_descriptors')
| None =>
    Some
      ([], final_descriptors')
end =
Some
  (sufX, _final_descriptors')
Hproject_tr: equivocators_trace_project
  _final_descriptors' tr =
Some (preX, idescriptors)
HtrX: trX = preX ++ sufX
eqv: index
His_tr: ∀ eqv : index,
  equivocator_state_n (is eqv)
  ≤ equivocator_state_n
      (finite_trace_last is tr eqv)
Hproper': proper_equivocator_descriptors IM
  final_descriptors'
  (finite_trace_last is tr)
Heq_final_descriptors': final_descriptors' =
equivocator_descriptors_update
  IM descriptors (projT1 l)
  (final_descriptors'
     (projT1 l))
Heq_ltr: finite_trace_last is tr =
state_update (equivocator_IM IM) s
  (projT1 l)
  (finite_trace_last is tr (projT1 l))
Hex_new: previous_state_descriptor_prop
  (IM (projT1 l)) (descriptors (projT1 l))
  (finite_trace_last is tr (projT1 l))
  (final_descriptors' (projT1 l))
Hx: match oitem with
| Some itemx =>
    ∀ sx : composite_state IM,
      sx =
      equivocators_state_project IM
        final_descriptors'
        (finite_trace_last is tr)
      → (let (i, li) := VLSM.l itemx in
         valid li (sx i, input itemx))
        ∧ (let (i, li) := VLSM.l itemx in
           let (si', om') :=
             transition li (sx i, input itemx) in
           (state_update IM sx i si', om')) =
          (destination itemx, output itemx)
| None =>
    equivocators_state_project IM descriptors s =
    equivocators_state_project IM
      final_descriptors'
      (finite_trace_last is tr)
end
Hitemx: match oitem with
| Some itemx =>
    (∃ Hex : existing_equivocator_label
               (IM
                  (projT1
                     (VLSM.l
                      {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                      |})))
               (projT2
                  (VLSM.l
                     {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                     |})),
       existT
         (projT1
            (VLSM.l
               {|
                 l := l;
                 input := iom;
                 destination := s;
                 output := oom
               |}))
         (existing_equivocator_label_extract
            (IM
               (projT1
                  (VLSM.l
                     {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                     |})))
            (projT2
               (VLSM.l
                  {|
                    l := l;
                    input := iom;
                    destination := s;
                    output := oom
                  |})) Hex) = 
       VLSM.l itemx)
    ∧ input
        {|
          l := l;
          input := iom;
          destination := s;
          output := oom
        |} = input itemx
      ∧ output
          {|
            l := l;
            input := iom;
            destination := s;
            output := oom
          |} = output itemx
        ∧ equivocators_state_project IM
            descriptors
            (destination
               {|
                 l := l;
                 input := iom;
                 destination := s;
                 output := oom
               |}) = destination itemx
          ∧ final_descriptors'
              (projT1
                 (VLSM.l
                    {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                    |})) =
            equivocator_label_descriptor
              (projT2
                 (VLSM.l
                    {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                    |}))
| None => True
end
Hprojectx: equivocators_transition_item_project
  descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some (oitem, final_descriptors')
Htr_x: ∀ eqv : index,
  equivocator_state_n
    (finite_trace_last is tr eqv)
  ≤ equivocator_state_n
      (destination
         {|
           l := l;
           input := iom;
           destination := s;
           output := oom
         |} eqv)
Hv: valid l (finite_trace_last is tr, iom)
Ht: transition l (finite_trace_last is tr, iom) =
(s, oom)
previous_state_descriptor_prop (IM eqv)
  (final_descriptors' eqv) (is eqv) (idescriptors eqv)
      apply IHtr with preX.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
idescriptors: equivocator_descriptors IM
tr: list
  (composite_transition_item (equivocator_IM IM))
is: composite_state (equivocator_IM IM)
final_descriptors': equivocator_descriptors IM
IHtr: ∀ trX : list (composite_transition_item IM),
  equivocators_trace_project final_descriptors'
    tr = Some (trX, idescriptors)
  → ∀ eqv : index,
      previous_state_descriptor_prop 
        (IM eqv) (final_descriptors' eqv)
        (is eqv) (idescriptors eqv)
Htr: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
  is (finite_trace_last is tr) tr
s: state
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
iom, oom: option message
l: label
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
descriptors: equivocator_descriptors IM
Hdescriptors: proper_equivocator_descriptors IM
  descriptors
  (destination
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |})
trX: list (composite_transition_item IM)
oitem: option (composite_transition_item IM)
preX, sufX: list (composite_transition_item IM)
_final_descriptors': equivocator_descriptors IM
H_final_descriptors': match oitem with
| Some item' =>
    Some
      ([item'],
       final_descriptors')
| None =>
    Some
      ([], final_descriptors')
end =
Some
  (sufX, _final_descriptors')
Hproject_tr: equivocators_trace_project
  _final_descriptors' tr =
Some (preX, idescriptors)
HtrX: trX = preX ++ sufX
eqv: index
His_tr: ∀ eqv : index,
  equivocator_state_n (is eqv)
  ≤ equivocator_state_n
      (finite_trace_last is tr eqv)
Hproper': proper_equivocator_descriptors IM
  final_descriptors'
  (finite_trace_last is tr)
Heq_final_descriptors': final_descriptors' =
equivocator_descriptors_update
  IM descriptors (projT1 l)
  (final_descriptors'
     (projT1 l))
Heq_ltr: finite_trace_last is tr =
state_update (equivocator_IM IM) s
  (projT1 l)
  (finite_trace_last is tr (projT1 l))
Hex_new: previous_state_descriptor_prop
  (IM (projT1 l)) (descriptors (projT1 l))
  (finite_trace_last is tr (projT1 l))
  (final_descriptors' (projT1 l))
Hx: match oitem with
| Some itemx =>
    ∀ sx : composite_state IM,
      sx =
      equivocators_state_project IM
        final_descriptors'
        (finite_trace_last is tr)
      → (let (i, li) := VLSM.l itemx in
         valid li (sx i, input itemx))
        ∧ (let (i, li) := VLSM.l itemx in
           let (si', om') :=
             transition li (sx i, input itemx) in
           (state_update IM sx i si', om')) =
          (destination itemx, output itemx)
| None =>
    equivocators_state_project IM descriptors s =
    equivocators_state_project IM
      final_descriptors'
      (finite_trace_last is tr)
end
Hitemx: match oitem with
| Some itemx =>
    (∃ Hex : existing_equivocator_label
               (IM
                  (projT1
                     (VLSM.l
                      {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                      |})))
               (projT2
                  (VLSM.l
                     {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                     |})),
       existT
         (projT1
            (VLSM.l
               {|
                 l := l;
                 input := iom;
                 destination := s;
                 output := oom
               |}))
         (existing_equivocator_label_extract
            (IM
               (projT1
                  (VLSM.l
                     {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                     |})))
            (projT2
               (VLSM.l
                  {|
                    l := l;
                    input := iom;
                    destination := s;
                    output := oom
                  |})) Hex) = 
       VLSM.l itemx)
    ∧ input
        {|
          l := l;
          input := iom;
          destination := s;
          output := oom
        |} = input itemx
      ∧ output
          {|
            l := l;
            input := iom;
            destination := s;
            output := oom
          |} = output itemx
        ∧ equivocators_state_project IM
            descriptors
            (destination
               {|
                 l := l;
                 input := iom;
                 destination := s;
                 output := oom
               |}) = destination itemx
          ∧ final_descriptors'
              (projT1
                 (VLSM.l
                    {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                    |})) =
            equivocator_label_descriptor
              (projT2
                 (VLSM.l
                    {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                    |}))
| None => True
end
Hprojectx: equivocators_transition_item_project
  descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some (oitem, final_descriptors')
Htr_x: ∀ eqv : index,
  equivocator_state_n
    (finite_trace_last is tr eqv)
  ≤ equivocator_state_n
      (destination
         {|
           l := l;
           input := iom;
           destination := s;
           output := oom
         |} eqv)
Hv: valid l (finite_trace_last is tr, iom)
Ht: transition l (finite_trace_last is tr, iom) =
(s, oom)
equivocators_trace_project final_descriptors' tr =
Some (preX, idescriptors)
      by destruct oitem; inversion H_final_descriptors'; subst.
    }message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
idescriptors: equivocator_descriptors IM
tr: list
  (composite_transition_item (equivocator_IM IM))
is: composite_state (equivocator_IM IM)
final_descriptors': equivocator_descriptors IM
IHtr: ∀ trX : list (composite_transition_item IM),
  equivocators_trace_project final_descriptors'
    tr = Some (trX, idescriptors)
  → ∀ eqv : index,
      previous_state_descriptor_prop 
        (IM eqv) (final_descriptors' eqv)
        (is eqv) (idescriptors eqv)
Htr: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
  is (finite_trace_last is tr) tr
s: state
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
iom, oom: option message
l: label
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
descriptors: equivocator_descriptors IM
Hdescriptors: proper_equivocator_descriptors IM
  descriptors
  (destination
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |})
trX: list (composite_transition_item IM)
oitem: option (composite_transition_item IM)
Hproject_tr: ∃ (preX
   sufX : list
            (composite_transition_item
               IM)) (eqv_descriptors' : 
                     equivocator_descriptors
                      IM),
  match oitem with
  | Some item' =>
      Some ([item'], final_descriptors')
  | None =>
      Some ([], final_descriptors')
  end = Some (sufX, eqv_descriptors')
  ∧ equivocators_trace_project
      eqv_descriptors' tr =
    Some (preX, idescriptors)
    ∧ trX = preX ++ sufX
eqv: index
His_tr: ∀ eqv : index,
  equivocator_state_n (is eqv)
  ≤ equivocator_state_n
      (finite_trace_last is tr eqv)
Hproper': proper_equivocator_descriptors IM
  final_descriptors'
  (finite_trace_last is tr)
Heq_final_descriptors': final_descriptors' =
equivocator_descriptors_update
  IM descriptors (projT1 l)
  (final_descriptors'
     (projT1 l))
Heq_ltr: finite_trace_last is tr =
state_update (equivocator_IM IM) s
  (projT1 l)
  (finite_trace_last is tr (projT1 l))
Hex_new: previous_state_descriptor_prop
  (IM (projT1 l)) (descriptors (projT1 l))
  (finite_trace_last is tr (projT1 l))
  (final_descriptors' (projT1 l))
Hx: match oitem with
| Some itemx =>
    ∀ sx : composite_state IM,
      sx =
      equivocators_state_project IM
        final_descriptors'
        (finite_trace_last is tr)
      → (let (i, li) := VLSM.l itemx in
         valid li (sx i, input itemx))
        ∧ (let (i, li) := VLSM.l itemx in
           let (si', om') :=
             transition li (sx i, input itemx) in
           (state_update IM sx i si', om')) =
          (destination itemx, output itemx)
| None =>
    equivocators_state_project IM descriptors s =
    equivocators_state_project IM
      final_descriptors'
      (finite_trace_last is tr)
end
Hitemx: match oitem with
| Some itemx =>
    (∃ Hex : existing_equivocator_label
               (IM
                  (projT1
                     (VLSM.l
                      {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                      |})))
               (projT2
                  (VLSM.l
                     {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                     |})),
       existT
         (projT1
            (VLSM.l
               {|
                 l := l;
                 input := iom;
                 destination := s;
                 output := oom
               |}))
         (existing_equivocator_label_extract
            (IM
               (projT1
                  (VLSM.l
                     {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                     |})))
            (projT2
               (VLSM.l
                  {|
                    l := l;
                    input := iom;
                    destination := s;
                    output := oom
                  |})) Hex) = 
       VLSM.l itemx)
    ∧ input
        {|
          l := l;
          input := iom;
          destination := s;
          output := oom
        |} = input itemx
      ∧ output
          {|
            l := l;
            input := iom;
            destination := s;
            output := oom
          |} = output itemx
        ∧ equivocators_state_project IM
            descriptors
            (destination
               {|
                 l := l;
                 input := iom;
                 destination := s;
                 output := oom
               |}) = destination itemx
          ∧ final_descriptors'
              (projT1
                 (VLSM.l
                    {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                    |})) =
            equivocator_label_descriptor
              (projT2
                 (VLSM.l
                    {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                    |}))
| None => True
end
Hprojectx: equivocators_transition_item_project
  descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some (oitem, final_descriptors')
Htr_x: ∀ eqv : index,
  equivocator_state_n
    (finite_trace_last is tr eqv)
  ≤ equivocator_state_n
      (destination
         {|
           l := l;
           input := iom;
           destination := s;
           output := oom
         |} eqv)
Hv: valid l (finite_trace_last is tr, iom)
Ht: transition l (finite_trace_last is tr, iom) =
(s, oom)
Hex_new': previous_state_descriptor_prop 
  (IM eqv) (final_descriptors' eqv)
  (is eqv) (idescriptors eqv)
previous_state_descriptor_prop (IM eqv)
  (descriptors eqv) (is eqv) (idescriptors eqv)

    destruct l as (i, li).message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
idescriptors: equivocator_descriptors IM
tr: list
  (composite_transition_item (equivocator_IM IM))
is: composite_state (equivocator_IM IM)
final_descriptors': equivocator_descriptors IM
IHtr: ∀ trX : list (composite_transition_item IM),
  equivocators_trace_project final_descriptors'
    tr = Some (trX, idescriptors)
  → ∀ eqv : index,
      previous_state_descriptor_prop 
        (IM eqv) (final_descriptors' eqv)
        (is eqv) (idescriptors eqv)
Htr: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
  is (finite_trace_last is tr) tr
s: state
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
iom, oom: option message
i: index
li: label (equivocator_IM IM i)
descriptors: equivocator_descriptors IM
Hdescriptors: proper_equivocator_descriptors IM
  descriptors
  (destination
     {|
       l := existT i li;
       input := iom;
       destination := s;
       output := oom
     |})
trX: list (composite_transition_item IM)
oitem: option (composite_transition_item IM)
Hproject_tr: ∃ (preX
   sufX : list
            (composite_transition_item
               IM)) (eqv_descriptors' : 
                     equivocator_descriptors
                      IM),
  match oitem with
  | Some item' =>
      Some ([item'], final_descriptors')
  | None =>
      Some ([], final_descriptors')
  end = Some (sufX, eqv_descriptors')
  ∧ equivocators_trace_project
      eqv_descriptors' tr =
    Some (preX, idescriptors)
    ∧ trX = preX ++ sufX
eqv: index
His_tr: ∀ eqv : index,
  equivocator_state_n (is eqv)
  ≤ equivocator_state_n
      (finite_trace_last is tr eqv)
Hproper': proper_equivocator_descriptors IM
  final_descriptors'
  (finite_trace_last is tr)
Heq_final_descriptors': final_descriptors' =
equivocator_descriptors_update
  IM descriptors
  (projT1 (existT i li))
  (final_descriptors'
     (projT1 (existT i li)))
Heq_ltr: finite_trace_last is tr =
state_update (equivocator_IM IM) s
  (projT1 (existT i li))
  (finite_trace_last is tr
     (projT1 (existT i li)))
Hex_new: previous_state_descriptor_prop
  (IM (projT1 (existT i li)))
  (descriptors (projT1 (existT i li)))
  (finite_trace_last is tr
     (projT1 (existT i li)))
  (final_descriptors' (projT1 (existT i li)))
Hx: match oitem with
| Some itemx =>
    ∀ sx : composite_state IM,
      sx =
      equivocators_state_project IM
        final_descriptors'
        (finite_trace_last is tr)
      → (let (i, li) := l itemx in
         valid li (sx i, input itemx))
        ∧ (let (i, li) := l itemx in
           let (si', om') :=
             transition li (sx i, input itemx) in
           (state_update IM sx i si', om')) =
          (destination itemx, output itemx)
| None =>
    equivocators_state_project IM descriptors s =
    equivocators_state_project IM
      final_descriptors'
      (finite_trace_last is tr)
end
Hitemx: match oitem with
| Some itemx =>
    (∃ Hex : existing_equivocator_label
               (IM
                  (projT1
                     (l
                      {|
                      l := existT i li;
                      input := iom;
                      destination := s;
                      output := oom
                      |})))
               (projT2
                  (l
                     {|
                      l := existT i li;
                      input := iom;
                      destination := s;
                      output := oom
                     |})),
       existT
         (projT1
            (l
               {|
                 l := existT i li;
                 input := iom;
                 destination := s;
                 output := oom
               |}))
         (existing_equivocator_label_extract
            (IM
               (projT1
                  (l
                     {|
                      l := existT i li;
                      input := iom;
                      destination := s;
                      output := oom
                     |})))
            (projT2
               (l
                  {|
                    l := existT i li;
                    input := iom;
                    destination := s;
                    output := oom
                  |})) Hex) = 
       l itemx)
    ∧ input
        {|
          l := existT i li;
          input := iom;
          destination := s;
          output := oom
        |} = input itemx
      ∧ output
          {|
            l := existT i li;
            input := iom;
            destination := s;
            output := oom
          |} = output itemx
        ∧ equivocators_state_project IM
            descriptors
            (destination
               {|
                 l := existT i li;
                 input := iom;
                 destination := s;
                 output := oom
               |}) = destination itemx
          ∧ final_descriptors'
              (projT1
                 (l
                    {|
                      l := existT i li;
                      input := iom;
                      destination := s;
                      output := oom
                    |})) =
            equivocator_label_descriptor
              (projT2
                 (l
                    {|
                      l := existT i li;
                      input := iom;
                      destination := s;
                      output := oom
                    |}))
| None => True
end
Hprojectx: equivocators_transition_item_project
  descriptors
  {|
    l := existT i li;
    input := iom;
    destination := s;
    output := oom
  |} = Some (oitem, final_descriptors')
Htr_x: ∀ eqv : index,
  equivocator_state_n
    (finite_trace_last is tr eqv)
  ≤ equivocator_state_n
      (destination
         {|
           l := existT i li;
           input := iom;
           destination := s;
           output := oom
         |} eqv)
Hv: valid (existT i li)
  (finite_trace_last is tr, iom)
Ht: transition (existT i li)
  (finite_trace_last is tr, iom) = (
s, oom)
Hex_new': previous_state_descriptor_prop 
  (IM eqv) (final_descriptors' eqv)
  (is eqv) (idescriptors eqv)
previous_state_descriptor_prop (IM eqv)
  (descriptors eqv) (is eqv) (idescriptors eqv) simpl in *.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
idescriptors: equivocator_descriptors IM
tr: list
  (composite_transition_item (equivocator_IM IM))
is: composite_state (equivocator_IM IM)
final_descriptors': equivocator_descriptors IM
IHtr: ∀ trX : list (composite_transition_item IM),
  equivocators_trace_project final_descriptors'
    tr = Some (trX, idescriptors)
  → ∀ eqv : index,
      previous_state_descriptor_prop 
        (IM eqv) (final_descriptors' eqv)
        (is eqv) (idescriptors eqv)
Htr: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
  is (finite_trace_last is tr) tr
s: composite_state (equivocator_IM IM)
iom, oom: option message
i: index
li: EquivocatorLabel (IM i)
descriptors: equivocator_descriptors IM
Hdescriptors: proper_equivocator_descriptors IM
  descriptors s
trX: list (composite_transition_item IM)
oitem: option (composite_transition_item IM)
Hproject_tr: ∃ (preX
   sufX : list
            (composite_transition_item
               IM)) (eqv_descriptors' : 
                     equivocator_descriptors
                      IM),
  match oitem with
  | Some item' =>
      Some ([item'], final_descriptors')
  | None =>
      Some ([], final_descriptors')
  end = Some (sufX, eqv_descriptors')
  ∧ equivocators_trace_project
      eqv_descriptors' tr =
    Some (preX, idescriptors)
    ∧ trX = preX ++ sufX
eqv: index
His_tr: ∀ eqv : index,
  equivocator_state_n (is eqv)
  ≤ equivocator_state_n
      (finite_trace_last is tr eqv)
Hproper': proper_equivocator_descriptors IM
  final_descriptors'
  (finite_trace_last is tr)
Heq_final_descriptors': final_descriptors' =
equivocator_descriptors_update
  IM descriptors i
  (final_descriptors' i)
Heq_ltr: finite_trace_last is tr =
state_update (equivocator_IM IM) s i
  (finite_trace_last is tr i)
Hex_new: previous_state_descriptor_prop 
  (IM i) (descriptors i)
  (finite_trace_last is tr i)
  (final_descriptors' i)
Hx: match oitem with
| Some itemx =>
    ∀ sx : composite_state IM,
      sx =
      equivocators_state_project IM
        final_descriptors'
        (finite_trace_last is tr)
      → (let (i, li) := l itemx in
         valid li (sx i, input itemx))
        ∧ (let (i, li) := l itemx in
           let (si', om') :=
             transition li (sx i, input itemx) in
           (state_update IM sx i si', om')) =
          (destination itemx, output itemx)
| None =>
    equivocators_state_project IM descriptors s =
    equivocators_state_project IM
      final_descriptors'
      (finite_trace_last is tr)
end
Hitemx: match oitem with
| Some itemx =>
    (∃ Hex : existing_equivocator_label
               (IM i) li,
       existT i
         (existing_equivocator_label_extract
            (IM i) li Hex) = 
       l itemx)
    ∧ iom = input itemx
      ∧ oom = output itemx
        ∧ equivocators_state_project IM
            descriptors s = 
          destination itemx
          ∧ final_descriptors' i =
            equivocator_label_descriptor li
| None => True
end
Hprojectx: equivocators_transition_item_project
  descriptors
  {|
    l := existT i li;
    input := iom;
    destination := s;
    output := oom
  |} = Some (oitem, final_descriptors')
Htr_x: ∀ eqv : index,
  equivocator_state_n
    (finite_trace_last is tr eqv)
  ≤ equivocator_state_n (s eqv)
Hv: equivocator_valid (IM i) li
  (finite_trace_last is tr i, iom)
Ht: (let (si', om') :=
   equivocator_transition 
     (IM i) li (finite_trace_last is tr i, iom) in
 (state_update (equivocator_IM IM)
    (finite_trace_last is tr) i si', om')) =
(s, oom)
Hex_new': previous_state_descriptor_prop 
  (IM eqv) (final_descriptors' eqv)
  (is eqv) (idescriptors eqv)
previous_state_descriptor_prop (IM eqv)
  (descriptors eqv) (is eqv) (idescriptors eqv)
    destruct (decide (i = eqv)).message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
idescriptors: equivocator_descriptors IM
tr: list
  (composite_transition_item (equivocator_IM IM))
is: composite_state (equivocator_IM IM)
final_descriptors': equivocator_descriptors IM
IHtr: ∀ trX : list (composite_transition_item IM),
  equivocators_trace_project final_descriptors'
    tr = Some (trX, idescriptors)
  → ∀ eqv : index,
      previous_state_descriptor_prop 
        (IM eqv) (final_descriptors' eqv)
        (is eqv) (idescriptors eqv)
Htr: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
  is (finite_trace_last is tr) tr
s: composite_state (equivocator_IM IM)
iom, oom: option message
i: index
li: EquivocatorLabel (IM i)
descriptors: equivocator_descriptors IM
Hdescriptors: proper_equivocator_descriptors IM
  descriptors s
trX: list (composite_transition_item IM)
oitem: option (composite_transition_item IM)
Hproject_tr: ∃ (preX
   sufX : list
            (composite_transition_item
               IM)) (eqv_descriptors' : 
                     equivocator_descriptors
                      IM),
  match oitem with
  | Some item' =>
      Some ([item'], final_descriptors')
  | None =>
      Some ([], final_descriptors')
  end = Some (sufX, eqv_descriptors')
  ∧ equivocators_trace_project
      eqv_descriptors' tr =
    Some (preX, idescriptors)
    ∧ trX = preX ++ sufX
eqv: index
His_tr: ∀ eqv : index,
  equivocator_state_n (is eqv)
  ≤ equivocator_state_n
      (finite_trace_last is tr eqv)
Hproper': proper_equivocator_descriptors IM
  final_descriptors'
  (finite_trace_last is tr)
Heq_final_descriptors': final_descriptors' =
equivocator_descriptors_update
  IM descriptors i
  (final_descriptors' i)
Heq_ltr: finite_trace_last is tr =
state_update (equivocator_IM IM) s i
  (finite_trace_last is tr i)
Hex_new: previous_state_descriptor_prop 
  (IM i) (descriptors i)
  (finite_trace_last is tr i)
  (final_descriptors' i)
Hx: match oitem with
| Some itemx =>
    ∀ sx : composite_state IM,
      sx =
      equivocators_state_project IM
        final_descriptors'
        (finite_trace_last is tr)
      → (let (i, li) := l itemx in
         valid li (sx i, input itemx))
        ∧ (let (i, li) := l itemx in
           let (si', om') :=
             transition li (sx i, input itemx) in
           (state_update IM sx i si', om')) =
          (destination itemx, output itemx)
| None =>
    equivocators_state_project IM descriptors s =
    equivocators_state_project IM
      final_descriptors'
      (finite_trace_last is tr)
end
Hitemx: match oitem with
| Some itemx =>
    (∃ Hex : existing_equivocator_label
               (IM i) li,
       existT i
         (existing_equivocator_label_extract
            (IM i) li Hex) = 
       l itemx)
    ∧ iom = input itemx
      ∧ oom = output itemx
        ∧ equivocators_state_project IM
            descriptors s = 
          destination itemx
          ∧ final_descriptors' i =
            equivocator_label_descriptor li
| None => True
end
Hprojectx: equivocators_transition_item_project
  descriptors
  {|
    l := existT i li;
    input := iom;
    destination := s;
    output := oom
  |} = Some (oitem, final_descriptors')
Htr_x: ∀ eqv : index,
  equivocator_state_n
    (finite_trace_last is tr eqv)
  ≤ equivocator_state_n (s eqv)
Hv: equivocator_valid (IM i) li
  (finite_trace_last is tr i, iom)
Ht: (let (si', om') :=
   equivocator_transition 
     (IM i) li (finite_trace_last is tr i, iom) in
 (state_update (equivocator_IM IM)
    (finite_trace_last is tr) i si', om')) =
(s, oom)
Hex_new': previous_state_descriptor_prop 
  (IM eqv) (final_descriptors' eqv)
  (is eqv) (idescriptors eqv)
e: i = eqv
previous_state_descriptor_prop (IM eqv)
  (descriptors eqv) (is eqv) (idescriptors eqv)
message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
idescriptors: equivocator_descriptors IM
tr: list
  (composite_transition_item (equivocator_IM IM))
is: composite_state (equivocator_IM IM)
final_descriptors': equivocator_descriptors IM
IHtr: ∀ trX : list (composite_transition_item IM),
  equivocators_trace_project final_descriptors'
    tr = Some (trX, idescriptors)
  → ∀ eqv : index,
      previous_state_descriptor_prop 
        (IM eqv) (final_descriptors' eqv)
        (is eqv) (idescriptors eqv)
Htr: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
  is (finite_trace_last is tr) tr
s: composite_state (equivocator_IM IM)
iom, oom: option message
i: index
li: EquivocatorLabel (IM i)
descriptors: equivocator_descriptors IM
Hdescriptors: proper_equivocator_descriptors IM
  descriptors s
trX: list (composite_transition_item IM)
oitem: option (composite_transition_item IM)
Hproject_tr: ∃ (preX
   sufX : list
            (composite_transition_item
               IM)) (eqv_descriptors' : 
                     equivocator_descriptors
                      IM),
  match oitem with
  | Some item' =>
      Some ([item'], final_descriptors')
  | None =>
      Some ([], final_descriptors')
  end = Some (sufX, eqv_descriptors')
  ∧ equivocators_trace_project
      eqv_descriptors' tr =
    Some (preX, idescriptors)
    ∧ trX = preX ++ sufX
eqv: index
His_tr: ∀ eqv : index,
  equivocator_state_n (is eqv)
  ≤ equivocator_state_n
      (finite_trace_last is tr eqv)
Hproper': proper_equivocator_descriptors IM
  final_descriptors'
  (finite_trace_last is tr)
Heq_final_descriptors': final_descriptors' =
equivocator_descriptors_update
  IM descriptors i
  (final_descriptors' i)
Heq_ltr: finite_trace_last is tr =
state_update (equivocator_IM IM) s i
  (finite_trace_last is tr i)
Hex_new: previous_state_descriptor_prop 
  (IM i) (descriptors i)
  (finite_trace_last is tr i)
  (final_descriptors' i)
Hx: match oitem with
| Some itemx =>
    ∀ sx : composite_state IM,
      sx =
      equivocators_state_project IM
        final_descriptors'
        (finite_trace_last is tr)
      → (let (i, li) := l itemx in
         valid li (sx i, input itemx))
        ∧ (let (i, li) := l itemx in
           let (si', om') :=
             transition li (sx i, input itemx) in
           (state_update IM sx i si', om')) =
          (destination itemx, output itemx)
| None =>
    equivocators_state_project IM descriptors s =
    equivocators_state_project IM
      final_descriptors'
      (finite_trace_last is tr)
end
Hitemx: match oitem with
| Some itemx =>
    (∃ Hex : existing_equivocator_label
               (IM i) li,
       existT i
         (existing_equivocator_label_extract
            (IM i) li Hex) = 
       l itemx)
    ∧ iom = input itemx
      ∧ oom = output itemx
        ∧ equivocators_state_project IM
            descriptors s = 
          destination itemx
          ∧ final_descriptors' i =
            equivocator_label_descriptor li
| None => True
end
Hprojectx: equivocators_transition_item_project
  descriptors
  {|
    l := existT i li;
    input := iom;
    destination := s;
    output := oom
  |} = Some (oitem, final_descriptors')
Htr_x: ∀ eqv : index,
  equivocator_state_n
    (finite_trace_last is tr eqv)
  ≤ equivocator_state_n (s eqv)
Hv: equivocator_valid (IM i) li
  (finite_trace_last is tr i, iom)
Ht: (let (si', om') :=
   equivocator_transition 
     (IM i) li (finite_trace_last is tr i, iom) in
 (state_update (equivocator_IM IM)
    (finite_trace_last is tr) i si', om')) =
(s, oom)
Hex_new': previous_state_descriptor_prop 
  (IM eqv) (final_descriptors' eqv)
  (is eqv) (idescriptors eqv)
n: i ≠ eqv
previous_state_descriptor_prop 
  (IM eqv) (descriptors eqv) 
  (is eqv) (idescriptors eqv)
    +message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
idescriptors: equivocator_descriptors IM
tr: list
  (composite_transition_item (equivocator_IM IM))
is: composite_state (equivocator_IM IM)
final_descriptors': equivocator_descriptors IM
IHtr: ∀ trX : list (composite_transition_item IM),
  equivocators_trace_project final_descriptors'
    tr = Some (trX, idescriptors)
  → ∀ eqv : index,
      previous_state_descriptor_prop 
        (IM eqv) (final_descriptors' eqv)
        (is eqv) (idescriptors eqv)
Htr: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
  is (finite_trace_last is tr) tr
s: composite_state (equivocator_IM IM)
iom, oom: option message
i: index
li: EquivocatorLabel (IM i)
descriptors: equivocator_descriptors IM
Hdescriptors: proper_equivocator_descriptors IM
  descriptors s
trX: list (composite_transition_item IM)
oitem: option (composite_transition_item IM)
Hproject_tr: ∃ (preX
   sufX : list
            (composite_transition_item
               IM)) (eqv_descriptors' : 
                     equivocator_descriptors
                      IM),
  match oitem with
  | Some item' =>
      Some ([item'], final_descriptors')
  | None =>
      Some ([], final_descriptors')
  end = Some (sufX, eqv_descriptors')
  ∧ equivocators_trace_project
      eqv_descriptors' tr =
    Some (preX, idescriptors)
    ∧ trX = preX ++ sufX
eqv: index
His_tr: ∀ eqv : index,
  equivocator_state_n (is eqv)
  ≤ equivocator_state_n
      (finite_trace_last is tr eqv)
Hproper': proper_equivocator_descriptors IM
  final_descriptors'
  (finite_trace_last is tr)
Heq_final_descriptors': final_descriptors' =
equivocator_descriptors_update
  IM descriptors i
  (final_descriptors' i)
Heq_ltr: finite_trace_last is tr =
state_update (equivocator_IM IM) s i
  (finite_trace_last is tr i)
Hex_new: previous_state_descriptor_prop 
  (IM i) (descriptors i)
  (finite_trace_last is tr i)
  (final_descriptors' i)
Hx: match oitem with
| Some itemx =>
    ∀ sx : composite_state IM,
      sx =
      equivocators_state_project IM
        final_descriptors'
        (finite_trace_last is tr)
      → (let (i, li) := l itemx in
         valid li (sx i, input itemx))
        ∧ (let (i, li) := l itemx in
           let (si', om') :=
             transition li (sx i, input itemx) in
           (state_update IM sx i si', om')) =
          (destination itemx, output itemx)
| None =>
    equivocators_state_project IM descriptors s =
    equivocators_state_project IM
      final_descriptors'
      (finite_trace_last is tr)
end
Hitemx: match oitem with
| Some itemx =>
    (∃ Hex : existing_equivocator_label
               (IM i) li,
       existT i
         (existing_equivocator_label_extract
            (IM i) li Hex) = 
       l itemx)
    ∧ iom = input itemx
      ∧ oom = output itemx
        ∧ equivocators_state_project IM
            descriptors s = 
          destination itemx
          ∧ final_descriptors' i =
            equivocator_label_descriptor li
| None => True
end
Hprojectx: equivocators_transition_item_project
  descriptors
  {|
    l := existT i li;
    input := iom;
    destination := s;
    output := oom
  |} = Some (oitem, final_descriptors')
Htr_x: ∀ eqv : index,
  equivocator_state_n
    (finite_trace_last is tr eqv)
  ≤ equivocator_state_n (s eqv)
Hv: equivocator_valid (IM i) li
  (finite_trace_last is tr i, iom)
Ht: (let (si', om') :=
   equivocator_transition 
     (IM i) li (finite_trace_last is tr i, iom) in
 (state_update (equivocator_IM IM)
    (finite_trace_last is tr) i si', om')) =
(s, oom)
Hex_new': previous_state_descriptor_prop 
  (IM eqv) (final_descriptors' eqv)
  (is eqv) (idescriptors eqv)
e: i = eqv
previous_state_descriptor_prop (IM eqv)
  (descriptors eqv) (is eqv) (idescriptors eqv) subst.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
idescriptors: equivocator_descriptors IM
tr: list
  (composite_transition_item (equivocator_IM IM))
is: composite_state (equivocator_IM IM)
final_descriptors': equivocator_descriptors IM
IHtr: ∀ trX : list (composite_transition_item IM),
  equivocators_trace_project final_descriptors'
    tr = Some (trX, idescriptors)
  → ∀ eqv : index,
      previous_state_descriptor_prop 
        (IM eqv) (final_descriptors' eqv)
        (is eqv) (idescriptors eqv)
Htr: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
  is (finite_trace_last is tr) tr
s: composite_state (equivocator_IM IM)
iom, oom: option message
eqv: index
li: EquivocatorLabel (IM eqv)
descriptors: equivocator_descriptors IM
Hdescriptors: proper_equivocator_descriptors IM
  descriptors s
trX: list (composite_transition_item IM)
oitem: option (composite_transition_item IM)
Hproject_tr: ∃ (preX
   sufX : list
            (composite_transition_item
               IM)) (eqv_descriptors' : 
                     equivocator_descriptors
                      IM),
  match oitem with
  | Some item' =>
      Some ([item'], final_descriptors')
  | None =>
      Some ([], final_descriptors')
  end = Some (sufX, eqv_descriptors')
  ∧ equivocators_trace_project
      eqv_descriptors' tr =
    Some (preX, idescriptors)
    ∧ trX = preX ++ sufX
His_tr: ∀ eqv : index,
  equivocator_state_n (is eqv)
  ≤ equivocator_state_n
      (finite_trace_last is tr eqv)
Hproper': proper_equivocator_descriptors IM
  final_descriptors'
  (finite_trace_last is tr)
Hex_new: previous_state_descriptor_prop 
  (IM eqv) (descriptors eqv)
  (finite_trace_last is tr eqv)
  (final_descriptors' eqv)
Heq_ltr: finite_trace_last is tr =
state_update (equivocator_IM IM) s eqv
  (finite_trace_last is tr eqv)
Heq_final_descriptors': final_descriptors' =
equivocator_descriptors_update
  IM descriptors eqv
  (final_descriptors' eqv)
Hx: match oitem with
| Some itemx =>
    ∀ sx : composite_state IM,
      sx =
      equivocators_state_project IM
        final_descriptors'
        (finite_trace_last is tr)
      → (let (i, li) := l itemx in
         valid li (sx i, input itemx))
        ∧ (let (i, li) := l itemx in
           let (si', om') :=
             transition li (sx i, input itemx) in
           (state_update IM sx i si', om')) =
          (destination itemx, output itemx)
| None =>
    equivocators_state_project IM descriptors s =
    equivocators_state_project IM
      final_descriptors'
      (finite_trace_last is tr)
end
Hprojectx: equivocators_transition_item_project
  descriptors
  {|
    l := existT eqv li;
    input := iom;
    destination := s;
    output := oom
  |} = Some (oitem, final_descriptors')
Hitemx: match oitem with
| Some itemx =>
    (∃ Hex : existing_equivocator_label
               (IM eqv) li,
       existT eqv
         (existing_equivocator_label_extract
            (IM eqv) li Hex) = 
       l itemx)
    ∧ iom = input itemx
      ∧ oom = output itemx
        ∧ equivocators_state_project IM
            descriptors s = 
          destination itemx
          ∧ final_descriptors' eqv =
            equivocator_label_descriptor li
| None => True
end
Htr_x: ∀ eqv : index,
  equivocator_state_n
    (finite_trace_last is tr eqv)
  ≤ equivocator_state_n (s eqv)
Ht: (let (si', om') :=
   equivocator_transition 
     (IM eqv) li
     (finite_trace_last is tr eqv, iom) in
 (state_update (equivocator_IM IM)
    (finite_trace_last is tr) eqv si', om')) =
(s, oom)
Hv: equivocator_valid (IM eqv) li
  (finite_trace_last is tr eqv, iom)
Hex_new': previous_state_descriptor_prop 
  (IM eqv) (final_descriptors' eqv)
  (is eqv) (idescriptors eqv)
previous_state_descriptor_prop (IM eqv)
  (descriptors eqv) (is eqv) (idescriptors eqv) specialize (His_tr eqv).message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
idescriptors: equivocator_descriptors IM
tr: list
  (composite_transition_item (equivocator_IM IM))
is: composite_state (equivocator_IM IM)
final_descriptors': equivocator_descriptors IM
IHtr: ∀ trX : list (composite_transition_item IM),
  equivocators_trace_project final_descriptors'
    tr = Some (trX, idescriptors)
  → ∀ eqv : index,
      previous_state_descriptor_prop 
        (IM eqv) (final_descriptors' eqv)
        (is eqv) (idescriptors eqv)
Htr: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
  is (finite_trace_last is tr) tr
s: composite_state (equivocator_IM IM)
iom, oom: option message
eqv: index
li: EquivocatorLabel (IM eqv)
descriptors: equivocator_descriptors IM
Hdescriptors: proper_equivocator_descriptors IM
  descriptors s
trX: list (composite_transition_item IM)
oitem: option (composite_transition_item IM)
Hproject_tr: ∃ (preX
   sufX : list
            (composite_transition_item
               IM)) (eqv_descriptors' : 
                     equivocator_descriptors
                      IM),
  match oitem with
  | Some item' =>
      Some ([item'], final_descriptors')
  | None =>
      Some ([], final_descriptors')
  end = Some (sufX, eqv_descriptors')
  ∧ equivocators_trace_project
      eqv_descriptors' tr =
    Some (preX, idescriptors)
    ∧ trX = preX ++ sufX
His_tr: equivocator_state_n (is eqv)
≤ equivocator_state_n
    (finite_trace_last is tr eqv)
Hproper': proper_equivocator_descriptors IM
  final_descriptors'
  (finite_trace_last is tr)
Hex_new: previous_state_descriptor_prop 
  (IM eqv) (descriptors eqv)
  (finite_trace_last is tr eqv)
  (final_descriptors' eqv)
Heq_ltr: finite_trace_last is tr =
state_update (equivocator_IM IM) s eqv
  (finite_trace_last is tr eqv)
Heq_final_descriptors': final_descriptors' =
equivocator_descriptors_update
  IM descriptors eqv
  (final_descriptors' eqv)
Hx: match oitem with
| Some itemx =>
    ∀ sx : composite_state IM,
      sx =
      equivocators_state_project IM
        final_descriptors'
        (finite_trace_last is tr)
      → (let (i, li) := l itemx in
         valid li (sx i, input itemx))
        ∧ (let (i, li) := l itemx in
           let (si', om') :=
             transition li (sx i, input itemx) in
           (state_update IM sx i si', om')) =
          (destination itemx, output itemx)
| None =>
    equivocators_state_project IM descriptors s =
    equivocators_state_project IM
      final_descriptors'
      (finite_trace_last is tr)
end
Hprojectx: equivocators_transition_item_project
  descriptors
  {|
    l := existT eqv li;
    input := iom;
    destination := s;
    output := oom
  |} = Some (oitem, final_descriptors')
Hitemx: match oitem with
| Some itemx =>
    (∃ Hex : existing_equivocator_label
               (IM eqv) li,
       existT eqv
         (existing_equivocator_label_extract
            (IM eqv) li Hex) = 
       l itemx)
    ∧ iom = input itemx
      ∧ oom = output itemx
        ∧ equivocators_state_project IM
            descriptors s = 
          destination itemx
          ∧ final_descriptors' eqv =
            equivocator_label_descriptor li
| None => True
end
Htr_x: ∀ eqv : index,
  equivocator_state_n
    (finite_trace_last is tr eqv)
  ≤ equivocator_state_n (s eqv)
Ht: (let (si', om') :=
   equivocator_transition 
     (IM eqv) li
     (finite_trace_last is tr eqv, iom) in
 (state_update (equivocator_IM IM)
    (finite_trace_last is tr) eqv si', om')) =
(s, oom)
Hv: equivocator_valid (IM eqv) li
  (finite_trace_last is tr eqv, iom)
Hex_new': previous_state_descriptor_prop 
  (IM eqv) (final_descriptors' eqv)
  (is eqv) (idescriptors eqv)
previous_state_descriptor_prop (IM eqv)
  (descriptors eqv) (is eqv) (idescriptors eqv) specialize (Htr_x eqv).message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
idescriptors: equivocator_descriptors IM
tr: list
  (composite_transition_item (equivocator_IM IM))
is: composite_state (equivocator_IM IM)
final_descriptors': equivocator_descriptors IM
IHtr: ∀ trX : list (composite_transition_item IM),
  equivocators_trace_project final_descriptors'
    tr = Some (trX, idescriptors)
  → ∀ eqv : index,
      previous_state_descriptor_prop 
        (IM eqv) (final_descriptors' eqv)
        (is eqv) (idescriptors eqv)
Htr: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
  is (finite_trace_last is tr) tr
s: composite_state (equivocator_IM IM)
iom, oom: option message
eqv: index
li: EquivocatorLabel (IM eqv)
descriptors: equivocator_descriptors IM
Hdescriptors: proper_equivocator_descriptors IM
  descriptors s
trX: list (composite_transition_item IM)
oitem: option (composite_transition_item IM)
Hproject_tr: ∃ (preX
   sufX : list
            (composite_transition_item
               IM)) (eqv_descriptors' : 
                     equivocator_descriptors
                      IM),
  match oitem with
  | Some item' =>
      Some ([item'], final_descriptors')
  | None =>
      Some ([], final_descriptors')
  end = Some (sufX, eqv_descriptors')
  ∧ equivocators_trace_project
      eqv_descriptors' tr =
    Some (preX, idescriptors)
    ∧ trX = preX ++ sufX
His_tr: equivocator_state_n (is eqv)
≤ equivocator_state_n
    (finite_trace_last is tr eqv)
Hproper': proper_equivocator_descriptors IM
  final_descriptors'
  (finite_trace_last is tr)
Hex_new: previous_state_descriptor_prop 
  (IM eqv) (descriptors eqv)
  (finite_trace_last is tr eqv)
  (final_descriptors' eqv)
Heq_ltr: finite_trace_last is tr =
state_update (equivocator_IM IM) s eqv
  (finite_trace_last is tr eqv)
Heq_final_descriptors': final_descriptors' =
equivocator_descriptors_update
  IM descriptors eqv
  (final_descriptors' eqv)
Hx: match oitem with
| Some itemx =>
    ∀ sx : composite_state IM,
      sx =
      equivocators_state_project IM
        final_descriptors'
        (finite_trace_last is tr)
      → (let (i, li) := l itemx in
         valid li (sx i, input itemx))
        ∧ (let (i, li) := l itemx in
           let (si', om') :=
             transition li (sx i, input itemx) in
           (state_update IM sx i si', om')) =
          (destination itemx, output itemx)
| None =>
    equivocators_state_project IM descriptors s =
    equivocators_state_project IM
      final_descriptors'
      (finite_trace_last is tr)
end
Hprojectx: equivocators_transition_item_project
  descriptors
  {|
    l := existT eqv li;
    input := iom;
    destination := s;
    output := oom
  |} = Some (oitem, final_descriptors')
Hitemx: match oitem with
| Some itemx =>
    (∃ Hex : existing_equivocator_label
               (IM eqv) li,
       existT eqv
         (existing_equivocator_label_extract
            (IM eqv) li Hex) = 
       l itemx)
    ∧ iom = input itemx
      ∧ oom = output itemx
        ∧ equivocators_state_project IM
            descriptors s = 
          destination itemx
          ∧ final_descriptors' eqv =
            equivocator_label_descriptor li
| None => True
end
Htr_x: equivocator_state_n
  (finite_trace_last is tr eqv)
≤ equivocator_state_n (s eqv)
Ht: (let (si', om') :=
   equivocator_transition 
     (IM eqv) li
     (finite_trace_last is tr eqv, iom) in
 (state_update (equivocator_IM IM)
    (finite_trace_last is tr) eqv si', om')) =
(s, oom)
Hv: equivocator_valid (IM eqv) li
  (finite_trace_last is tr eqv, iom)
Hex_new': previous_state_descriptor_prop 
  (IM eqv) (final_descriptors' eqv)
  (is eqv) (idescriptors eqv)
previous_state_descriptor_prop (IM eqv)
  (descriptors eqv) (is eqv) (idescriptors eqv)
      destruct (descriptors eqv) eqn: Hvin_desc_eqv;
        [by simpl in Hex_new; rewrite Hex_new in Hex_new' |].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
idescriptors: equivocator_descriptors IM
tr: list
  (composite_transition_item (equivocator_IM IM))
is: composite_state (equivocator_IM IM)
final_descriptors': equivocator_descriptors IM
IHtr: ∀ trX : list (composite_transition_item IM),
  equivocators_trace_project final_descriptors'
    tr = Some (trX, idescriptors)
  → ∀ eqv : index,
      previous_state_descriptor_prop 
        (IM eqv) (final_descriptors' eqv)
        (is eqv) (idescriptors eqv)
Htr: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
  is (finite_trace_last is tr) tr
s: composite_state (equivocator_IM IM)
iom, oom: option message
eqv: index
li: EquivocatorLabel (IM eqv)
descriptors: equivocator_descriptors IM
Hdescriptors: proper_equivocator_descriptors IM
  descriptors s
trX: list (composite_transition_item IM)
oitem: option (composite_transition_item IM)
Hproject_tr: ∃ (preX
   sufX : list
            (composite_transition_item
               IM)) (eqv_descriptors' : 
                     equivocator_descriptors
                      IM),
  match oitem with
  | Some item' =>
      Some ([item'], final_descriptors')
  | None =>
      Some ([], final_descriptors')
  end = Some (sufX, eqv_descriptors')
  ∧ equivocators_trace_project
      eqv_descriptors' tr =
    Some (preX, idescriptors)
    ∧ trX = preX ++ sufX
His_tr: equivocator_state_n (is eqv)
≤ equivocator_state_n
    (finite_trace_last is tr eqv)
Hproper': proper_equivocator_descriptors IM
  final_descriptors'
  (finite_trace_last is tr)
n: nat
Hvin_desc_eqv: descriptors eqv = Existing n
Hex_new: previous_state_descriptor_prop 
  (IM eqv) (Existing n)
  (finite_trace_last is tr eqv)
  (final_descriptors' eqv)
Heq_ltr: finite_trace_last is tr =
state_update (equivocator_IM IM) s eqv
  (finite_trace_last is tr eqv)
Heq_final_descriptors': final_descriptors' =
equivocator_descriptors_update
  IM descriptors eqv
  (final_descriptors' eqv)
Hx: match oitem with
| Some itemx =>
    ∀ sx : composite_state IM,
      sx =
      equivocators_state_project IM
        final_descriptors'
        (finite_trace_last is tr)
      → (let (i, li) := l itemx in
         valid li (sx i, input itemx))
        ∧ (let (i, li) := l itemx in
           let (si', om') :=
             transition li (sx i, input itemx) in
           (state_update IM sx i si', om')) =
          (destination itemx, output itemx)
| None =>
    equivocators_state_project IM descriptors s =
    equivocators_state_project IM
      final_descriptors'
      (finite_trace_last is tr)
end
Hprojectx: equivocators_transition_item_project
  descriptors
  {|
    l := existT eqv li;
    input := iom;
    destination := s;
    output := oom
  |} = Some (oitem, final_descriptors')
Hitemx: match oitem with
| Some itemx =>
    (∃ Hex : existing_equivocator_label
               (IM eqv) li,
       existT eqv
         (existing_equivocator_label_extract
            (IM eqv) li Hex) = 
       l itemx)
    ∧ iom = input itemx
      ∧ oom = output itemx
        ∧ equivocators_state_project IM
            descriptors s = 
          destination itemx
          ∧ final_descriptors' eqv =
            equivocator_label_descriptor li
| None => True
end
Htr_x: equivocator_state_n
  (finite_trace_last is tr eqv)
≤ equivocator_state_n (s eqv)
Ht: (let (si', om') :=
   equivocator_transition 
     (IM eqv) li
     (finite_trace_last is tr eqv, iom) in
 (state_update (equivocator_IM IM)
    (finite_trace_last is tr) eqv si', om')) =
(s, oom)
Hv: equivocator_valid (IM eqv) li
  (finite_trace_last is tr eqv, iom)
Hex_new': previous_state_descriptor_prop 
  (IM eqv) (final_descriptors' eqv)
  (is eqv) (idescriptors eqv)
previous_state_descriptor_prop (IM eqv) (Existing n)
  (is eqv) (idescriptors eqv)
      destruct (final_descriptors' eqv) eqn: Hfin_desc_eqv'.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
idescriptors: equivocator_descriptors IM
tr: list
  (composite_transition_item (equivocator_IM IM))
is: composite_state (equivocator_IM IM)
final_descriptors': equivocator_descriptors IM
IHtr: ∀ trX : list (composite_transition_item IM),
  equivocators_trace_project final_descriptors'
    tr = Some (trX, idescriptors)
  → ∀ eqv : index,
      previous_state_descriptor_prop 
        (IM eqv) (final_descriptors' eqv)
        (is eqv) (idescriptors eqv)
Htr: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
  is (finite_trace_last is tr) tr
s: composite_state (equivocator_IM IM)
iom, oom: option message
eqv: index
li: EquivocatorLabel (IM eqv)
descriptors: equivocator_descriptors IM
Hdescriptors: proper_equivocator_descriptors IM
  descriptors s
trX: list (composite_transition_item IM)
oitem: option (composite_transition_item IM)
Hproject_tr: ∃ (preX
   sufX : list
            (composite_transition_item
               IM)) (eqv_descriptors' : 
                     equivocator_descriptors
                      IM),
  match oitem with
  | Some item' =>
      Some ([item'], final_descriptors')
  | None =>
      Some ([], final_descriptors')
  end = Some (sufX, eqv_descriptors')
  ∧ equivocators_trace_project
      eqv_descriptors' tr =
    Some (preX, idescriptors)
    ∧ trX = preX ++ sufX
His_tr: equivocator_state_n (is eqv)
≤ equivocator_state_n
    (finite_trace_last is tr eqv)
Hproper': proper_equivocator_descriptors IM
  final_descriptors'
  (finite_trace_last is tr)
n: nat
Hvin_desc_eqv: descriptors eqv = Existing n
s0: state (IM eqv)
Hfin_desc_eqv': final_descriptors' eqv =
NewMachine s0
Hex_new: previous_state_descriptor_prop 
  (IM eqv) (Existing n)
  (finite_trace_last is tr eqv)
  (NewMachine s0)
Heq_ltr: finite_trace_last is tr =
state_update (equivocator_IM IM) s eqv
  (finite_trace_last is tr eqv)
Heq_final_descriptors': final_descriptors' =
equivocator_descriptors_update
  IM descriptors eqv
  (NewMachine s0)
Hx: match oitem with
| Some itemx =>
    ∀ sx : composite_state IM,
      sx =
      equivocators_state_project IM
        final_descriptors'
        (finite_trace_last is tr)
      → (let (i, li) := l itemx in
         valid li (sx i, input itemx))
        ∧ (let (i, li) := l itemx in
           let (si', om') :=
             transition li (sx i, input itemx) in
           (state_update IM sx i si', om')) =
          (destination itemx, output itemx)
| None =>
    equivocators_state_project IM descriptors s =
    equivocators_state_project IM
      final_descriptors'
      (finite_trace_last is tr)
end
Hprojectx: equivocators_transition_item_project
  descriptors
  {|
    l := existT eqv li;
    input := iom;
    destination := s;
    output := oom
  |} = Some (oitem, final_descriptors')
Hitemx: match oitem with
| Some itemx =>
    (∃ Hex : existing_equivocator_label
               (IM eqv) li,
       existT eqv
         (existing_equivocator_label_extract
            (IM eqv) li Hex) = 
       l itemx)
    ∧ iom = input itemx
      ∧ oom = output itemx
        ∧ equivocators_state_project IM
            descriptors s = 
          destination itemx
          ∧ NewMachine s0 =
            equivocator_label_descriptor li
| None => True
end
Htr_x: equivocator_state_n
  (finite_trace_last is tr eqv)
≤ equivocator_state_n (s eqv)
Ht: (let (si', om') :=
   equivocator_transition 
     (IM eqv) li
     (finite_trace_last is tr eqv, iom) in
 (state_update (equivocator_IM IM)
    (finite_trace_last is tr) eqv si', om')) =
(s, oom)
Hv: equivocator_valid (IM eqv) li
  (finite_trace_last is tr eqv, iom)
Hex_new': previous_state_descriptor_prop 
  (IM eqv) (NewMachine s0) 
  (is eqv) (idescriptors eqv)
previous_state_descriptor_prop (IM eqv) (Existing n)
  (is eqv) (idescriptors eqv)
message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
idescriptors: equivocator_descriptors IM
tr: list
  (composite_transition_item (equivocator_IM IM))
is: composite_state (equivocator_IM IM)
final_descriptors': equivocator_descriptors IM
IHtr: ∀ trX : list (composite_transition_item IM),
  equivocators_trace_project final_descriptors'
    tr = Some (trX, idescriptors)
  → ∀ eqv : index,
      previous_state_descriptor_prop 
        (IM eqv) (final_descriptors' eqv)
        (is eqv) (idescriptors eqv)
Htr: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
  is (finite_trace_last is tr) tr
s: composite_state (equivocator_IM IM)
iom, oom: option message
eqv: index
li: EquivocatorLabel (IM eqv)
descriptors: equivocator_descriptors IM
Hdescriptors: proper_equivocator_descriptors IM
  descriptors s
trX: list (composite_transition_item IM)
oitem: option (composite_transition_item IM)
Hproject_tr: ∃ (preX
   sufX : list
            (composite_transition_item
               IM)) (eqv_descriptors' : 
                     equivocator_descriptors
                      IM),
  match oitem with
  | Some item' =>
      Some ([item'], final_descriptors')
  | None =>
      Some ([], final_descriptors')
  end = Some (sufX, eqv_descriptors')
  ∧ equivocators_trace_project
      eqv_descriptors' tr =
    Some (preX, idescriptors)
    ∧ trX = preX ++ sufX
His_tr: equivocator_state_n (is eqv)
≤ equivocator_state_n
    (finite_trace_last is tr eqv)
Hproper': proper_equivocator_descriptors IM
  final_descriptors'
  (finite_trace_last is tr)
n: nat
Hvin_desc_eqv: descriptors eqv = Existing n
n0: nat
Hfin_desc_eqv': final_descriptors' eqv = Existing n0
Hex_new: previous_state_descriptor_prop 
  (IM eqv) (Existing n)
  (finite_trace_last is tr eqv)
  (Existing n0)
Heq_ltr: finite_trace_last is tr =
state_update (equivocator_IM IM) s eqv
  (finite_trace_last is tr eqv)
Heq_final_descriptors': final_descriptors' =
equivocator_descriptors_update
  IM descriptors eqv
  (Existing n0)
Hx: match oitem with
| Some itemx =>
    ∀ sx : composite_state IM,
      sx =
      equivocators_state_project IM
        final_descriptors'
        (finite_trace_last is tr)
      → (let (i, li) := l itemx in
         valid li (sx i, input itemx))
        ∧ (let (i, li) := l itemx in
           let (si', om') :=
             transition li (sx i, input itemx) in
           (state_update IM sx i si', om')) =
          (destination itemx, output itemx)
| None =>
    equivocators_state_project IM descriptors s =
    equivocators_state_project IM
      final_descriptors'
      (finite_trace_last is tr)
end
Hprojectx: equivocators_transition_item_project
  descriptors
  {|
    l := existT eqv li;
    input := iom;
    destination := s;
    output := oom
  |} = Some (oitem, final_descriptors')
Hitemx: match oitem with
| Some itemx =>
    (∃ Hex : existing_equivocator_label
               (IM eqv) li,
       existT eqv
         (existing_equivocator_label_extract
            (IM eqv) li Hex) = 
       l itemx)
    ∧ iom = input itemx
      ∧ oom = output itemx
        ∧ equivocators_state_project IM
            descriptors s = 
          destination itemx
          ∧ Existing n0 =
            equivocator_label_descriptor li
| None => True
end
Htr_x: equivocator_state_n
  (finite_trace_last is tr eqv)
≤ equivocator_state_n (s eqv)
Ht: (let (si', om') :=
   equivocator_transition 
     (IM eqv) li
     (finite_trace_last is tr eqv, iom) in
 (state_update (equivocator_IM IM)
    (finite_trace_last is tr) eqv si', om')) =
(s, oom)
Hv: equivocator_valid (IM eqv) li
  (finite_trace_last is tr eqv, iom)
Hex_new': previous_state_descriptor_prop 
  (IM eqv) (Existing n0) 
  (is eqv) (idescriptors eqv)
previous_state_descriptor_prop 
  (IM eqv) (Existing n) 
  (is eqv) (idescriptors eqv)
      *message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
idescriptors: equivocator_descriptors IM
tr: list
  (composite_transition_item (equivocator_IM IM))
is: composite_state (equivocator_IM IM)
final_descriptors': equivocator_descriptors IM
IHtr: ∀ trX : list (composite_transition_item IM),
  equivocators_trace_project final_descriptors'
    tr = Some (trX, idescriptors)
  → ∀ eqv : index,
      previous_state_descriptor_prop 
        (IM eqv) (final_descriptors' eqv)
        (is eqv) (idescriptors eqv)
Htr: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
  is (finite_trace_last is tr) tr
s: composite_state (equivocator_IM IM)
iom, oom: option message
eqv: index
li: EquivocatorLabel (IM eqv)
descriptors: equivocator_descriptors IM
Hdescriptors: proper_equivocator_descriptors IM
  descriptors s
trX: list (composite_transition_item IM)
oitem: option (composite_transition_item IM)
Hproject_tr: ∃ (preX
   sufX : list
            (composite_transition_item
               IM)) (eqv_descriptors' : 
                     equivocator_descriptors
                      IM),
  match oitem with
  | Some item' =>
      Some ([item'], final_descriptors')
  | None =>
      Some ([], final_descriptors')
  end = Some (sufX, eqv_descriptors')
  ∧ equivocators_trace_project
      eqv_descriptors' tr =
    Some (preX, idescriptors)
    ∧ trX = preX ++ sufX
His_tr: equivocator_state_n (is eqv)
≤ equivocator_state_n
    (finite_trace_last is tr eqv)
Hproper': proper_equivocator_descriptors IM
  final_descriptors'
  (finite_trace_last is tr)
n: nat
Hvin_desc_eqv: descriptors eqv = Existing n
s0: state (IM eqv)
Hfin_desc_eqv': final_descriptors' eqv =
NewMachine s0
Hex_new: previous_state_descriptor_prop 
  (IM eqv) (Existing n)
  (finite_trace_last is tr eqv)
  (NewMachine s0)
Heq_ltr: finite_trace_last is tr =
state_update (equivocator_IM IM) s eqv
  (finite_trace_last is tr eqv)
Heq_final_descriptors': final_descriptors' =
equivocator_descriptors_update
  IM descriptors eqv
  (NewMachine s0)
Hx: match oitem with
| Some itemx =>
    ∀ sx : composite_state IM,
      sx =
      equivocators_state_project IM
        final_descriptors'
        (finite_trace_last is tr)
      → (let (i, li) := l itemx in
         valid li (sx i, input itemx))
        ∧ (let (i, li) := l itemx in
           let (si', om') :=
             transition li (sx i, input itemx) in
           (state_update IM sx i si', om')) =
          (destination itemx, output itemx)
| None =>
    equivocators_state_project IM descriptors s =
    equivocators_state_project IM
      final_descriptors'
      (finite_trace_last is tr)
end
Hprojectx: equivocators_transition_item_project
  descriptors
  {|
    l := existT eqv li;
    input := iom;
    destination := s;
    output := oom
  |} = Some (oitem, final_descriptors')
Hitemx: match oitem with
| Some itemx =>
    (∃ Hex : existing_equivocator_label
               (IM eqv) li,
       existT eqv
         (existing_equivocator_label_extract
            (IM eqv) li Hex) = 
       l itemx)
    ∧ iom = input itemx
      ∧ oom = output itemx
        ∧ equivocators_state_project IM
            descriptors s = 
          destination itemx
          ∧ NewMachine s0 =
            equivocator_label_descriptor li
| None => True
end
Htr_x: equivocator_state_n
  (finite_trace_last is tr eqv)
≤ equivocator_state_n (s eqv)
Ht: (let (si', om') :=
   equivocator_transition 
     (IM eqv) li
     (finite_trace_last is tr eqv, iom) in
 (state_update (equivocator_IM IM)
    (finite_trace_last is tr) eqv si', om')) =
(s, oom)
Hv: equivocator_valid (IM eqv) li
  (finite_trace_last is tr eqv, iom)
Hex_new': previous_state_descriptor_prop 
  (IM eqv) (NewMachine s0) 
  (is eqv) (idescriptors eqv)
previous_state_descriptor_prop (IM eqv) (Existing n)
  (is eqv) (idescriptors eqv) by simpl in Hex_new, Hex_new'; rewrite Hex_new'; simpl;  lia.
      *message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
idescriptors: equivocator_descriptors IM
tr: list
  (composite_transition_item (equivocator_IM IM))
is: composite_state (equivocator_IM IM)
final_descriptors': equivocator_descriptors IM
IHtr: ∀ trX : list (composite_transition_item IM),
  equivocators_trace_project final_descriptors'
    tr = Some (trX, idescriptors)
  → ∀ eqv : index,
      previous_state_descriptor_prop 
        (IM eqv) (final_descriptors' eqv)
        (is eqv) (idescriptors eqv)
Htr: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
  is (finite_trace_last is tr) tr
s: composite_state (equivocator_IM IM)
iom, oom: option message
eqv: index
li: EquivocatorLabel (IM eqv)
descriptors: equivocator_descriptors IM
Hdescriptors: proper_equivocator_descriptors IM
  descriptors s
trX: list (composite_transition_item IM)
oitem: option (composite_transition_item IM)
Hproject_tr: ∃ (preX
   sufX : list
            (composite_transition_item
               IM)) (eqv_descriptors' : 
                     equivocator_descriptors
                      IM),
  match oitem with
  | Some item' =>
      Some ([item'], final_descriptors')
  | None =>
      Some ([], final_descriptors')
  end = Some (sufX, eqv_descriptors')
  ∧ equivocators_trace_project
      eqv_descriptors' tr =
    Some (preX, idescriptors)
    ∧ trX = preX ++ sufX
His_tr: equivocator_state_n (is eqv)
≤ equivocator_state_n
    (finite_trace_last is tr eqv)
Hproper': proper_equivocator_descriptors IM
  final_descriptors'
  (finite_trace_last is tr)
n: nat
Hvin_desc_eqv: descriptors eqv = Existing n
n0: nat
Hfin_desc_eqv': final_descriptors' eqv = Existing n0
Hex_new: previous_state_descriptor_prop 
  (IM eqv) (Existing n)
  (finite_trace_last is tr eqv)
  (Existing n0)
Heq_ltr: finite_trace_last is tr =
state_update (equivocator_IM IM) s eqv
  (finite_trace_last is tr eqv)
Heq_final_descriptors': final_descriptors' =
equivocator_descriptors_update
  IM descriptors eqv
  (Existing n0)
Hx: match oitem with
| Some itemx =>
    ∀ sx : composite_state IM,
      sx =
      equivocators_state_project IM
        final_descriptors'
        (finite_trace_last is tr)
      → (let (i, li) := l itemx in
         valid li (sx i, input itemx))
        ∧ (let (i, li) := l itemx in
           let (si', om') :=
             transition li (sx i, input itemx) in
           (state_update IM sx i si', om')) =
          (destination itemx, output itemx)
| None =>
    equivocators_state_project IM descriptors s =
    equivocators_state_project IM
      final_descriptors'
      (finite_trace_last is tr)
end
Hprojectx: equivocators_transition_item_project
  descriptors
  {|
    l := existT eqv li;
    input := iom;
    destination := s;
    output := oom
  |} = Some (oitem, final_descriptors')
Hitemx: match oitem with
| Some itemx =>
    (∃ Hex : existing_equivocator_label
               (IM eqv) li,
       existT eqv
         (existing_equivocator_label_extract
            (IM eqv) li Hex) = 
       l itemx)
    ∧ iom = input itemx
      ∧ oom = output itemx
        ∧ equivocators_state_project IM
            descriptors s = 
          destination itemx
          ∧ Existing n0 =
            equivocator_label_descriptor li
| None => True
end
Htr_x: equivocator_state_n
  (finite_trace_last is tr eqv)
≤ equivocator_state_n (s eqv)
Ht: (let (si', om') :=
   equivocator_transition 
     (IM eqv) li
     (finite_trace_last is tr eqv, iom) in
 (state_update (equivocator_IM IM)
    (finite_trace_last is tr) eqv si', om')) =
(s, oom)
Hv: equivocator_valid (IM eqv) li
  (finite_trace_last is tr eqv, iom)
Hex_new': previous_state_descriptor_prop 
  (IM eqv) (Existing n0) 
  (is eqv) (idescriptors eqv)
previous_state_descriptor_prop (IM eqv) (Existing n)
  (is eqv) (idescriptors eqv) by destruct (idescriptors eqv); simpl in *; lia.
    +message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
idescriptors: equivocator_descriptors IM
tr: list
  (composite_transition_item (equivocator_IM IM))
is: composite_state (equivocator_IM IM)
final_descriptors': equivocator_descriptors IM
IHtr: ∀ trX : list (composite_transition_item IM),
  equivocators_trace_project final_descriptors'
    tr = Some (trX, idescriptors)
  → ∀ eqv : index,
      previous_state_descriptor_prop 
        (IM eqv) (final_descriptors' eqv)
        (is eqv) (idescriptors eqv)
Htr: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
  is (finite_trace_last is tr) tr
s: composite_state (equivocator_IM IM)
iom, oom: option message
i: index
li: EquivocatorLabel (IM i)
descriptors: equivocator_descriptors IM
Hdescriptors: proper_equivocator_descriptors IM
  descriptors s
trX: list (composite_transition_item IM)
oitem: option (composite_transition_item IM)
Hproject_tr: ∃ (preX
   sufX : list
            (composite_transition_item
               IM)) (eqv_descriptors' : 
                     equivocator_descriptors
                      IM),
  match oitem with
  | Some item' =>
      Some ([item'], final_descriptors')
  | None =>
      Some ([], final_descriptors')
  end = Some (sufX, eqv_descriptors')
  ∧ equivocators_trace_project
      eqv_descriptors' tr =
    Some (preX, idescriptors)
    ∧ trX = preX ++ sufX
eqv: index
His_tr: ∀ eqv : index,
  equivocator_state_n (is eqv)
  ≤ equivocator_state_n
      (finite_trace_last is tr eqv)
Hproper': proper_equivocator_descriptors IM
  final_descriptors'
  (finite_trace_last is tr)
Heq_final_descriptors': final_descriptors' =
equivocator_descriptors_update
  IM descriptors i
  (final_descriptors' i)
Heq_ltr: finite_trace_last is tr =
state_update (equivocator_IM IM) s i
  (finite_trace_last is tr i)
Hex_new: previous_state_descriptor_prop 
  (IM i) (descriptors i)
  (finite_trace_last is tr i)
  (final_descriptors' i)
Hx: match oitem with
| Some itemx =>
    ∀ sx : composite_state IM,
      sx =
      equivocators_state_project IM
        final_descriptors'
        (finite_trace_last is tr)
      → (let (i, li) := l itemx in
         valid li (sx i, input itemx))
        ∧ (let (i, li) := l itemx in
           let (si', om') :=
             transition li (sx i, input itemx) in
           (state_update IM sx i si', om')) =
          (destination itemx, output itemx)
| None =>
    equivocators_state_project IM descriptors s =
    equivocators_state_project IM
      final_descriptors'
      (finite_trace_last is tr)
end
Hitemx: match oitem with
| Some itemx =>
    (∃ Hex : existing_equivocator_label
               (IM i) li,
       existT i
         (existing_equivocator_label_extract
            (IM i) li Hex) = 
       l itemx)
    ∧ iom = input itemx
      ∧ oom = output itemx
        ∧ equivocators_state_project IM
            descriptors s = 
          destination itemx
          ∧ final_descriptors' i =
            equivocator_label_descriptor li
| None => True
end
Hprojectx: equivocators_transition_item_project
  descriptors
  {|
    l := existT i li;
    input := iom;
    destination := s;
    output := oom
  |} = Some (oitem, final_descriptors')
Htr_x: ∀ eqv : index,
  equivocator_state_n
    (finite_trace_last is tr eqv)
  ≤ equivocator_state_n (s eqv)
Hv: equivocator_valid (IM i) li
  (finite_trace_last is tr i, iom)
Ht: (let (si', om') :=
   equivocator_transition 
     (IM i) li (finite_trace_last is tr i, iom) in
 (state_update (equivocator_IM IM)
    (finite_trace_last is tr) i si', om')) =
(s, oom)
Hex_new': previous_state_descriptor_prop 
  (IM eqv) (final_descriptors' eqv)
  (is eqv) (idescriptors eqv)
n: i ≠ eqv
previous_state_descriptor_prop (IM eqv)
  (descriptors eqv) (is eqv) (idescriptors eqv) by rewrite Heq_final_descriptors' in Hex_new'; state_update_simpl.
Qed.

Lemma equivocators_trace_project_preserves_equivocating_indices_final
  (descriptors idescriptors : equivocator_descriptors IM)
  (tr : list (composite_transition_item (equivocator_IM IM)))
  (trX : list (composite_transition_item IM))
  (is s : composite_state (equivocator_IM IM))
  (Htr : finite_constrained_trace_from_to (free_composite_vlsm (equivocator_IM IM)) is s tr)
  (Hdescriptors : not_equivocating_equivocator_descriptors IM descriptors s)
  (Hproject_tr : equivocators_trace_project descriptors tr = Some (trX, idescriptors))
  :
    set_union
      (equivocating_indices IM (enum index) is)
      (newmachine_descriptors_list IM (enum index) idescriptors)
    ⊆
    equivocating_indices IM (enum index) s.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
descriptors, idescriptors: equivocator_descriptors IM
tr: list
  (composite_transition_item (equivocator_IM IM))
trX: list (composite_transition_item IM)
is, s: composite_state (equivocator_IM IM)
Htr: finite_constrained_trace_from_to
  (free_composite_vlsm (equivocator_IM IM)) is s
  tr
Hdescriptors: not_equivocating_equivocator_descriptors
  IM descriptors s
Hproject_tr: equivocators_trace_project descriptors
  tr = Some (trX, idescriptors)
set_union (equivocating_indices IM (enum index) is)
  (newmachine_descriptors_list IM (enum index)
     idescriptors)
⊆ equivocating_indices IM (enum index) s
Proof.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
descriptors, idescriptors: equivocator_descriptors IM
tr: list
  (composite_transition_item (equivocator_IM IM))
trX: list (composite_transition_item IM)
is, s: composite_state (equivocator_IM IM)
Htr: finite_constrained_trace_from_to
  (free_composite_vlsm (equivocator_IM IM)) is s
  tr
Hdescriptors: not_equivocating_equivocator_descriptors
  IM descriptors s
Hproject_tr: equivocators_trace_project descriptors
  tr = Some (trX, idescriptors)
set_union (equivocating_indices IM (enum index) is)
  (newmachine_descriptors_list IM (enum index)
     idescriptors)
⊆ equivocating_indices IM (enum index) s
  apply not_equivocating_equivocator_descriptors_proper in Hdescriptors as Hproper.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
descriptors, idescriptors: equivocator_descriptors IM
tr: list
  (composite_transition_item (equivocator_IM IM))
trX: list (composite_transition_item IM)
is, s: composite_state (equivocator_IM IM)
Htr: finite_constrained_trace_from_to
  (free_composite_vlsm (equivocator_IM IM)) is s
  tr
Hdescriptors: not_equivocating_equivocator_descriptors
  IM descriptors s
Hproject_tr: equivocators_trace_project descriptors
  tr = Some (trX, idescriptors)
Hproper: proper_equivocator_descriptors IM
  descriptors s
set_union (equivocating_indices IM (enum index) is)
  (newmachine_descriptors_list IM (enum index)
     idescriptors)
⊆ equivocating_indices IM (enum index) s
  specialize
    (equivocators_trace_project_preserves_equivocating_indices _ _ _ _ _ _
      Htr Hproper Hproject_tr)
    as Hincl.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
descriptors, idescriptors: equivocator_descriptors IM
tr: list
  (composite_transition_item (equivocator_IM IM))
trX: list (composite_transition_item IM)
is, s: composite_state (equivocator_IM IM)
Htr: finite_constrained_trace_from_to
  (free_composite_vlsm (equivocator_IM IM)) is s
  tr
Hdescriptors: not_equivocating_equivocator_descriptors
  IM descriptors s
Hproject_tr: equivocators_trace_project descriptors
  tr = Some (trX, idescriptors)
Hproper: proper_equivocator_descriptors IM
  descriptors s
Hincl: set_union
  (equivocating_indices IM (enum index) is)
  (newmachine_descriptors_list IM 
     (enum index) idescriptors)
⊆ set_union
    (equivocating_indices IM (enum index) s)
    (newmachine_descriptors_list IM
       (enum index) descriptors)
set_union (equivocating_indices IM (enum index) is)
  (newmachine_descriptors_list IM (enum index)
     idescriptors)
⊆ equivocating_indices IM (enum index) s
  intros eqv Heqv.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
descriptors, idescriptors: equivocator_descriptors IM
tr: list
  (composite_transition_item (equivocator_IM IM))
trX: list (composite_transition_item IM)
is, s: composite_state (equivocator_IM IM)
Htr: finite_constrained_trace_from_to
  (free_composite_vlsm (equivocator_IM IM)) is s
  tr
Hdescriptors: not_equivocating_equivocator_descriptors
  IM descriptors s
Hproject_tr: equivocators_trace_project descriptors
  tr = Some (trX, idescriptors)
Hproper: proper_equivocator_descriptors IM
  descriptors s
Hincl: set_union
  (equivocating_indices IM (enum index) is)
  (newmachine_descriptors_list IM 
     (enum index) idescriptors)
⊆ set_union
    (equivocating_indices IM (enum index) s)
    (newmachine_descriptors_list IM
       (enum index) descriptors)
eqv: index
Heqv: eqv
∈ set_union
    (equivocating_indices IM (enum index) is)
    (newmachine_descriptors_list IM
       (enum index) idescriptors)
eqv ∈ equivocating_indices IM (enum index) s specialize (Hincl eqv Heqv).message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
descriptors, idescriptors: equivocator_descriptors IM
tr: list
  (composite_transition_item (equivocator_IM IM))
trX: list (composite_transition_item IM)
is, s: composite_state (equivocator_IM IM)
Htr: finite_constrained_trace_from_to
  (free_composite_vlsm (equivocator_IM IM)) is s
  tr
Hdescriptors: not_equivocating_equivocator_descriptors
  IM descriptors s
Hproject_tr: equivocators_trace_project descriptors
  tr = Some (trX, idescriptors)
Hproper: proper_equivocator_descriptors IM
  descriptors s
eqv: index
Hincl: eqv
∈ set_union
    (equivocating_indices IM (enum index) s)
    (newmachine_descriptors_list IM
       (enum index) descriptors)
Heqv: eqv
∈ set_union
    (equivocating_indices IM (enum index) is)
    (newmachine_descriptors_list IM
       (enum index) idescriptors)
eqv ∈ equivocating_indices IM (enum index) s
  apply set_union_iff in Hincl.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
descriptors, idescriptors: equivocator_descriptors IM
tr: list
  (composite_transition_item (equivocator_IM IM))
trX: list (composite_transition_item IM)
is, s: composite_state (equivocator_IM IM)
Htr: finite_constrained_trace_from_to
  (free_composite_vlsm (equivocator_IM IM)) is s
  tr
Hdescriptors: not_equivocating_equivocator_descriptors
  IM descriptors s
Hproject_tr: equivocators_trace_project descriptors
  tr = Some (trX, idescriptors)
Hproper: proper_equivocator_descriptors IM
  descriptors s
eqv: index
Hincl: eqv ∈ equivocating_indices IM (enum index) s
∨ eqv
  ∈ newmachine_descriptors_list IM
      (enum index) descriptors
Heqv: eqv
∈ set_union
    (equivocating_indices IM (enum index) is)
    (newmachine_descriptors_list IM
       (enum index) idescriptors)
eqv ∈ equivocating_indices IM (enum index) s
  clear Heqv.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
descriptors, idescriptors: equivocator_descriptors IM
tr: list
  (composite_transition_item (equivocator_IM IM))
trX: list (composite_transition_item IM)
is, s: composite_state (equivocator_IM IM)
Htr: finite_constrained_trace_from_to
  (free_composite_vlsm (equivocator_IM IM)) is s
  tr
Hdescriptors: not_equivocating_equivocator_descriptors
  IM descriptors s
Hproject_tr: equivocators_trace_project descriptors
  tr = Some (trX, idescriptors)
Hproper: proper_equivocator_descriptors IM
  descriptors s
eqv: index
Hincl: eqv ∈ equivocating_indices IM (enum index) s
∨ eqv
  ∈ newmachine_descriptors_list IM
      (enum index) descriptors
eqv ∈ equivocating_indices IM (enum index) s
  destruct Hincl as [| Heqv]; [done |].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
descriptors, idescriptors: equivocator_descriptors IM
tr: list
  (composite_transition_item (equivocator_IM IM))
trX: list (composite_transition_item IM)
is, s: composite_state (equivocator_IM IM)
Htr: finite_constrained_trace_from_to
  (free_composite_vlsm (equivocator_IM IM)) is s
  tr
Hdescriptors: not_equivocating_equivocator_descriptors
  IM descriptors s
Hproject_tr: equivocators_trace_project descriptors
  tr = Some (trX, idescriptors)
Hproper: proper_equivocator_descriptors IM
  descriptors s
eqv: index
Heqv: eqv
∈ newmachine_descriptors_list IM 
    (enum index) descriptors
eqv ∈ equivocating_indices IM (enum index) s
  specialize (Hdescriptors eqv).message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
descriptors, idescriptors: equivocator_descriptors IM
tr: list
  (composite_transition_item (equivocator_IM IM))
trX: list (composite_transition_item IM)
is, s: composite_state (equivocator_IM IM)
Htr: finite_constrained_trace_from_to
  (free_composite_vlsm (equivocator_IM IM)) is s
  tr
eqv: index
Hdescriptors: existing_descriptor (IM eqv)
  (descriptors eqv) (s eqv)
Hproject_tr: equivocators_trace_project descriptors
  tr = Some (trX, idescriptors)
Hproper: proper_equivocator_descriptors IM
  descriptors s
Heqv: eqv
∈ newmachine_descriptors_list IM 
    (enum index) descriptors
eqv ∈ equivocating_indices IM (enum index) s
  apply elem_of_list_filter in Heqv.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
descriptors, idescriptors: equivocator_descriptors IM
tr: list
  (composite_transition_item (equivocator_IM IM))
trX: list (composite_transition_item IM)
is, s: composite_state (equivocator_IM IM)
Htr: finite_constrained_trace_from_to
  (free_composite_vlsm (equivocator_IM IM)) is s
  tr
eqv: index
Hdescriptors: existing_descriptor (IM eqv)
  (descriptors eqv) (s eqv)
Hproject_tr: equivocators_trace_project descriptors
  tr = Some (trX, idescriptors)
Hproper: proper_equivocator_descriptors IM
  descriptors s
Heqv: is_newmachine_descriptor 
  (IM eqv) (descriptors eqv) ∧ 
eqv ∈ enum index
eqv ∈ equivocating_indices IM (enum index) s
  destruct Heqv as [Heqv Hin].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
descriptors, idescriptors: equivocator_descriptors IM
tr: list
  (composite_transition_item (equivocator_IM IM))
trX: list (composite_transition_item IM)
is, s: composite_state (equivocator_IM IM)
Htr: finite_constrained_trace_from_to
  (free_composite_vlsm (equivocator_IM IM)) is s
  tr
eqv: index
Hdescriptors: existing_descriptor (IM eqv)
  (descriptors eqv) (s eqv)
Hproject_tr: equivocators_trace_project descriptors
  tr = Some (trX, idescriptors)
Hproper: proper_equivocator_descriptors IM
  descriptors s
Heqv: is_newmachine_descriptor 
  (IM eqv) (descriptors eqv)
Hin: eqv ∈ enum index
eqv ∈ equivocating_indices IM (enum index) s
  by destruct (descriptors eqv).
Qed.

We can project a trace over the composition of equivocators in two ways: (1) first project on a equivocator component, then project the equivocator to the original component (2) first projects to the composition of original components, then project to one of them

The result below says that the two ways lead to the same result.

Lemma equivocators_trace_project_finite_trace_projection_list_commute
  (i : index)
  (final_descriptors initial_descriptors : equivocator_descriptors IM)
  (eqv_initial : MachineDescriptor (IM i))
  (tr : list (composite_transition_item (equivocator_IM IM)))
  (trX : list (composite_transition_item IM))
  (trXi : list (transition_item (IM i)))
  (eqv_final := final_descriptors i)
  (Hproject_tr : equivocators_trace_project final_descriptors tr = Some (trX, initial_descriptors))
  (Hproject_tri :
    equivocator_vlsm_trace_project (IM i)
      (finite_trace_projection_list (equivocator_IM IM) i tr) eqv_final
    = Some (trXi, eqv_initial))
  : initial_descriptors i = eqv_initial /\
    finite_trace_projection_list IM i trX = trXi.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
i: index
final_descriptors, initial_descriptors: equivocator_descriptors IM
eqv_initial: MachineDescriptor (IM i)
tr: list
  (composite_transition_item (equivocator_IM IM))
trX: list (composite_transition_item IM)
trXi: list transition_item
eqv_final:= final_descriptors i: MachineDescriptor (IM i)
Hproject_tr: equivocators_trace_project
  final_descriptors tr =
Some (trX, initial_descriptors)
Hproject_tri: equivocator_vlsm_trace_project 
  (IM i)
  (finite_trace_projection_list
     (equivocator_IM IM) i tr)
  eqv_final = Some (trXi, eqv_initial)
initial_descriptors i = eqv_initial
∧ finite_trace_projection_list IM i trX = trXi
Proof.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
i: index
final_descriptors, initial_descriptors: equivocator_descriptors IM
eqv_initial: MachineDescriptor (IM i)
tr: list
  (composite_transition_item (equivocator_IM IM))
trX: list (composite_transition_item IM)
trXi: list transition_item
eqv_final:= final_descriptors i: MachineDescriptor (IM i)
Hproject_tr: equivocators_trace_project
  final_descriptors tr =
Some (trX, initial_descriptors)
Hproject_tri: equivocator_vlsm_trace_project 
  (IM i)
  (finite_trace_projection_list
     (equivocator_IM IM) i tr)
  eqv_final = Some (trXi, eqv_initial)
initial_descriptors i = eqv_initial
∧ finite_trace_projection_list IM i trX = trXi
  generalize dependent trXi.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
i: index
final_descriptors, initial_descriptors: equivocator_descriptors IM
eqv_initial: MachineDescriptor (IM i)
tr: list
  (composite_transition_item (equivocator_IM IM))
trX: list (composite_transition_item IM)
eqv_final:= final_descriptors i: MachineDescriptor (IM i)
Hproject_tr: equivocators_trace_project
  final_descriptors tr =
Some (trX, initial_descriptors)
∀ trXi : list transition_item,
  equivocator_vlsm_trace_project (IM i)
    (finite_trace_projection_list (equivocator_IM IM)
       i tr) eqv_final = Some (trXi, eqv_initial)
  → initial_descriptors i = eqv_initial
    ∧ finite_trace_projection_list IM i trX = trXi generalize dependent trX.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
i: index
final_descriptors, initial_descriptors: equivocator_descriptors IM
eqv_initial: MachineDescriptor (IM i)
tr: list
  (composite_transition_item (equivocator_IM IM))
eqv_final:= final_descriptors i: MachineDescriptor (IM i)
∀ trX : list (composite_transition_item IM),
  equivocators_trace_project final_descriptors tr =
  Some (trX, initial_descriptors)
  → ∀ trXi : list transition_item,
      equivocator_vlsm_trace_project (IM i)
        (finite_trace_projection_list
           (equivocator_IM IM) i tr) eqv_final =
      Some (trXi, eqv_initial)
      → initial_descriptors i = eqv_initial
        ∧ finite_trace_projection_list IM i trX = trXi
  generalize dependent final_descriptors.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
i: index
initial_descriptors: equivocator_descriptors IM
eqv_initial: MachineDescriptor (IM i)
tr: list
  (composite_transition_item (equivocator_IM IM))
∀ (final_descriptors : equivocator_descriptors IM) 
  (eqv_final := final_descriptors i) (trX : list
                                              (composite_transition_item
                                                 IM)),
  equivocators_trace_project final_descriptors tr =
  Some (trX, initial_descriptors)
  → ∀ trXi : list transition_item,
      equivocator_vlsm_trace_project (IM i)
        (finite_trace_projection_list
           (equivocator_IM IM) i tr) eqv_final =
      Some (trXi, eqv_initial)
      → initial_descriptors i = eqv_initial
        ∧ finite_trace_projection_list IM i trX = trXi
  induction tr using rev_ind; intros.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
i: index
initial_descriptors: equivocator_descriptors IM
eqv_initial: MachineDescriptor (IM i)
final_descriptors: equivocator_descriptors IM
eqv_final:= final_descriptors i: MachineDescriptor (IM i)
trX: list (composite_transition_item IM)
Hproject_tr: equivocators_trace_project
  final_descriptors [] =
Some (trX, initial_descriptors)
trXi: list transition_item
Hproject_tri: equivocator_vlsm_trace_project 
  (IM i)
  (finite_trace_projection_list
     (equivocator_IM IM) i [])
  eqv_final = Some (trXi, eqv_initial)
initial_descriptors i = eqv_initial
∧ finite_trace_projection_list IM i trX = trXi
message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
i: index
initial_descriptors: equivocator_descriptors IM
eqv_initial: MachineDescriptor (IM i)
x: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
IHtr: ∀ (final_descriptors : 
   equivocator_descriptors IM) 
  (eqv_final := final_descriptors i) 
  (trX : list (composite_transition_item IM)),
  equivocators_trace_project final_descriptors
    tr = Some (trX, initial_descriptors)
  → ∀ trXi : list transition_item,
      equivocator_vlsm_trace_project 
        (IM i)
        (finite_trace_projection_list
           (equivocator_IM IM) i tr) eqv_final =
      Some (trXi, eqv_initial)
      → initial_descriptors i = eqv_initial
        ∧ finite_trace_projection_list IM i trX =
          trXi
final_descriptors: equivocator_descriptors IM
eqv_final:= final_descriptors i: MachineDescriptor (IM i)
trX: list (composite_transition_item IM)
Hproject_tr: equivocators_trace_project
  final_descriptors (tr ++ [x]) =
Some (trX, initial_descriptors)
trXi: list transition_item
Hproject_tri: equivocator_vlsm_trace_project 
  (IM i)
  (finite_trace_projection_list
     (equivocator_IM IM) i 
     (tr ++ [x])) eqv_final =
Some (trXi, eqv_initial)
initial_descriptors i = eqv_initial
∧ finite_trace_projection_list IM i trX = trXi
  -message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
i: index
initial_descriptors: equivocator_descriptors IM
eqv_initial: MachineDescriptor (IM i)
final_descriptors: equivocator_descriptors IM
eqv_final:= final_descriptors i: MachineDescriptor (IM i)
trX: list (composite_transition_item IM)
Hproject_tr: equivocators_trace_project
  final_descriptors [] =
Some (trX, initial_descriptors)
trXi: list transition_item
Hproject_tri: equivocator_vlsm_trace_project 
  (IM i)
  (finite_trace_projection_list
     (equivocator_IM IM) i [])
  eqv_final = Some (trXi, eqv_initial)
initial_descriptors i = eqv_initial
∧ finite_trace_projection_list IM i trX = trXi by inversion Hproject_tr; inversion Hproject_tri; subst.
  -message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
i: index
initial_descriptors: equivocator_descriptors IM
eqv_initial: MachineDescriptor (IM i)
x: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
IHtr: ∀ (final_descriptors : 
   equivocator_descriptors IM) 
  (eqv_final := final_descriptors i) 
  (trX : list (composite_transition_item IM)),
  equivocators_trace_project final_descriptors
    tr = Some (trX, initial_descriptors)
  → ∀ trXi : list transition_item,
      equivocator_vlsm_trace_project 
        (IM i)
        (finite_trace_projection_list
           (equivocator_IM IM) i tr) eqv_final =
      Some (trXi, eqv_initial)
      → initial_descriptors i = eqv_initial
        ∧ finite_trace_projection_list IM i trX =
          trXi
final_descriptors: equivocator_descriptors IM
eqv_final:= final_descriptors i: MachineDescriptor (IM i)
trX: list (composite_transition_item IM)
Hproject_tr: equivocators_trace_project
  final_descriptors (tr ++ [x]) =
Some (trX, initial_descriptors)
trXi: list transition_item
Hproject_tri: equivocator_vlsm_trace_project 
  (IM i)
  (finite_trace_projection_list
     (equivocator_IM IM) i 
     (tr ++ [x])) eqv_final =
Some (trXi, eqv_initial)
initial_descriptors i = eqv_initial
∧ finite_trace_projection_list IM i trX = trXi unfold equivocators_trace_project in Hproject_tr.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
i: index
initial_descriptors: equivocator_descriptors IM
eqv_initial: MachineDescriptor (IM i)
x: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
IHtr: ∀ (final_descriptors : 
   equivocator_descriptors IM) 
  (eqv_final := final_descriptors i) 
  (trX : list (composite_transition_item IM)),
  equivocators_trace_project final_descriptors
    tr = Some (trX, initial_descriptors)
  → ∀ trXi : list transition_item,
      equivocator_vlsm_trace_project 
        (IM i)
        (finite_trace_projection_list
           (equivocator_IM IM) i tr) eqv_final =
      Some (trXi, eqv_initial)
      → initial_descriptors i = eqv_initial
        ∧ finite_trace_projection_list IM i trX =
          trXi
final_descriptors: equivocator_descriptors IM
eqv_final:= final_descriptors i: MachineDescriptor (IM i)
trX: list (composite_transition_item IM)
Hproject_tr: foldr equivocators_trace_project_folder
  (Some ([], final_descriptors))
  (tr ++ [x]) =
Some (trX, initial_descriptors)
trXi: list transition_item
Hproject_tri: equivocator_vlsm_trace_project 
  (IM i)
  (finite_trace_projection_list
     (equivocator_IM IM) i 
     (tr ++ [x])) eqv_final =
Some (trXi, eqv_initial)
initial_descriptors i = eqv_initial
∧ finite_trace_projection_list IM i trX = trXi
    rewrite fold_right_app in Hproject_tr.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
i: index
initial_descriptors: equivocator_descriptors IM
eqv_initial: MachineDescriptor (IM i)
x: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
IHtr: ∀ (final_descriptors : 
   equivocator_descriptors IM) 
  (eqv_final := final_descriptors i) 
  (trX : list (composite_transition_item IM)),
  equivocators_trace_project final_descriptors
    tr = Some (trX, initial_descriptors)
  → ∀ trXi : list transition_item,
      equivocator_vlsm_trace_project 
        (IM i)
        (finite_trace_projection_list
           (equivocator_IM IM) i tr) eqv_final =
      Some (trXi, eqv_initial)
      → initial_descriptors i = eqv_initial
        ∧ finite_trace_projection_list IM i trX =
          trXi
final_descriptors: equivocator_descriptors IM
eqv_final:= final_descriptors i: MachineDescriptor (IM i)
trX: list (composite_transition_item IM)
Hproject_tr: foldr equivocators_trace_project_folder
  (foldr
     equivocators_trace_project_folder
     (Some ([], final_descriptors)) [x])
  tr = Some (trX, initial_descriptors)
trXi: list transition_item
Hproject_tri: equivocator_vlsm_trace_project 
  (IM i)
  (finite_trace_projection_list
     (equivocator_IM IM) i 
     (tr ++ [x])) eqv_final =
Some (trXi, eqv_initial)
initial_descriptors i = eqv_initial
∧ finite_trace_projection_list IM i trX = trXi
    match type of Hproject_tr with
    | fold_right _ ?i _ = _ => destruct i as [(projectx, final_descriptors') |] eqn: Hproject_x
    end
    ; [| by rewrite equivocators_trace_project_fold_None in Hproject_tr; inversion Hproject_tr].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
i: index
initial_descriptors: equivocator_descriptors IM
eqv_initial: MachineDescriptor (IM i)
x: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
IHtr: ∀ (final_descriptors : 
   equivocator_descriptors IM) 
  (eqv_final := final_descriptors i) 
  (trX : list (composite_transition_item IM)),
  equivocators_trace_project final_descriptors
    tr = Some (trX, initial_descriptors)
  → ∀ trXi : list transition_item,
      equivocator_vlsm_trace_project 
        (IM i)
        (finite_trace_projection_list
           (equivocator_IM IM) i tr) eqv_final =
      Some (trXi, eqv_initial)
      → initial_descriptors i = eqv_initial
        ∧ finite_trace_projection_list IM i trX =
          trXi
final_descriptors: equivocator_descriptors IM
eqv_final:= final_descriptors i: MachineDescriptor (IM i)
trX, projectx: list (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hproject_x: foldr equivocators_trace_project_folder
  (Some ([], final_descriptors)) [x] =
Some (projectx, final_descriptors')
Hproject_tr: foldr equivocators_trace_project_folder
  (Some (projectx, final_descriptors'))
  tr = Some (trX, initial_descriptors)
trXi: list transition_item
Hproject_tri: equivocator_vlsm_trace_project 
  (IM i)
  (finite_trace_projection_list
     (equivocator_IM IM) i 
     (tr ++ [x])) eqv_final =
Some (trXi, eqv_initial)
initial_descriptors i = eqv_initial
∧ finite_trace_projection_list IM i trX = trXi
    apply equivocators_trace_project_folder_additive_iff in Hproject_tr.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
i: index
initial_descriptors: equivocator_descriptors IM
eqv_initial: MachineDescriptor (IM i)
x: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
IHtr: ∀ (final_descriptors : 
   equivocator_descriptors IM) 
  (eqv_final := final_descriptors i) 
  (trX : list (composite_transition_item IM)),
  equivocators_trace_project final_descriptors
    tr = Some (trX, initial_descriptors)
  → ∀ trXi : list transition_item,
      equivocator_vlsm_trace_project 
        (IM i)
        (finite_trace_projection_list
           (equivocator_IM IM) i tr) eqv_final =
      Some (trXi, eqv_initial)
      → initial_descriptors i = eqv_initial
        ∧ finite_trace_projection_list IM i trX =
          trXi
final_descriptors: equivocator_descriptors IM
eqv_final:= final_descriptors i: MachineDescriptor (IM i)
trX, projectx: list (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hproject_x: foldr equivocators_trace_project_folder
  (Some ([], final_descriptors)) [x] =
Some (projectx, final_descriptors')
Hproject_tr: ∃ trX0 : list
           (composite_transition_item IM),
  foldr
    equivocators_trace_project_folder
    (Some ([], final_descriptors')) tr =
  Some (trX0, initial_descriptors)
  ∧ trX = trX0 ++ projectx
trXi: list transition_item
Hproject_tri: equivocator_vlsm_trace_project 
  (IM i)
  (finite_trace_projection_list
     (equivocator_IM IM) i 
     (tr ++ [x])) eqv_final =
Some (trXi, eqv_initial)
initial_descriptors i = eqv_initial
∧ finite_trace_projection_list IM i trX = trXi
    destruct Hproject_tr as [trX0 [HtrX0 HtrX]].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
i: index
initial_descriptors: equivocator_descriptors IM
eqv_initial: MachineDescriptor (IM i)
x: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
IHtr: ∀ (final_descriptors : 
   equivocator_descriptors IM) 
  (eqv_final := final_descriptors i) 
  (trX : list (composite_transition_item IM)),
  equivocators_trace_project final_descriptors
    tr = Some (trX, initial_descriptors)
  → ∀ trXi : list transition_item,
      equivocator_vlsm_trace_project 
        (IM i)
        (finite_trace_projection_list
           (equivocator_IM IM) i tr) eqv_final =
      Some (trXi, eqv_initial)
      → initial_descriptors i = eqv_initial
        ∧ finite_trace_projection_list IM i trX =
          trXi
final_descriptors: equivocator_descriptors IM
eqv_final:= final_descriptors i: MachineDescriptor (IM i)
trX, projectx: list (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hproject_x: foldr equivocators_trace_project_folder
  (Some ([], final_descriptors)) [x] =
Some (projectx, final_descriptors')
trX0: list (composite_transition_item IM)
HtrX0: foldr equivocators_trace_project_folder
  (Some ([], final_descriptors')) tr =
Some (trX0, initial_descriptors)
HtrX: trX = trX0 ++ projectx
trXi: list transition_item
Hproject_tri: equivocator_vlsm_trace_project 
  (IM i)
  (finite_trace_projection_list
     (equivocator_IM IM) i 
     (tr ++ [x])) eqv_final =
Some (trXi, eqv_initial)
initial_descriptors i = eqv_initial
∧ finite_trace_projection_list IM i trX = trXi
    specialize (IHtr _ _ HtrX0).message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
i: index
initial_descriptors: equivocator_descriptors IM
eqv_initial: MachineDescriptor (IM i)
x: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
final_descriptors': equivocator_descriptors IM
trX0: list (composite_transition_item IM)
IHtr: ∀ trXi : list transition_item,
  equivocator_vlsm_trace_project 
    (IM i)
    (finite_trace_projection_list
       (equivocator_IM IM) i tr)
    (final_descriptors' i) =
  Some (trXi, eqv_initial)
  → initial_descriptors i = eqv_initial
    ∧ finite_trace_projection_list IM i trX0 =
      trXi
final_descriptors: equivocator_descriptors IM
eqv_final:= final_descriptors i: MachineDescriptor (IM i)
trX, projectx: list (composite_transition_item IM)
Hproject_x: foldr equivocators_trace_project_folder
  (Some ([], final_descriptors)) [x] =
Some (projectx, final_descriptors')
HtrX0: foldr equivocators_trace_project_folder
  (Some ([], final_descriptors')) tr =
Some (trX0, initial_descriptors)
HtrX: trX = trX0 ++ projectx
trXi: list transition_item
Hproject_tri: equivocator_vlsm_trace_project 
  (IM i)
  (finite_trace_projection_list
     (equivocator_IM IM) i 
     (tr ++ [x])) eqv_final =
Some (trXi, eqv_initial)
initial_descriptors i = eqv_initial
∧ finite_trace_projection_list IM i trX = trXi
    unfold finite_trace_projection_list in Hproject_tri.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
i: index
initial_descriptors: equivocator_descriptors IM
eqv_initial: MachineDescriptor (IM i)
x: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
final_descriptors': equivocator_descriptors IM
trX0: list (composite_transition_item IM)
IHtr: ∀ trXi : list transition_item,
  equivocator_vlsm_trace_project 
    (IM i)
    (finite_trace_projection_list
       (equivocator_IM IM) i tr)
    (final_descriptors' i) =
  Some (trXi, eqv_initial)
  → initial_descriptors i = eqv_initial
    ∧ finite_trace_projection_list IM i trX0 =
      trXi
final_descriptors: equivocator_descriptors IM
eqv_final:= final_descriptors i: MachineDescriptor (IM i)
trX, projectx: list (composite_transition_item IM)
Hproject_x: foldr equivocators_trace_project_folder
  (Some ([], final_descriptors)) [x] =
Some (projectx, final_descriptors')
HtrX0: foldr equivocators_trace_project_folder
  (Some ([], final_descriptors')) tr =
Some (trX0, initial_descriptors)
HtrX: trX = trX0 ++ projectx
trXi: list transition_item
Hproject_tri: equivocator_vlsm_trace_project 
  (IM i)
  (pre_VLSM_projection_finite_trace_project
     (composite_type
        (equivocator_IM IM))
     (equivocator_IM IM i)
     (composite_project_label
        (equivocator_IM IM) i)
     (λ s : state
              (composite_type
                 (equivocator_IM IM)),
        s i) (tr ++ [x])) eqv_final =
Some (trXi, eqv_initial)
initial_descriptors i = eqv_initial
∧ finite_trace_projection_list IM i trX = trXi
    rewrite @pre_VLSM_projection_finite_trace_project_app in Hproject_tri.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
i: index
initial_descriptors: equivocator_descriptors IM
eqv_initial: MachineDescriptor (IM i)
x: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
final_descriptors': equivocator_descriptors IM
trX0: list (composite_transition_item IM)
IHtr: ∀ trXi : list transition_item,
  equivocator_vlsm_trace_project 
    (IM i)
    (finite_trace_projection_list
       (equivocator_IM IM) i tr)
    (final_descriptors' i) =
  Some (trXi, eqv_initial)
  → initial_descriptors i = eqv_initial
    ∧ finite_trace_projection_list IM i trX0 =
      trXi
final_descriptors: equivocator_descriptors IM
eqv_final:= final_descriptors i: MachineDescriptor (IM i)
trX, projectx: list (composite_transition_item IM)
Hproject_x: foldr equivocators_trace_project_folder
  (Some ([], final_descriptors)) [x] =
Some (projectx, final_descriptors')
HtrX0: foldr equivocators_trace_project_folder
  (Some ([], final_descriptors')) tr =
Some (trX0, initial_descriptors)
HtrX: trX = trX0 ++ projectx
trXi: list transition_item
Hproject_tri: equivocator_vlsm_trace_project 
  (IM i)
  (pre_VLSM_projection_finite_trace_project
     (composite_type
        (equivocator_IM IM))
     (equivocator_IM IM i)
     (composite_project_label
        (equivocator_IM IM) i)
     (λ s : state
              (composite_type
                 (equivocator_IM IM)),
        s i) tr ++
   pre_VLSM_projection_finite_trace_project
     (composite_type
        (equivocator_IM IM))
     (equivocator_IM IM i)
     (composite_project_label
        (equivocator_IM IM) i)
     (λ s : state
              (composite_type
                 (equivocator_IM IM)),
        s i) [x]) eqv_final =
Some (trXi, eqv_initial)
initial_descriptors i = eqv_initial
∧ finite_trace_projection_list IM i trX = trXi
    apply equivocator_vlsm_trace_project_app in Hproject_tri.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
i: index
initial_descriptors: equivocator_descriptors IM
eqv_initial: MachineDescriptor (IM i)
x: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
final_descriptors': equivocator_descriptors IM
trX0: list (composite_transition_item IM)
IHtr: ∀ trXi : list transition_item,
  equivocator_vlsm_trace_project 
    (IM i)
    (finite_trace_projection_list
       (equivocator_IM IM) i tr)
    (final_descriptors' i) =
  Some (trXi, eqv_initial)
  → initial_descriptors i = eqv_initial
    ∧ finite_trace_projection_list IM i trX0 =
      trXi
final_descriptors: equivocator_descriptors IM
eqv_final:= final_descriptors i: MachineDescriptor (IM i)
trX, projectx: list (composite_transition_item IM)
Hproject_x: foldr equivocators_trace_project_folder
  (Some ([], final_descriptors)) [x] =
Some (projectx, final_descriptors')
HtrX0: foldr equivocators_trace_project_folder
  (Some ([], final_descriptors')) tr =
Some (trX0, initial_descriptors)
HtrX: trX = trX0 ++ projectx
trXi: list transition_item
Hproject_tri: ∃ (dmiddle : MachineDescriptor (IM i)) 
  (prefix
   suffix : list transition_item) 
  (_ : equivocator_vlsm_trace_project
         (IM i)
         (pre_VLSM_projection_finite_trace_project
            (composite_type
               (equivocator_IM IM))
            (equivocator_IM IM i)
            (composite_project_label
               (equivocator_IM IM) i)
            (λ s : state
                     (composite_type
                      (equivocator_IM
                      IM)), 
               s i) tr) dmiddle =
       Some (prefix, eqv_initial)) 
  (_ : equivocator_vlsm_trace_project
         (IM i)
         (pre_VLSM_projection_finite_trace_project
            (composite_type
               (equivocator_IM IM))
            (equivocator_IM IM i)
            (composite_project_label
               (equivocator_IM IM) i)
            (λ s : state
                     (composite_type
                      (equivocator_IM
                      IM)), 
               s i) [x]) eqv_final =
       Some (suffix, dmiddle)),
  trXi = prefix ++ suffix
initial_descriptors i = eqv_initial
∧ finite_trace_projection_list IM i trX = trXi
    destruct Hproject_tri as [eqv_final' [trXi' [project_xi [HtrXi' [Hproject_xi HeqtrXi]]]]].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
i: index
initial_descriptors: equivocator_descriptors IM
eqv_initial: MachineDescriptor (IM i)
x: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
final_descriptors': equivocator_descriptors IM
trX0: list (composite_transition_item IM)
IHtr: ∀ trXi : list transition_item,
  equivocator_vlsm_trace_project 
    (IM i)
    (finite_trace_projection_list
       (equivocator_IM IM) i tr)
    (final_descriptors' i) =
  Some (trXi, eqv_initial)
  → initial_descriptors i = eqv_initial
    ∧ finite_trace_projection_list IM i trX0 =
      trXi
final_descriptors: equivocator_descriptors IM
eqv_final:= final_descriptors i: MachineDescriptor (IM i)
trX, projectx: list (composite_transition_item IM)
Hproject_x: foldr equivocators_trace_project_folder
  (Some ([], final_descriptors)) [x] =
Some (projectx, final_descriptors')
HtrX0: foldr equivocators_trace_project_folder
  (Some ([], final_descriptors')) tr =
Some (trX0, initial_descriptors)
HtrX: trX = trX0 ++ projectx
trXi: list transition_item
eqv_final': MachineDescriptor (IM i)
trXi', project_xi: list transition_item
HtrXi': equivocator_vlsm_trace_project 
  (IM i)
  (pre_VLSM_projection_finite_trace_project
     (composite_type (equivocator_IM IM))
     (equivocator_IM IM i)
     (composite_project_label
        (equivocator_IM IM) i)
     (λ s : state
              (composite_type
                 (equivocator_IM IM)), 
        s i) tr) eqv_final' =
Some (trXi', eqv_initial)
Hproject_xi: equivocator_vlsm_trace_project 
  (IM i)
  (pre_VLSM_projection_finite_trace_project
     (composite_type (equivocator_IM IM))
     (equivocator_IM IM i)
     (composite_project_label
        (equivocator_IM IM) i)
     (λ s : state
              (composite_type
                 (equivocator_IM IM)),
        s i) [x]) eqv_final =
Some (project_xi, eqv_final')
HeqtrXi: trXi = trXi' ++ project_xi
initial_descriptors i = eqv_initial
∧ finite_trace_projection_list IM i trX = trXi
    assert (Hfinal'i : final_descriptors' i = eqv_final' /\
      finite_trace_projection_list IM i projectx = project_xi).message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
i: index
initial_descriptors: equivocator_descriptors IM
eqv_initial: MachineDescriptor (IM i)
x: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
final_descriptors': equivocator_descriptors IM
trX0: list (composite_transition_item IM)
IHtr: ∀ trXi : list transition_item,
  equivocator_vlsm_trace_project 
    (IM i)
    (finite_trace_projection_list
       (equivocator_IM IM) i tr)
    (final_descriptors' i) =
  Some (trXi, eqv_initial)
  → initial_descriptors i = eqv_initial
    ∧ finite_trace_projection_list IM i trX0 =
      trXi
final_descriptors: equivocator_descriptors IM
eqv_final:= final_descriptors i: MachineDescriptor (IM i)
trX, projectx: list (composite_transition_item IM)
Hproject_x: foldr equivocators_trace_project_folder
  (Some ([], final_descriptors)) [x] =
Some (projectx, final_descriptors')
HtrX0: foldr equivocators_trace_project_folder
  (Some ([], final_descriptors')) tr =
Some (trX0, initial_descriptors)
HtrX: trX = trX0 ++ projectx
trXi: list transition_item
eqv_final': MachineDescriptor (IM i)
trXi', project_xi: list transition_item
HtrXi': equivocator_vlsm_trace_project 
  (IM i)
  (pre_VLSM_projection_finite_trace_project
     (composite_type (equivocator_IM IM))
     (equivocator_IM IM i)
     (composite_project_label
        (equivocator_IM IM) i)
     (λ s : state
              (composite_type
                 (equivocator_IM IM)), 
        s i) tr) eqv_final' =
Some (trXi', eqv_initial)
Hproject_xi: equivocator_vlsm_trace_project 
  (IM i)
  (pre_VLSM_projection_finite_trace_project
     (composite_type (equivocator_IM IM))
     (equivocator_IM IM i)
     (composite_project_label
        (equivocator_IM IM) i)
     (λ s : state
              (composite_type
                 (equivocator_IM IM)),
        s i) [x]) eqv_final =
Some (project_xi, eqv_final')
HeqtrXi: trXi = trXi' ++ project_xi
final_descriptors' i = eqv_final'
∧ finite_trace_projection_list IM i projectx =
  project_xi
message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
i: index
initial_descriptors: equivocator_descriptors IM
eqv_initial: MachineDescriptor (IM i)
x: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
final_descriptors': equivocator_descriptors IM
trX0: list (composite_transition_item IM)
IHtr: ∀ trXi : list transition_item,
  equivocator_vlsm_trace_project 
    (IM i)
    (finite_trace_projection_list
       (equivocator_IM IM) i tr)
    (final_descriptors' i) =
  Some (trXi, eqv_initial)
  → initial_descriptors i = eqv_initial
    ∧ finite_trace_projection_list IM i trX0 =
      trXi
final_descriptors: equivocator_descriptors IM
eqv_final:= final_descriptors i: MachineDescriptor (IM i)
trX, projectx: list (composite_transition_item IM)
Hproject_x: foldr equivocators_trace_project_folder
  (Some ([], final_descriptors)) [x] =
Some (projectx, final_descriptors')
HtrX0: foldr equivocators_trace_project_folder
  (Some ([], final_descriptors')) tr =
Some (trX0, initial_descriptors)
HtrX: trX = trX0 ++ projectx
trXi: list transition_item
eqv_final': MachineDescriptor (IM i)
trXi', project_xi: list transition_item
HtrXi': equivocator_vlsm_trace_project 
  (IM i)
  (pre_VLSM_projection_finite_trace_project
     (composite_type (equivocator_IM IM))
     (equivocator_IM IM i)
     (composite_project_label
        (equivocator_IM IM) i)
     (λ s : state
              (composite_type
                 (equivocator_IM IM)), 
        s i) tr) eqv_final' =
Some (trXi', eqv_initial)
Hproject_xi: equivocator_vlsm_trace_project 
  (IM i)
  (pre_VLSM_projection_finite_trace_project
     (composite_type (equivocator_IM IM))
     (equivocator_IM IM i)
     (composite_project_label
        (equivocator_IM IM) i)
     (λ s : state
              (composite_type
                 (equivocator_IM IM)),
        s i) [x]) eqv_final =
Some (project_xi, eqv_final')
HeqtrXi: trXi = trXi' ++ project_xi
Hfinal'i: final_descriptors' i = eqv_final'
∧ finite_trace_projection_list IM i
    projectx = project_xi
initial_descriptors i = eqv_initial
∧ finite_trace_projection_list IM i trX = trXi
    {message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
i: index
initial_descriptors: equivocator_descriptors IM
eqv_initial: MachineDescriptor (IM i)
x: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
final_descriptors': equivocator_descriptors IM
trX0: list (composite_transition_item IM)
IHtr: ∀ trXi : list transition_item,
  equivocator_vlsm_trace_project 
    (IM i)
    (finite_trace_projection_list
       (equivocator_IM IM) i tr)
    (final_descriptors' i) =
  Some (trXi, eqv_initial)
  → initial_descriptors i = eqv_initial
    ∧ finite_trace_projection_list IM i trX0 =
      trXi
final_descriptors: equivocator_descriptors IM
eqv_final:= final_descriptors i: MachineDescriptor (IM i)
trX, projectx: list (composite_transition_item IM)
Hproject_x: foldr equivocators_trace_project_folder
  (Some ([], final_descriptors)) [x] =
Some (projectx, final_descriptors')
HtrX0: foldr equivocators_trace_project_folder
  (Some ([], final_descriptors')) tr =
Some (trX0, initial_descriptors)
HtrX: trX = trX0 ++ projectx
trXi: list transition_item
eqv_final': MachineDescriptor (IM i)
trXi', project_xi: list transition_item
HtrXi': equivocator_vlsm_trace_project 
  (IM i)
  (pre_VLSM_projection_finite_trace_project
     (composite_type (equivocator_IM IM))
     (equivocator_IM IM i)
     (composite_project_label
        (equivocator_IM IM) i)
     (λ s : state
              (composite_type
                 (equivocator_IM IM)), 
        s i) tr) eqv_final' =
Some (trXi', eqv_initial)
Hproject_xi: equivocator_vlsm_trace_project 
  (IM i)
  (pre_VLSM_projection_finite_trace_project
     (composite_type (equivocator_IM IM))
     (equivocator_IM IM i)
     (composite_project_label
        (equivocator_IM IM) i)
     (λ s : state
              (composite_type
                 (equivocator_IM IM)),
        s i) [x]) eqv_final =
Some (project_xi, eqv_final')
HeqtrXi: trXi = trXi' ++ project_xi
final_descriptors' i = eqv_final'
∧ finite_trace_projection_list IM i projectx =
  project_xi clear - Hproject_x Hproject_xi.message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
i: index
x: composite_transition_item (equivocator_IM IM)
final_descriptors', final_descriptors: equivocator_descriptors IM
eqv_final:= final_descriptors i: MachineDescriptor (IM i)
projectx: list (composite_transition_item IM)
Hproject_x: foldr equivocators_trace_project_folder
  (Some ([], final_descriptors)) [x] =
Some (projectx, final_descriptors')
eqv_final': MachineDescriptor (IM i)
project_xi: list transition_item
Hproject_xi: equivocator_vlsm_trace_project 
  (IM i)
  (pre_VLSM_projection_finite_trace_project
     (composite_type (equivocator_IM IM))
     (equivocator_IM IM i)
     (composite_project_label
        (equivocator_IM IM) i)
     (λ s : state
              (composite_type
                 (equivocator_IM IM)),
        s i) [x]) eqv_final =
Some (project_xi, eqv_final')
final_descriptors' i = eqv_final'
∧ finite_trace_projection_list IM i projectx =
  project_xi
      simpl in *.message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
i: index
x: composite_transition_item (equivocator_IM IM)
final_descriptors', final_descriptors: equivocator_descriptors IM
eqv_final:= final_descriptors i: MachineDescriptor (IM i)
projectx: list (composite_transition_item IM)
Hproject_x: match
  equivocators_transition_item_project
    final_descriptors x
with
| Some (Some item', odescriptor) =>
    Some ([item'], odescriptor)
| Some (None, odescriptor) =>
    Some ([], odescriptor)
| None => None
end = Some (projectx, final_descriptors')
eqv_final': MachineDescriptor (IM i)
project_xi: list transition_item
Hproject_xi: equivocator_vlsm_trace_project 
  (IM i)
  match
    pre_VLSM_projection_transition_item_project
      (composite_type
         (equivocator_IM IM))
      (equivocator_type (IM i))
      (composite_project_label
         (equivocator_IM IM) i)
      (λ s : composite_state
               (equivocator_IM IM), 
         s i) x
  with
  | Some y => [y]
  | None => []
  end eqv_final =
Some (project_xi, eqv_final')
final_descriptors' i = eqv_final'
∧ finite_trace_projection_list IM i projectx =
  project_xi
      destruct (equivocators_transition_item_project final_descriptors x)
        as [(ox, final') |] eqn: Hpr_item_x
      ; [| by congruence].message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
i: index
x: composite_transition_item (equivocator_IM IM)
final_descriptors', final_descriptors: equivocator_descriptors IM
eqv_final:= final_descriptors i: MachineDescriptor (IM i)
projectx: list (composite_transition_item IM)
ox: option (composite_transition_item IM)
final': equivocator_descriptors IM
Hpr_item_x: equivocators_transition_item_project
  final_descriptors x = 
Some (ox, final')
Hproject_x: match ox with
| Some item' => Some ([item'], final')
| None => Some ([], final')
end = Some (projectx, final_descriptors')
eqv_final': MachineDescriptor (IM i)
project_xi: list transition_item
Hproject_xi: equivocator_vlsm_trace_project 
  (IM i)
  match
    pre_VLSM_projection_transition_item_project
      (composite_type
         (equivocator_IM IM))
      (equivocator_type (IM i))
      (composite_project_label
         (equivocator_IM IM) i)
      (λ s : composite_state
               (equivocator_IM IM), 
         s i) x
  with
  | Some y => [y]
  | None => []
  end eqv_final =
Some (project_xi, eqv_final')
final_descriptors' i = eqv_final'
∧ finite_trace_projection_list IM i projectx =
  project_xi
      unfold equivocators_transition_item_project in Hpr_item_x.message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
i: index
x: composite_transition_item (equivocator_IM IM)
final_descriptors', final_descriptors: equivocator_descriptors IM
eqv_final:= final_descriptors i: MachineDescriptor (IM i)
projectx: list (composite_transition_item IM)
ox: option (composite_transition_item IM)
final': equivocator_descriptors IM
Hpr_item_x: match
  equivocator_vlsm_transition_item_project
    (IM (projT1 (l x)))
    (composite_transition_item_projection
       (equivocator_IM IM) x)
    (final_descriptors (projT1 (l x)))
with
| Some (Some item', deqv') =>
    Some
      (Some
         {|
           l :=
             existT (projT1 (l x))
               (l item');
           input := input x;
           destination :=
             equivocators_state_project
               IM final_descriptors
               (destination x);
           output := output x
         |},
       equivocator_descriptors_update IM
         final_descriptors 
         (projT1 (l x)) deqv')
| Some (None, deqv') =>
    Some
      (None,
       equivocator_descriptors_update IM
         final_descriptors 
         (projT1 (l x)) deqv')
| None => None
end = Some (ox, final')
Hproject_x: match ox with
| Some item' => Some ([item'], final')
| None => Some ([], final')
end = Some (projectx, final_descriptors')
eqv_final': MachineDescriptor (IM i)
project_xi: list transition_item
Hproject_xi: equivocator_vlsm_trace_project 
  (IM i)
  match
    pre_VLSM_projection_transition_item_project
      (composite_type
         (equivocator_IM IM))
      (equivocator_type (IM i))
      (composite_project_label
         (equivocator_IM IM) i)
      (λ s : composite_state
               (equivocator_IM IM), 
         s i) x
  with
  | Some y => [y]
  | None => []
  end eqv_final =
Some (project_xi, eqv_final')
final_descriptors' i = eqv_final'
∧ finite_trace_projection_list IM i projectx =
  project_xi
      destruct (decide (i = projT1 (l x))).message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
i: index
x: composite_transition_item (equivocator_IM IM)
final_descriptors', final_descriptors: equivocator_descriptors IM
eqv_final:= final_descriptors i: MachineDescriptor (IM i)
projectx: list (composite_transition_item IM)
ox: option (composite_transition_item IM)
final': equivocator_descriptors IM
Hpr_item_x: match
  equivocator_vlsm_transition_item_project
    (IM (projT1 (l x)))
    (composite_transition_item_projection
       (equivocator_IM IM) x)
    (final_descriptors (projT1 (l x)))
with
| Some (Some item', deqv') =>
    Some
      (Some
         {|
           l :=
             existT (projT1 (l x))
               (l item');
           input := input x;
           destination :=
             equivocators_state_project
               IM final_descriptors
               (destination x);
           output := output x
         |},
       equivocator_descriptors_update IM
         final_descriptors 
         (projT1 (l x)) deqv')
| Some (None, deqv') =>
    Some
      (None,
       equivocator_descriptors_update IM
         final_descriptors 
         (projT1 (l x)) deqv')
| None => None
end = Some (ox, final')
Hproject_x: match ox with
| Some item' => Some ([item'], final')
| None => Some ([], final')
end = Some (projectx, final_descriptors')
eqv_final': MachineDescriptor (IM i)
project_xi: list transition_item
Hproject_xi: equivocator_vlsm_trace_project 
  (IM i)
  match
    pre_VLSM_projection_transition_item_project
      (composite_type
         (equivocator_IM IM))
      (equivocator_type (IM i))
      (composite_project_label
         (equivocator_IM IM) i)
      (λ s : composite_state
               (equivocator_IM IM), 
         s i) x
  with
  | Some y => [y]
  | None => []
  end eqv_final =
Some (project_xi, eqv_final')
e: i = projT1 (l x)
final_descriptors' i = eqv_final'
∧ finite_trace_projection_list IM i projectx =
  project_xi
message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
i: index
x: composite_transition_item (equivocator_IM IM)
final_descriptors', final_descriptors: equivocator_descriptors IM
eqv_final:= final_descriptors i: MachineDescriptor (IM i)
projectx: list (composite_transition_item IM)
ox: option (composite_transition_item IM)
final': equivocator_descriptors IM
Hpr_item_x: match
  equivocator_vlsm_transition_item_project
    (IM (projT1 (l x)))
    (composite_transition_item_projection
       (equivocator_IM IM) x)
    (final_descriptors (projT1 (l x)))
with
| Some (Some item', deqv') =>
    Some
      (Some
         {|
           l :=
             existT (projT1 (l x))
               (l item');
           input := input x;
           destination :=
             equivocators_state_project
               IM final_descriptors
               (destination x);
           output := output x
         |},
       equivocator_descriptors_update IM
         final_descriptors 
         (projT1 (l x)) deqv')
| Some (None, deqv') =>
    Some
      (None,
       equivocator_descriptors_update IM
         final_descriptors 
         (projT1 (l x)) deqv')
| None => None
end = Some (ox, final')
Hproject_x: match ox with
| Some item' => Some ([item'], final')
| None => Some ([], final')
end = Some (projectx, final_descriptors')
eqv_final': MachineDescriptor (IM i)
project_xi: list transition_item
Hproject_xi: equivocator_vlsm_trace_project 
  (IM i)
  match
    pre_VLSM_projection_transition_item_project
      (composite_type
         (equivocator_IM IM))
      (equivocator_type (IM i))
      (composite_project_label
         (equivocator_IM IM) i)
      (λ s : composite_state
               (equivocator_IM IM), 
         s i) x
  with
  | Some y => [y]
  | None => []
  end eqv_final =
Some (project_xi, eqv_final')
n: i ≠ projT1 (l x)
final_descriptors' i = eqv_final'
∧ finite_trace_projection_list IM i projectx =
  project_xi
      -message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
i: index
x: composite_transition_item (equivocator_IM IM)
final_descriptors', final_descriptors: equivocator_descriptors IM
eqv_final:= final_descriptors i: MachineDescriptor (IM i)
projectx: list (composite_transition_item IM)
ox: option (composite_transition_item IM)
final': equivocator_descriptors IM
Hpr_item_x: match
  equivocator_vlsm_transition_item_project
    (IM (projT1 (l x)))
    (composite_transition_item_projection
       (equivocator_IM IM) x)
    (final_descriptors (projT1 (l x)))
with
| Some (Some item', deqv') =>
    Some
      (Some
         {|
           l :=
             existT (projT1 (l x))
               (l item');
           input := input x;
           destination :=
             equivocators_state_project
               IM final_descriptors
               (destination x);
           output := output x
         |},
       equivocator_descriptors_update IM
         final_descriptors 
         (projT1 (l x)) deqv')
| Some (None, deqv') =>
    Some
      (None,
       equivocator_descriptors_update IM
         final_descriptors 
         (projT1 (l x)) deqv')
| None => None
end = Some (ox, final')
Hproject_x: match ox with
| Some item' => Some ([item'], final')
| None => Some ([], final')
end = Some (projectx, final_descriptors')
eqv_final': MachineDescriptor (IM i)
project_xi: list transition_item
Hproject_xi: equivocator_vlsm_trace_project 
  (IM i)
  match
    pre_VLSM_projection_transition_item_project
      (composite_type
         (equivocator_IM IM))
      (equivocator_type (IM i))
      (composite_project_label
         (equivocator_IM IM) i)
      (λ s : composite_state
               (equivocator_IM IM), 
         s i) x
  with
  | Some y => [y]
  | None => []
  end eqv_final =
Some (project_xi, eqv_final')
e: i = projT1 (l x)
final_descriptors' i = eqv_final'
∧ finite_trace_projection_list IM i projectx =
  project_xi subst i.message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
x: composite_transition_item (equivocator_IM IM)
final_descriptors', final_descriptors: equivocator_descriptors IM
eqv_final:= final_descriptors (projT1 (l x)): MachineDescriptor (IM (projT1 (l x)))
projectx: list (composite_transition_item IM)
ox: option (composite_transition_item IM)
final': equivocator_descriptors IM
Hpr_item_x: match
  equivocator_vlsm_transition_item_project
    (IM (projT1 (l x)))
    (composite_transition_item_projection
       (equivocator_IM IM) x)
    (final_descriptors (projT1 (l x)))
with
| Some (Some item', deqv') =>
    Some
      (Some
         {|
           l :=
             existT (projT1 (l x))
               (l item');
           input := input x;
           destination :=
             equivocators_state_project
               IM final_descriptors
               (destination x);
           output := output x
         |},
       equivocator_descriptors_update IM
         final_descriptors 
         (projT1 (l x)) deqv')
| Some (None, deqv') =>
    Some
      (None,
       equivocator_descriptors_update IM
         final_descriptors 
         (projT1 (l x)) deqv')
| None => None
end = Some (ox, final')
Hproject_x: match ox with
| Some item' => Some ([item'], final')
| None => Some ([], final')
end = Some (projectx, final_descriptors')
project_xi: list transition_item
eqv_final': MachineDescriptor (IM (projT1 (l x)))
Hproject_xi: equivocator_vlsm_trace_project
  (IM (projT1 (l x)))
  match
    pre_VLSM_projection_transition_item_project
      (composite_type
         (equivocator_IM IM))
      (equivocator_type
         (IM (projT1 (l x))))
      (composite_project_label
         (equivocator_IM IM)
         (projT1 (l x)))
      (λ s : composite_state
               (equivocator_IM IM),
         s (projT1 (l x))) x
  with
  | Some y => [y]
  | None => []
  end eqv_final =
Some (project_xi, eqv_final')
final_descriptors' (projT1 (l x)) = eqv_final'
∧ finite_trace_projection_list IM (projT1 (l x))
    projectx = project_xi
        rewrite (composite_transition_item_projection_iff (equivocator_IM IM) x)
         in Hproject_xi.message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
x: composite_transition_item (equivocator_IM IM)
final_descriptors', final_descriptors: equivocator_descriptors IM
eqv_final:= final_descriptors (projT1 (l x)): MachineDescriptor (IM (projT1 (l x)))
projectx: list (composite_transition_item IM)
ox: option (composite_transition_item IM)
final': equivocator_descriptors IM
Hpr_item_x: match
  equivocator_vlsm_transition_item_project
    (IM (projT1 (l x)))
    (composite_transition_item_projection
       (equivocator_IM IM) x)
    (final_descriptors (projT1 (l x)))
with
| Some (Some item', deqv') =>
    Some
      (Some
         {|
           l :=
             existT (projT1 (l x))
               (l item');
           input := input x;
           destination :=
             equivocators_state_project
               IM final_descriptors
               (destination x);
           output := output x
         |},
       equivocator_descriptors_update IM
         final_descriptors 
         (projT1 (l x)) deqv')
| Some (None, deqv') =>
    Some
      (None,
       equivocator_descriptors_update IM
         final_descriptors 
         (projT1 (l x)) deqv')
| None => None
end = Some (ox, final')
Hproject_x: match ox with
| Some item' => Some ([item'], final')
| None => Some ([], final')
end = Some (projectx, final_descriptors')
project_xi: list transition_item
eqv_final': MachineDescriptor (IM (projT1 (l x)))
Hproject_xi: equivocator_vlsm_trace_project
  (IM (projT1 (l x)))
  [composite_transition_item_projection
     (equivocator_IM IM) x] eqv_final =
Some (project_xi, eqv_final')
final_descriptors' (projT1 (l x)) = eqv_final'
∧ finite_trace_projection_list IM (projT1 (l x))
    projectx = project_xi
        simpl in Hproject_xi.message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
x: composite_transition_item (equivocator_IM IM)
final_descriptors', final_descriptors: equivocator_descriptors IM
eqv_final:= final_descriptors (projT1 (l x)): MachineDescriptor (IM (projT1 (l x)))
projectx: list (composite_transition_item IM)
ox: option (composite_transition_item IM)
final': equivocator_descriptors IM
Hpr_item_x: match
  equivocator_vlsm_transition_item_project
    (IM (projT1 (l x)))
    (composite_transition_item_projection
       (equivocator_IM IM) x)
    (final_descriptors (projT1 (l x)))
with
| Some (Some item', deqv') =>
    Some
      (Some
         {|
           l :=
             existT (projT1 (l x))
               (l item');
           input := input x;
           destination :=
             equivocators_state_project
               IM final_descriptors
               (destination x);
           output := output x
         |},
       equivocator_descriptors_update IM
         final_descriptors 
         (projT1 (l x)) deqv')
| Some (None, deqv') =>
    Some
      (None,
       equivocator_descriptors_update IM
         final_descriptors 
         (projT1 (l x)) deqv')
| None => None
end = Some (ox, final')
Hproject_x: match ox with
| Some item' => Some ([item'], final')
| None => Some ([], final')
end = Some (projectx, final_descriptors')
project_xi: list transition_item
eqv_final': MachineDescriptor (IM (projT1 (l x)))
Hproject_xi: match
  equivocator_vlsm_transition_item_project
    (IM (projT1 (l x)))
    (composite_transition_item_projection
       (equivocator_IM IM) x) eqv_final
with
| Some (Some item', odescriptor) =>
    Some ([item'], odescriptor)
| Some (None, odescriptor) =>
    Some ([], odescriptor)
| None => None
end = Some (project_xi, eqv_final')
final_descriptors' (projT1 (l x)) = eqv_final'
∧ finite_trace_projection_list IM (projT1 (l x))
    projectx = project_xi
        subst eqv_final.message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
x: composite_transition_item (equivocator_IM IM)
final_descriptors', final_descriptors: equivocator_descriptors IM
projectx: list (composite_transition_item IM)
ox: option (composite_transition_item IM)
final': equivocator_descriptors IM
Hpr_item_x: match
  equivocator_vlsm_transition_item_project
    (IM (projT1 (l x)))
    (composite_transition_item_projection
       (equivocator_IM IM) x)
    (final_descriptors (projT1 (l x)))
with
| Some (Some item', deqv') =>
    Some
      (Some
         {|
           l :=
             existT (projT1 (l x))
               (l item');
           input := input x;
           destination :=
             equivocators_state_project
               IM final_descriptors
               (destination x);
           output := output x
         |},
       equivocator_descriptors_update IM
         final_descriptors 
         (projT1 (l x)) deqv')
| Some (None, deqv') =>
    Some
      (None,
       equivocator_descriptors_update IM
         final_descriptors 
         (projT1 (l x)) deqv')
| None => None
end = Some (ox, final')
Hproject_x: match ox with
| Some item' => Some ([item'], final')
| None => Some ([], final')
end = Some (projectx, final_descriptors')
project_xi: list transition_item
eqv_final': MachineDescriptor (IM (projT1 (l x)))
Hproject_xi: match
  equivocator_vlsm_transition_item_project
    (IM (projT1 (l x)))
    (composite_transition_item_projection
       (equivocator_IM IM) x)
    (final_descriptors (projT1 (l x)))
with
| Some (Some item', odescriptor) =>
    Some ([item'], odescriptor)
| Some (None, odescriptor) =>
    Some ([], odescriptor)
| None => None
end = Some (project_xi, eqv_final')
final_descriptors' (projT1 (l x)) = eqv_final'
∧ finite_trace_projection_list IM (projT1 (l x))
    projectx = project_xi
        destruct (equivocator_vlsm_transition_item_project _ _ _)
          as [(oitem', descriptor') |] eqn: Heqpr_item_x
        ; [| done].message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
x: composite_transition_item (equivocator_IM IM)
final_descriptors', final_descriptors: equivocator_descriptors IM
projectx: list (composite_transition_item IM)
ox: option (composite_transition_item IM)
final': equivocator_descriptors IM
oitem': option transition_item
descriptor': MachineDescriptor (IM (projT1 (l x)))
Heqpr_item_x: equivocator_vlsm_transition_item_project
  (IM (projT1 (l x)))
  (composite_transition_item_projection
     (equivocator_IM IM) x)
  (final_descriptors (projT1 (l x))) =
Some (oitem', descriptor')
Hpr_item_x: match oitem' with
| Some item' =>
    Some
      (Some
         {|
           l :=
             existT (projT1 (l x))
               (l item');
           input := input x;
           destination :=
             equivocators_state_project
               IM final_descriptors
               (destination x);
           output := output x
         |},
       equivocator_descriptors_update IM
         final_descriptors 
         (projT1 (l x)) descriptor')
| None =>
    Some
      (None,
       equivocator_descriptors_update IM
         final_descriptors 
         (projT1 (l x)) descriptor')
end = Some (ox, final')
Hproject_x: match ox with
| Some item' => Some ([item'], final')
| None => Some ([], final')
end = Some (projectx, final_descriptors')
project_xi: list transition_item
eqv_final': MachineDescriptor (IM (projT1 (l x)))
Hproject_xi: match oitem' with
| Some item' =>
    Some ([item'], descriptor')
| None => Some ([], descriptor')
end = Some (project_xi, eqv_final')
final_descriptors' (projT1 (l x)) = eqv_final'
∧ finite_trace_projection_list IM (projT1 (l x))
    projectx = project_xi
        destruct oitem' as [item' |]
        ; inversion Hproject_xi; subst descriptor' project_xi; clear Hproject_xi
        ; inversion Hpr_item_x; subst; clear Hpr_item_x
        ; inversion Hproject_x; subst; clear Hproject_x
        ; state_update_simpl
        ; [| by split].message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
x: composite_transition_item (equivocator_IM IM)
final_descriptors: equivocator_descriptors IM
item': transition_item
eqv_final': MachineDescriptor (IM (projT1 (l x)))
Heqpr_item_x: equivocator_vlsm_transition_item_project
  (IM (projT1 (l x)))
  (composite_transition_item_projection
     (equivocator_IM IM) x)
  (final_descriptors (projT1 (l x))) =
Some (Some item', eqv_final')
eqv_final' = eqv_final'
∧ finite_trace_projection_list IM (projT1 (l x))
    [{|
       l := existT (projT1 (l x)) (l item');
       input := input x;
       destination :=
         equivocators_state_project IM
           final_descriptors (destination x);
       output := output x
     |}] = [item']
        split; [done |].message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
x: composite_transition_item (equivocator_IM IM)
final_descriptors: equivocator_descriptors IM
item': transition_item
eqv_final': MachineDescriptor (IM (projT1 (l x)))
Heqpr_item_x: equivocator_vlsm_transition_item_project
  (IM (projT1 (l x)))
  (composite_transition_item_projection
     (equivocator_IM IM) x)
  (final_descriptors (projT1 (l x))) =
Some (Some item', eqv_final')
finite_trace_projection_list IM (projT1 (l x))
  [{|
     l := existT (projT1 (l x)) (l item');
     input := input x;
     destination :=
       equivocators_state_project IM final_descriptors
         (destination x);
     output := output x
   |}] = [item']
        simpl.message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
x: composite_transition_item (equivocator_IM IM)
final_descriptors: equivocator_descriptors IM
item': transition_item
eqv_final': MachineDescriptor (IM (projT1 (l x)))
Heqpr_item_x: equivocator_vlsm_transition_item_project
  (IM (projT1 (l x)))
  (composite_transition_item_projection
     (equivocator_IM IM) x)
  (final_descriptors (projT1 (l x))) =
Some (Some item', eqv_final')
match
  pre_VLSM_projection_transition_item_project
    (composite_type IM) (IM (projT1 (l x)))
    (composite_project_label IM (projT1 (l x)))
    (λ s : composite_state IM, s (projT1 (l x)))
    {|
      l := existT (projT1 (l x)) (l item');
      input := input x;
      destination :=
        equivocators_state_project IM
          final_descriptors (destination x);
      output := output x
    |}
with
| Some y => [y]
| None => []
end = [item'] destruct x.message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
l: label (composite_type (equivocator_IM IM))
input: option message
destination: state
  (composite_type (equivocator_IM IM))
output: option message
final_descriptors: equivocator_descriptors IM
item': transition_item
eqv_final': MachineDescriptor
  (IM
     (projT1
        (VLSM.l
           {|
             l := l;
             input := input;
             destination := destination;
             output := output
           |})))
Heqpr_item_x: equivocator_vlsm_transition_item_project
  (IM
     (projT1
        (VLSM.l
           {|
             l := l;
             input := input;
             destination := destination;
             output := output
           |})))
  (composite_transition_item_projection
     (equivocator_IM IM)
     {|
       l := l;
       input := input;
       destination := destination;
       output := output
     |})
  (final_descriptors
     (projT1
        (VLSM.l
           {|
             l := l;
             input := input;
             destination := destination;
             output := output
           |}))) =
Some (Some item', eqv_final')
match
  pre_VLSM_projection_transition_item_project
    (composite_type IM)
    (IM
       (projT1
          (VLSM.l
             {|
               l := l;
               input := input;
               destination := destination;
               output := output
             |})))
    (composite_project_label IM
       (projT1
          (VLSM.l
             {|
               l := l;
               input := input;
               destination := destination;
               output := output
             |})))
    (λ s : composite_state IM,
       s
         (projT1
            (VLSM.l
               {|
                 l := l;
                 input := input;
                 destination := destination;
                 output := output
               |})))
    {|
      l :=
        existT
          (projT1
             (VLSM.l
                {|
                  l := l;
                  input := input;
                  destination := destination;
                  output := output
                |})) (VLSM.l item');
      input :=
        VLSM.input
          {|
            l := l;
            input := input;
            destination := destination;
            output := output
          |};
      destination :=
        equivocators_state_project IM
          final_descriptors
          (VLSM.destination
             {|
               l := l;
               input := input;
               destination := destination;
               output := output
             |});
      output :=
        VLSM.output
          {|
            l := l;
            input := input;
            destination := destination;
            output := output
          |}
    |}
with
| Some y => [y]
| None => []
end = [item'] simpl in *.message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
l: composite_label (equivocator_IM IM)
input: option message
destination: composite_state (equivocator_IM IM)
output: option message
final_descriptors: equivocator_descriptors IM
item': transition_item
eqv_final': MachineDescriptor (IM (projT1 l))
Heqpr_item_x: equivocator_vlsm_transition_item_project
  (IM (projT1 l))
  (composite_transition_item_projection
     (equivocator_IM IM)
     {|
       l := l;
       input := input;
       destination := destination;
       output := output
     |}) (final_descriptors (projT1 l)) =
Some (Some item', eqv_final')
match
  pre_VLSM_projection_transition_item_project
    (composite_type IM) (IM (projT1 l))
    (composite_project_label IM (projT1 l))
    (λ s : composite_state IM, s (projT1 l))
    {|
      l := existT (projT1 l) (VLSM.l item');
      input := input;
      destination :=
        equivocators_state_project IM
          final_descriptors destination;
      output := output
    |}
with
| Some y => [y]
| None => []
end = [item'] destruct l as (i, li).message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
i: index
li: label (equivocator_IM IM i)
input: option message
destination: composite_state (equivocator_IM IM)
output: option message
final_descriptors: equivocator_descriptors IM
item': transition_item
eqv_final': MachineDescriptor
  (IM (projT1 (existT i li)))
Heqpr_item_x: equivocator_vlsm_transition_item_project
  (IM (projT1 (existT i li)))
  (composite_transition_item_projection
     (equivocator_IM IM)
     {|
       l := existT i li;
       input := input;
       destination := destination;
       output := output
     |})
  (final_descriptors
     (projT1 (existT i li))) =
Some (Some item', eqv_final')
match
  pre_VLSM_projection_transition_item_project
    (composite_type IM) (IM (projT1 (existT i li)))
    (composite_project_label IM (projT1 (existT i li)))
    (λ s : composite_state IM,
       s (projT1 (existT i li)))
    {|
      l := existT (projT1 (existT i li)) (l item');
      input := input;
      destination :=
        equivocators_state_project IM
          final_descriptors destination;
      output := output
    |}
with
| Some y => [y]
| None => []
end = [item'] simpl in *.message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
i: index
li: EquivocatorLabel (IM i)
input: option message
destination: composite_state (equivocator_IM IM)
output: option message
final_descriptors: equivocator_descriptors IM
item': transition_item
eqv_final': MachineDescriptor (IM i)
Heqpr_item_x: equivocator_vlsm_transition_item_project
  (IM i)
  (composite_transition_item_projection
     (equivocator_IM IM)
     {|
       l := existT i li;
       input := input;
       destination := destination;
       output := output
     |}) (final_descriptors i) =
Some (Some item', eqv_final')
match
  pre_VLSM_projection_transition_item_project
    (composite_type IM) (IM i)
    (composite_project_label IM i)
    (λ s : composite_state IM, s i)
    {|
      l := existT i (l item');
      input := input;
      destination :=
        equivocators_state_project IM
          final_descriptors destination;
      output := output
    |}
with
| Some y => [y]
| None => []
end = [item']
        unfold pre_VLSM_projection_transition_item_project, composite_project_label.message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
i: index
li: EquivocatorLabel (IM i)
input: option message
destination: composite_state (equivocator_IM IM)
output: option message
final_descriptors: equivocator_descriptors IM
item': transition_item
eqv_final': MachineDescriptor (IM i)
Heqpr_item_x: equivocator_vlsm_transition_item_project
  (IM i)
  (composite_transition_item_projection
     (equivocator_IM IM)
     {|
       l := existT i li;
       input := input;
       destination := destination;
       output := output
     |}) (final_descriptors i) =
Some (Some item', eqv_final')
match
  match
    match
      decide
        (i =
         projT1
           (l
              {|
                l := existT i (l item');
                input := input;
                destination :=
                  equivocators_state_project IM
                    final_descriptors destination;
                output := output
              |}))
    with
    | left e =>
        Some
          (eq_rect_r (λ n : index, label (IM n))
             (projT2
                (l
                   {|
                     l := existT i (l item');
                     input := input;
                     destination :=
                       equivocators_state_project IM
                         final_descriptors destination;
                     output := output
                   |})) e)
    | right _ => None
    end
  with
  | Some lY =>
      Some
        {|
          l := lY;
          input :=
            VLSM.input
              {|
                l := existT i (l item');
                input := input;
                destination :=
                  equivocators_state_project IM
                    final_descriptors destination;
                output := output
              |};
          destination :=
            VLSM.destination
              {|
                l := existT i (l item');
                input := input;
                destination :=
                  equivocators_state_project IM
                    final_descriptors destination;
                output := output
              |} i;
          output :=
            VLSM.output
              {|
                l := existT i (l item');
                input := input;
                destination :=
                  equivocators_state_project IM
                    final_descriptors destination;
                output := output
              |}
        |}
  | None => None
  end
with
| Some y => [y]
| None => []
end = [item'] simpl.message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
i: index
li: EquivocatorLabel (IM i)
input: option message
destination: composite_state (equivocator_IM IM)
output: option message
final_descriptors: equivocator_descriptors IM
item': transition_item
eqv_final': MachineDescriptor (IM i)
Heqpr_item_x: equivocator_vlsm_transition_item_project
  (IM i)
  (composite_transition_item_projection
     (equivocator_IM IM)
     {|
       l := existT i li;
       input := input;
       destination := destination;
       output := output
     |}) (final_descriptors i) =
Some (Some item', eqv_final')
match
  match
    match decide (i = i) with
    | left e =>
        Some
          (eq_rect_r (λ n : index, label (IM n))
             (l item') e)
    | right _ => None
    end
  with
  | Some lY =>
      Some
        {|
          l := lY;
          input := input;
          destination :=
            equivocators_state_project IM
              final_descriptors destination i;
          output := output
        |}
  | None => None
  end
with
| Some y => [y]
| None => []
end = [item']
        destruct (decide (i = i)); [| by congruence].message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
i: index
li: EquivocatorLabel (IM i)
input: option message
destination: composite_state (equivocator_IM IM)
output: option message
final_descriptors: equivocator_descriptors IM
item': transition_item
eqv_final': MachineDescriptor (IM i)
Heqpr_item_x: equivocator_vlsm_transition_item_project
  (IM i)
  (composite_transition_item_projection
     (equivocator_IM IM)
     {|
       l := existT i li;
       input := input;
       destination := destination;
       output := output
     |}) (final_descriptors i) =
Some (Some item', eqv_final')
e: i = i
[{|
   l :=
     eq_rect_r (λ n : index, label (IM n)) (l item') e;
   input := input;
   destination :=
     equivocators_state_project IM final_descriptors
       destination i;
   output := output
 |}] = [item']
        f_equal.message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
i: index
li: EquivocatorLabel (IM i)
input: option message
destination: composite_state (equivocator_IM IM)
output: option message
final_descriptors: equivocator_descriptors IM
item': transition_item
eqv_final': MachineDescriptor (IM i)
Heqpr_item_x: equivocator_vlsm_transition_item_project
  (IM i)
  (composite_transition_item_projection
     (equivocator_IM IM)
     {|
       l := existT i li;
       input := input;
       destination := destination;
       output := output
     |}) (final_descriptors i) =
Some (Some item', eqv_final')
e: i = i
{|
  l :=
    eq_rect_r (λ n : index, label (IM n)) (l item') e;
  input := input;
  destination :=
    equivocators_state_project IM final_descriptors
      destination i;
  output := output
|} = item'
        replace e with (@eq_refl _ i) by (apply Eqdep_dec.UIP_dec; done).message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
i: index
li: EquivocatorLabel (IM i)
input: option message
destination: composite_state (equivocator_IM IM)
output: option message
final_descriptors: equivocator_descriptors IM
item': transition_item
eqv_final': MachineDescriptor (IM i)
Heqpr_item_x: equivocator_vlsm_transition_item_project
  (IM i)
  (composite_transition_item_projection
     (equivocator_IM IM)
     {|
       l := existT i li;
       input := input;
       destination := destination;
       output := output
     |}) (final_descriptors i) =
Some (Some item', eqv_final')
e: i = i
{|
  l :=
    eq_rect_r (λ n : index, label (IM n)) (l item')
      eq_refl;
  input := input;
  destination :=
    equivocators_state_project IM final_descriptors
      destination i;
  output := output
|} = item' clear e.message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
i: index
li: EquivocatorLabel (IM i)
input: option message
destination: composite_state (equivocator_IM IM)
output: option message
final_descriptors: equivocator_descriptors IM
item': transition_item
eqv_final': MachineDescriptor (IM i)
Heqpr_item_x: equivocator_vlsm_transition_item_project
  (IM i)
  (composite_transition_item_projection
     (equivocator_IM IM)
     {|
       l := existT i li;
       input := input;
       destination := destination;
       output := output
     |}) (final_descriptors i) =
Some (Some item', eqv_final')
{|
  l :=
    eq_rect_r (λ n : index, label (IM n)) (l item')
      eq_refl;
  input := input;
  destination :=
    equivocators_state_project IM final_descriptors
      destination i;
  output := output
|} = item'
        destruct item'.message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
i: index
li: EquivocatorLabel (IM i)
input: option message
destination: composite_state (equivocator_IM IM)
output: option message
final_descriptors: equivocator_descriptors IM
l: label (IM i)
input0: option message
destination0: state (IM i)
output0: option message
eqv_final': MachineDescriptor (IM i)
Heqpr_item_x: equivocator_vlsm_transition_item_project
  (IM i)
  (composite_transition_item_projection
     (equivocator_IM IM)
     {|
       l := existT i li;
       input := input;
       destination := destination;
       output := output
     |}) (final_descriptors i) =
Some
  (Some
     {|
       l := l;
       input := input0;
       destination := destination0;
       output := output0
     |}, eqv_final')
{|
  l :=
    eq_rect_r (λ n : index, label (IM n))
      (VLSM.l
         {|
           l := l;
           input := input0;
           destination := destination0;
           output := output0
         |}) eq_refl;
  input := input;
  destination :=
    equivocators_state_project IM final_descriptors
      destination i;
  output := output
|} =
{|
  l := l;
  input := input0;
  destination := destination0;
  output := output0
|}
        apply equivocator_transition_item_project_inv_characterization in Heqpr_item_x.message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
i: index
li: EquivocatorLabel (IM i)
input: option message
destination: composite_state (equivocator_IM IM)
output: option message
final_descriptors: equivocator_descriptors IM
l: label (IM i)
input0: option message
destination0: state (IM i)
output0: option message
eqv_final': MachineDescriptor (IM i)
Heqpr_item_x: (∃ Hex : existing_equivocator_label
           (IM i)
           (VLSM.l
              (composite_transition_item_projection
                 (equivocator_IM IM)
                 {|
                   l := existT i li;
                   input := input;
                   destination :=
                     destination;
                   output := output
                 |})),
   VLSM.l
     {|
       l := l;
       input := input0;
       destination := destination0;
       output := output0
     |} =
   existing_equivocator_label_extract
     (IM i)
     (VLSM.l
        (composite_transition_item_projection
           (equivocator_IM IM)
           {|
             l := existT i li;
             input := input;
             destination := destination;
             output := output
           |})) Hex)
∧ VLSM.input
    (composite_transition_item_projection
       (equivocator_IM IM)
       {|
         l := existT i li;
         input := input;
         destination := destination;
         output := output
       |}) =
  VLSM.input
    {|
      l := l;
      input := input0;
      destination := destination0;
      output := output0
    |}
  ∧ VLSM.output
      (composite_transition_item_projection
         (equivocator_IM IM)
         {|
           l := existT i li;
           input := input;
           destination := destination;
           output := output
         |}) =
    VLSM.output
      {|
        l := l;
        input := input0;
        destination := destination0;
        output := output0
      |}
    ∧ equivocator_state_descriptor_project
        (VLSM.destination
           (composite_transition_item_projection
              (equivocator_IM IM)
              {|
                l := existT i li;
                input := input;
                destination :=
                  destination;
                output := output
              |}))
        (final_descriptors i) =
      VLSM.destination
        {|
          l := l;
          input := input0;
          destination := destination0;
          output := output0
        |}
      ∧ eqv_final' =
        equivocator_label_descriptor
          (VLSM.l
             (composite_transition_item_projection
                (equivocator_IM IM)
                {|
                  l := existT i li;
                  input := input;
                  destination :=
                    destination;
                  output := output
                |}))
{|
  l :=
    eq_rect_r (λ n : index, label (IM n))
      (VLSM.l
         {|
           l := l;
           input := input0;
           destination := destination0;
           output := output0
         |}) eq_refl;
  input := input;
  destination :=
    equivocators_state_project IM final_descriptors
      destination i;
  output := output
|} =
{|
  l := l;
  input := input0;
  destination := destination0;
  output := output0
|}
        simpl in *.message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
i: index
li: EquivocatorLabel (IM i)
input: option message
destination: composite_state (equivocator_IM IM)
output: option message
final_descriptors: equivocator_descriptors IM
l: label (IM i)
input0: option message
destination0: state (IM i)
output0: option message
eqv_final': MachineDescriptor (IM i)
Heqpr_item_x: (∃ Hex : existing_equivocator_label
           (IM i)
           (eq_rect_r
              (λ n : index,
                 EquivocatorLabel
                   (IM n)) li eq_refl),
   l =
   existing_equivocator_label_extract
     (IM i)
     (eq_rect_r
        (λ n : index,
           EquivocatorLabel (IM n)) li
        eq_refl) Hex)
∧ input = input0
  ∧ output = output0
    ∧ equivocator_state_descriptor_project
        (destination i)
        (final_descriptors i) =
      destination0
      ∧ eqv_final' =
        equivocator_label_descriptor
          (eq_rect_r
             (λ n : index,
                EquivocatorLabel (IM n))
             li eq_refl)
{|
  l := eq_rect_r (λ n : index, label (IM n)) l eq_refl;
  input := input;
  destination :=
    equivocators_state_project IM final_descriptors
      destination i;
  output := output
|} =
{|
  l := l;
  input := input0;
  destination := destination0;
  output := output0
|}
        by destruct Heqpr_item_x as [Hl [-> [-> [<- _]]]].
      -message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
i: index
x: composite_transition_item (equivocator_IM IM)
final_descriptors', final_descriptors: equivocator_descriptors IM
eqv_final:= final_descriptors i: MachineDescriptor (IM i)
projectx: list (composite_transition_item IM)
ox: option (composite_transition_item IM)
final': equivocator_descriptors IM
Hpr_item_x: match
  equivocator_vlsm_transition_item_project
    (IM (projT1 (l x)))
    (composite_transition_item_projection
       (equivocator_IM IM) x)
    (final_descriptors (projT1 (l x)))
with
| Some (Some item', deqv') =>
    Some
      (Some
         {|
           l :=
             existT (projT1 (l x))
               (l item');
           input := input x;
           destination :=
             equivocators_state_project
               IM final_descriptors
               (destination x);
           output := output x
         |},
       equivocator_descriptors_update IM
         final_descriptors 
         (projT1 (l x)) deqv')
| Some (None, deqv') =>
    Some
      (None,
       equivocator_descriptors_update IM
         final_descriptors 
         (projT1 (l x)) deqv')
| None => None
end = Some (ox, final')
Hproject_x: match ox with
| Some item' => Some ([item'], final')
| None => Some ([], final')
end = Some (projectx, final_descriptors')
eqv_final': MachineDescriptor (IM i)
project_xi: list transition_item
Hproject_xi: equivocator_vlsm_trace_project 
  (IM i)
  match
    pre_VLSM_projection_transition_item_project
      (composite_type
         (equivocator_IM IM))
      (equivocator_type (IM i))
      (composite_project_label
         (equivocator_IM IM) i)
      (λ s : composite_state
               (equivocator_IM IM), 
         s i) x
  with
  | Some y => [y]
  | None => []
  end eqv_final =
Some (project_xi, eqv_final')
n: i ≠ projT1 (l x)
final_descriptors' i = eqv_final'
∧ finite_trace_projection_list IM i projectx =
  project_xi rewrite (composite_transition_item_projection_neq (equivocator_IM IM) i x)
         in Hproject_xi by congruence.message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
i: index
x: composite_transition_item (equivocator_IM IM)
final_descriptors', final_descriptors: equivocator_descriptors IM
eqv_final:= final_descriptors i: MachineDescriptor (IM i)
projectx: list (composite_transition_item IM)
ox: option (composite_transition_item IM)
final': equivocator_descriptors IM
Hpr_item_x: match
  equivocator_vlsm_transition_item_project
    (IM (projT1 (l x)))
    (composite_transition_item_projection
       (equivocator_IM IM) x)
    (final_descriptors (projT1 (l x)))
with
| Some (Some item', deqv') =>
    Some
      (Some
         {|
           l :=
             existT (projT1 (l x))
               (l item');
           input := input x;
           destination :=
             equivocators_state_project
               IM final_descriptors
               (destination x);
           output := output x
         |},
       equivocator_descriptors_update IM
         final_descriptors 
         (projT1 (l x)) deqv')
| Some (None, deqv') =>
    Some
      (None,
       equivocator_descriptors_update IM
         final_descriptors 
         (projT1 (l x)) deqv')
| None => None
end = Some (ox, final')
Hproject_x: match ox with
| Some item' => Some ([item'], final')
| None => Some ([], final')
end = Some (projectx, final_descriptors')
eqv_final': MachineDescriptor (IM i)
project_xi: list transition_item
Hproject_xi: equivocator_vlsm_trace_project 
  (IM i) [] eqv_final =
Some (project_xi, eqv_final')
n: i ≠ projT1 (l x)
final_descriptors' i = eqv_final'
∧ finite_trace_projection_list IM i projectx =
  project_xi
        simpl in Hproject_xi.message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
i: index
x: composite_transition_item (equivocator_IM IM)
final_descriptors', final_descriptors: equivocator_descriptors IM
eqv_final:= final_descriptors i: MachineDescriptor (IM i)
projectx: list (composite_transition_item IM)
ox: option (composite_transition_item IM)
final': equivocator_descriptors IM
Hpr_item_x: match
  equivocator_vlsm_transition_item_project
    (IM (projT1 (l x)))
    (composite_transition_item_projection
       (equivocator_IM IM) x)
    (final_descriptors (projT1 (l x)))
with
| Some (Some item', deqv') =>
    Some
      (Some
         {|
           l :=
             existT (projT1 (l x))
               (l item');
           input := input x;
           destination :=
             equivocators_state_project
               IM final_descriptors
               (destination x);
           output := output x
         |},
       equivocator_descriptors_update IM
         final_descriptors 
         (projT1 (l x)) deqv')
| Some (None, deqv') =>
    Some
      (None,
       equivocator_descriptors_update IM
         final_descriptors 
         (projT1 (l x)) deqv')
| None => None
end = Some (ox, final')
Hproject_x: match ox with
| Some item' => Some ([item'], final')
| None => Some ([], final')
end = Some (projectx, final_descriptors')
eqv_final': MachineDescriptor (IM i)
project_xi: list transition_item
Hproject_xi: Some ([], eqv_final) =
Some (project_xi, eqv_final')
n: i ≠ projT1 (l x)
final_descriptors' i = eqv_final'
∧ finite_trace_projection_list IM i projectx =
  project_xi
        subst eqv_final.message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
i: index
x: composite_transition_item (equivocator_IM IM)
final_descriptors', final_descriptors: equivocator_descriptors IM
projectx: list (composite_transition_item IM)
ox: option (composite_transition_item IM)
final': equivocator_descriptors IM
Hpr_item_x: match
  equivocator_vlsm_transition_item_project
    (IM (projT1 (l x)))
    (composite_transition_item_projection
       (equivocator_IM IM) x)
    (final_descriptors (projT1 (l x)))
with
| Some (Some item', deqv') =>
    Some
      (Some
         {|
           l :=
             existT (projT1 (l x))
               (l item');
           input := input x;
           destination :=
             equivocators_state_project
               IM final_descriptors
               (destination x);
           output := output x
         |},
       equivocator_descriptors_update IM
         final_descriptors 
         (projT1 (l x)) deqv')
| Some (None, deqv') =>
    Some
      (None,
       equivocator_descriptors_update IM
         final_descriptors 
         (projT1 (l x)) deqv')
| None => None
end = Some (ox, final')
Hproject_x: match ox with
| Some item' => Some ([item'], final')
| None => Some ([], final')
end = Some (projectx, final_descriptors')
eqv_final': MachineDescriptor (IM i)
project_xi: list transition_item
Hproject_xi: Some ([], final_descriptors i) =
Some (project_xi, eqv_final')
n: i ≠ projT1 (l x)
final_descriptors' i = eqv_final'
∧ finite_trace_projection_list IM i projectx =
  project_xi
        inversion Hproject_xi.message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
i: index
x: composite_transition_item (equivocator_IM IM)
final_descriptors', final_descriptors: equivocator_descriptors IM
projectx: list (composite_transition_item IM)
ox: option (composite_transition_item IM)
final': equivocator_descriptors IM
Hpr_item_x: match
  equivocator_vlsm_transition_item_project
    (IM (projT1 (l x)))
    (composite_transition_item_projection
       (equivocator_IM IM) x)
    (final_descriptors (projT1 (l x)))
with
| Some (Some item', deqv') =>
    Some
      (Some
         {|
           l :=
             existT (projT1 (l x))
               (l item');
           input := input x;
           destination :=
             equivocators_state_project
               IM final_descriptors
               (destination x);
           output := output x
         |},
       equivocator_descriptors_update IM
         final_descriptors 
         (projT1 (l x)) deqv')
| Some (None, deqv') =>
    Some
      (None,
       equivocator_descriptors_update IM
         final_descriptors 
         (projT1 (l x)) deqv')
| None => None
end = Some (ox, final')
Hproject_x: match ox with
| Some item' => Some ([item'], final')
| None => Some ([], final')
end = Some (projectx, final_descriptors')
eqv_final': MachineDescriptor (IM i)
project_xi: list transition_item
Hproject_xi: Some ([], final_descriptors i) =
Some (project_xi, eqv_final')
n: i ≠ projT1 (l x)
H0: [] = project_xi
H1: final_descriptors i = eqv_final'
final_descriptors' i = final_descriptors i
∧ finite_trace_projection_list IM i projectx = [] subst.message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
i: index
x: composite_transition_item (equivocator_IM IM)
final_descriptors', final_descriptors: equivocator_descriptors IM
projectx: list (composite_transition_item IM)
ox: option (composite_transition_item IM)
final': equivocator_descriptors IM
Hpr_item_x: match
  equivocator_vlsm_transition_item_project
    (IM (projT1 (l x)))
    (composite_transition_item_projection
       (equivocator_IM IM) x)
    (final_descriptors (projT1 (l x)))
with
| Some (Some item', deqv') =>
    Some
      (Some
         {|
           l :=
             existT (projT1 (l x))
               (l item');
           input := input x;
           destination :=
             equivocators_state_project
               IM final_descriptors
               (destination x);
           output := output x
         |},
       equivocator_descriptors_update IM
         final_descriptors 
         (projT1 (l x)) deqv')
| Some (None, deqv') =>
    Some
      (None,
       equivocator_descriptors_update IM
         final_descriptors 
         (projT1 (l x)) deqv')
| None => None
end = Some (ox, final')
Hproject_x: match ox with
| Some item' => Some ([item'], final')
| None => Some ([], final')
end = Some (projectx, final_descriptors')
Hproject_xi: Some ([], final_descriptors i) =
Some ([], final_descriptors i)
n: i ≠ projT1 (l x)
final_descriptors' i = final_descriptors i
∧ finite_trace_projection_list IM i projectx = [] clear Hproject_xi.message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
i: index
x: composite_transition_item (equivocator_IM IM)
final_descriptors', final_descriptors: equivocator_descriptors IM
projectx: list (composite_transition_item IM)
ox: option (composite_transition_item IM)
final': equivocator_descriptors IM
Hpr_item_x: match
  equivocator_vlsm_transition_item_project
    (IM (projT1 (l x)))
    (composite_transition_item_projection
       (equivocator_IM IM) x)
    (final_descriptors (projT1 (l x)))
with
| Some (Some item', deqv') =>
    Some
      (Some
         {|
           l :=
             existT (projT1 (l x))
               (l item');
           input := input x;
           destination :=
             equivocators_state_project
               IM final_descriptors
               (destination x);
           output := output x
         |},
       equivocator_descriptors_update IM
         final_descriptors 
         (projT1 (l x)) deqv')
| Some (None, deqv') =>
    Some
      (None,
       equivocator_descriptors_update IM
         final_descriptors 
         (projT1 (l x)) deqv')
| None => None
end = Some (ox, final')
Hproject_x: match ox with
| Some item' => Some ([item'], final')
| None => Some ([], final')
end = Some (projectx, final_descriptors')
n: i ≠ projT1 (l x)
final_descriptors' i = final_descriptors i
∧ finite_trace_projection_list IM i projectx = []
        destruct
          (equivocator_vlsm_transition_item_project _ _ _)
          as [(oitem', descriptor') |] eqn: Heqpr_item_x
        ; [| done].message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
i: index
x: composite_transition_item (equivocator_IM IM)
final_descriptors', final_descriptors: equivocator_descriptors IM
projectx: list (composite_transition_item IM)
ox: option (composite_transition_item IM)
final': equivocator_descriptors IM
oitem': option transition_item
descriptor': MachineDescriptor (IM (projT1 (l x)))
Heqpr_item_x: equivocator_vlsm_transition_item_project
  (IM (projT1 (l x)))
  (composite_transition_item_projection
     (equivocator_IM IM) x)
  (final_descriptors (projT1 (l x))) =
Some (oitem', descriptor')
Hpr_item_x: match oitem' with
| Some item' =>
    Some
      (Some
         {|
           l :=
             existT (projT1 (l x))
               (l item');
           input := input x;
           destination :=
             equivocators_state_project
               IM final_descriptors
               (destination x);
           output := output x
         |},
       equivocator_descriptors_update IM
         final_descriptors 
         (projT1 (l x)) descriptor')
| None =>
    Some
      (None,
       equivocator_descriptors_update IM
         final_descriptors 
         (projT1 (l x)) descriptor')
end = Some (ox, final')
Hproject_x: match ox with
| Some item' => Some ([item'], final')
| None => Some ([], final')
end = Some (projectx, final_descriptors')
n: i ≠ projT1 (l x)
final_descriptors' i = final_descriptors i
∧ finite_trace_projection_list IM i projectx = []
        destruct oitem' as [item' |]
        ; inversion Hpr_item_x; subst; clear Hpr_item_x
        ; inversion Hproject_x; subst; clear Hproject_x
        ; state_update_simpl
        ; [| by split].message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
i: index
x: composite_transition_item (equivocator_IM IM)
final_descriptors: equivocator_descriptors IM
item': transition_item
descriptor': MachineDescriptor (IM (projT1 (l x)))
Heqpr_item_x: equivocator_vlsm_transition_item_project
  (IM (projT1 (l x)))
  (composite_transition_item_projection
     (equivocator_IM IM) x)
  (final_descriptors (projT1 (l x))) =
Some (Some item', descriptor')
n: i ≠ projT1 (l x)
final_descriptors i = final_descriptors i
∧ finite_trace_projection_list IM i
    [{|
       l := existT (projT1 (l x)) (l item');
       input := input x;
       destination :=
         equivocators_state_project IM
           final_descriptors (destination x);
       output := output x
     |}] = []
        by simpl; rewrite (composite_transition_item_projection_neq IM i).
    }message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
i: index
initial_descriptors: equivocator_descriptors IM
eqv_initial: MachineDescriptor (IM i)
x: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
final_descriptors': equivocator_descriptors IM
trX0: list (composite_transition_item IM)
IHtr: ∀ trXi : list transition_item,
  equivocator_vlsm_trace_project 
    (IM i)
    (finite_trace_projection_list
       (equivocator_IM IM) i tr)
    (final_descriptors' i) =
  Some (trXi, eqv_initial)
  → initial_descriptors i = eqv_initial
    ∧ finite_trace_projection_list IM i trX0 =
      trXi
final_descriptors: equivocator_descriptors IM
eqv_final:= final_descriptors i: MachineDescriptor (IM i)
trX, projectx: list (composite_transition_item IM)
Hproject_x: foldr equivocators_trace_project_folder
  (Some ([], final_descriptors)) [x] =
Some (projectx, final_descriptors')
HtrX0: foldr equivocators_trace_project_folder
  (Some ([], final_descriptors')) tr =
Some (trX0, initial_descriptors)
HtrX: trX = trX0 ++ projectx
trXi: list transition_item
eqv_final': MachineDescriptor (IM i)
trXi', project_xi: list transition_item
HtrXi': equivocator_vlsm_trace_project 
  (IM i)
  (pre_VLSM_projection_finite_trace_project
     (composite_type (equivocator_IM IM))
     (equivocator_IM IM i)
     (composite_project_label
        (equivocator_IM IM) i)
     (λ s : state
              (composite_type
                 (equivocator_IM IM)), 
        s i) tr) eqv_final' =
Some (trXi', eqv_initial)
Hproject_xi: equivocator_vlsm_trace_project 
  (IM i)
  (pre_VLSM_projection_finite_trace_project
     (composite_type (equivocator_IM IM))
     (equivocator_IM IM i)
     (composite_project_label
        (equivocator_IM IM) i)
     (λ s : state
              (composite_type
                 (equivocator_IM IM)),
        s i) [x]) eqv_final =
Some (project_xi, eqv_final')
HeqtrXi: trXi = trXi' ++ project_xi
Hfinal'i: final_descriptors' i = eqv_final'
∧ finite_trace_projection_list IM i
    projectx = project_xi
initial_descriptors i = eqv_initial
∧ finite_trace_projection_list IM i trX = trXi
    destruct Hfinal'i as [Hfinal'i Hpr_xi].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
i: index
initial_descriptors: equivocator_descriptors IM
eqv_initial: MachineDescriptor (IM i)
x: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
final_descriptors': equivocator_descriptors IM
trX0: list (composite_transition_item IM)
IHtr: ∀ trXi : list transition_item,
  equivocator_vlsm_trace_project 
    (IM i)
    (finite_trace_projection_list
       (equivocator_IM IM) i tr)
    (final_descriptors' i) =
  Some (trXi, eqv_initial)
  → initial_descriptors i = eqv_initial
    ∧ finite_trace_projection_list IM i trX0 =
      trXi
final_descriptors: equivocator_descriptors IM
eqv_final:= final_descriptors i: MachineDescriptor (IM i)
trX, projectx: list (composite_transition_item IM)
Hproject_x: foldr equivocators_trace_project_folder
  (Some ([], final_descriptors)) [x] =
Some (projectx, final_descriptors')
HtrX0: foldr equivocators_trace_project_folder
  (Some ([], final_descriptors')) tr =
Some (trX0, initial_descriptors)
HtrX: trX = trX0 ++ projectx
trXi: list transition_item
eqv_final': MachineDescriptor (IM i)
trXi', project_xi: list transition_item
HtrXi': equivocator_vlsm_trace_project 
  (IM i)
  (pre_VLSM_projection_finite_trace_project
     (composite_type (equivocator_IM IM))
     (equivocator_IM IM i)
     (composite_project_label
        (equivocator_IM IM) i)
     (λ s : state
              (composite_type
                 (equivocator_IM IM)), 
        s i) tr) eqv_final' =
Some (trXi', eqv_initial)
Hproject_xi: equivocator_vlsm_trace_project 
  (IM i)
  (pre_VLSM_projection_finite_trace_project
     (composite_type (equivocator_IM IM))
     (equivocator_IM IM i)
     (composite_project_label
        (equivocator_IM IM) i)
     (λ s : state
              (composite_type
                 (equivocator_IM IM)),
        s i) [x]) eqv_final =
Some (project_xi, eqv_final')
HeqtrXi: trXi = trXi' ++ project_xi
Hfinal'i: final_descriptors' i = eqv_final'
Hpr_xi: finite_trace_projection_list IM i projectx =
project_xi
initial_descriptors i = eqv_initial
∧ finite_trace_projection_list IM i trX = trXi
    rewrite <- Hfinal'i in HtrXi'.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
i: index
initial_descriptors: equivocator_descriptors IM
eqv_initial: MachineDescriptor (IM i)
x: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
final_descriptors': equivocator_descriptors IM
trX0: list (composite_transition_item IM)
IHtr: ∀ trXi : list transition_item,
  equivocator_vlsm_trace_project 
    (IM i)
    (finite_trace_projection_list
       (equivocator_IM IM) i tr)
    (final_descriptors' i) =
  Some (trXi, eqv_initial)
  → initial_descriptors i = eqv_initial
    ∧ finite_trace_projection_list IM i trX0 =
      trXi
final_descriptors: equivocator_descriptors IM
eqv_final:= final_descriptors i: MachineDescriptor (IM i)
trX, projectx: list (composite_transition_item IM)
Hproject_x: foldr equivocators_trace_project_folder
  (Some ([], final_descriptors)) [x] =
Some (projectx, final_descriptors')
HtrX0: foldr equivocators_trace_project_folder
  (Some ([], final_descriptors')) tr =
Some (trX0, initial_descriptors)
HtrX: trX = trX0 ++ projectx
trXi: list transition_item
eqv_final': MachineDescriptor (IM i)
trXi', project_xi: list transition_item
HtrXi': equivocator_vlsm_trace_project 
  (IM i)
  (pre_VLSM_projection_finite_trace_project
     (composite_type (equivocator_IM IM))
     (equivocator_IM IM i)
     (composite_project_label
        (equivocator_IM IM) i)
     (λ s : state
              (composite_type
                 (equivocator_IM IM)), 
        s i) tr) (final_descriptors' i) =
Some (trXi', eqv_initial)
Hproject_xi: equivocator_vlsm_trace_project 
  (IM i)
  (pre_VLSM_projection_finite_trace_project
     (composite_type (equivocator_IM IM))
     (equivocator_IM IM i)
     (composite_project_label
        (equivocator_IM IM) i)
     (λ s : state
              (composite_type
                 (equivocator_IM IM)),
        s i) [x]) eqv_final =
Some (project_xi, eqv_final')
HeqtrXi: trXi = trXi' ++ project_xi
Hfinal'i: final_descriptors' i = eqv_final'
Hpr_xi: finite_trace_projection_list IM i projectx =
project_xi
initial_descriptors i = eqv_initial
∧ finite_trace_projection_list IM i trX = trXi
    specialize (IHtr _ HtrXi').message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
i: index
initial_descriptors: equivocator_descriptors IM
eqv_initial: MachineDescriptor (IM i)
x: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
final_descriptors': equivocator_descriptors IM
trX0: list (composite_transition_item IM)
trXi': list transition_item
IHtr: initial_descriptors i = eqv_initial
∧ finite_trace_projection_list IM i trX0 =
  trXi'
final_descriptors: equivocator_descriptors IM
eqv_final:= final_descriptors i: MachineDescriptor (IM i)
trX, projectx: list (composite_transition_item IM)
Hproject_x: foldr equivocators_trace_project_folder
  (Some ([], final_descriptors)) [x] =
Some (projectx, final_descriptors')
HtrX0: foldr equivocators_trace_project_folder
  (Some ([], final_descriptors')) tr =
Some (trX0, initial_descriptors)
HtrX: trX = trX0 ++ projectx
trXi: list transition_item
eqv_final': MachineDescriptor (IM i)
project_xi: list transition_item
HtrXi': equivocator_vlsm_trace_project 
  (IM i)
  (pre_VLSM_projection_finite_trace_project
     (composite_type (equivocator_IM IM))
     (equivocator_IM IM i)
     (composite_project_label
        (equivocator_IM IM) i)
     (λ s : state
              (composite_type
                 (equivocator_IM IM)), 
        s i) tr) (final_descriptors' i) =
Some (trXi', eqv_initial)
Hproject_xi: equivocator_vlsm_trace_project 
  (IM i)
  (pre_VLSM_projection_finite_trace_project
     (composite_type (equivocator_IM IM))
     (equivocator_IM IM i)
     (composite_project_label
        (equivocator_IM IM) i)
     (λ s : state
              (composite_type
                 (equivocator_IM IM)),
        s i) [x]) eqv_final =
Some (project_xi, eqv_final')
HeqtrXi: trXi = trXi' ++ project_xi
Hfinal'i: final_descriptors' i = eqv_final'
Hpr_xi: finite_trace_projection_list IM i projectx =
project_xi
initial_descriptors i = eqv_initial
∧ finite_trace_projection_list IM i trX = trXi
    destruct IHtr as [Heqv_initial Hpr_trXi'].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
i: index
initial_descriptors: equivocator_descriptors IM
eqv_initial: MachineDescriptor (IM i)
x: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
final_descriptors': equivocator_descriptors IM
trX0: list (composite_transition_item IM)
trXi': list transition_item
Heqv_initial: initial_descriptors i = eqv_initial
Hpr_trXi': finite_trace_projection_list IM i trX0 =
trXi'
final_descriptors: equivocator_descriptors IM
eqv_final:= final_descriptors i: MachineDescriptor (IM i)
trX, projectx: list (composite_transition_item IM)
Hproject_x: foldr equivocators_trace_project_folder
  (Some ([], final_descriptors)) [x] =
Some (projectx, final_descriptors')
HtrX0: foldr equivocators_trace_project_folder
  (Some ([], final_descriptors')) tr =
Some (trX0, initial_descriptors)
HtrX: trX = trX0 ++ projectx
trXi: list transition_item
eqv_final': MachineDescriptor (IM i)
project_xi: list transition_item
HtrXi': equivocator_vlsm_trace_project 
  (IM i)
  (pre_VLSM_projection_finite_trace_project
     (composite_type (equivocator_IM IM))
     (equivocator_IM IM i)
     (composite_project_label
        (equivocator_IM IM) i)
     (λ s : state
              (composite_type
                 (equivocator_IM IM)), 
        s i) tr) (final_descriptors' i) =
Some (trXi', eqv_initial)
Hproject_xi: equivocator_vlsm_trace_project 
  (IM i)
  (pre_VLSM_projection_finite_trace_project
     (composite_type (equivocator_IM IM))
     (equivocator_IM IM i)
     (composite_project_label
        (equivocator_IM IM) i)
     (λ s : state
              (composite_type
                 (equivocator_IM IM)),
        s i) [x]) eqv_final =
Some (project_xi, eqv_final')
HeqtrXi: trXi = trXi' ++ project_xi
Hfinal'i: final_descriptors' i = eqv_final'
Hpr_xi: finite_trace_projection_list IM i projectx =
project_xi
initial_descriptors i = eqv_initial
∧ finite_trace_projection_list IM i trX = trXi
    split; [done |].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
i: index
initial_descriptors: equivocator_descriptors IM
eqv_initial: MachineDescriptor (IM i)
x: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
final_descriptors': equivocator_descriptors IM
trX0: list (composite_transition_item IM)
trXi': list transition_item
Heqv_initial: initial_descriptors i = eqv_initial
Hpr_trXi': finite_trace_projection_list IM i trX0 =
trXi'
final_descriptors: equivocator_descriptors IM
eqv_final:= final_descriptors i: MachineDescriptor (IM i)
trX, projectx: list (composite_transition_item IM)
Hproject_x: foldr equivocators_trace_project_folder
  (Some ([], final_descriptors)) [x] =
Some (projectx, final_descriptors')
HtrX0: foldr equivocators_trace_project_folder
  (Some ([], final_descriptors')) tr =
Some (trX0, initial_descriptors)
HtrX: trX = trX0 ++ projectx
trXi: list transition_item
eqv_final': MachineDescriptor (IM i)
project_xi: list transition_item
HtrXi': equivocator_vlsm_trace_project 
  (IM i)
  (pre_VLSM_projection_finite_trace_project
     (composite_type (equivocator_IM IM))
     (equivocator_IM IM i)
     (composite_project_label
        (equivocator_IM IM) i)
     (λ s : state
              (composite_type
                 (equivocator_IM IM)), 
        s i) tr) (final_descriptors' i) =
Some (trXi', eqv_initial)
Hproject_xi: equivocator_vlsm_trace_project 
  (IM i)
  (pre_VLSM_projection_finite_trace_project
     (composite_type (equivocator_IM IM))
     (equivocator_IM IM i)
     (composite_project_label
        (equivocator_IM IM) i)
     (λ s : state
              (composite_type
                 (equivocator_IM IM)),
        s i) [x]) eqv_final =
Some (project_xi, eqv_final')
HeqtrXi: trXi = trXi' ++ project_xi
Hfinal'i: final_descriptors' i = eqv_final'
Hpr_xi: finite_trace_projection_list IM i projectx =
project_xi
finite_trace_projection_list IM i trX = trXi
    subst.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
i: index
initial_descriptors: equivocator_descriptors IM
x: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
final_descriptors': equivocator_descriptors IM
trX0: list (composite_transition_item IM)
final_descriptors: equivocator_descriptors IM
eqv_final:= final_descriptors i: MachineDescriptor (IM i)
projectx: list (composite_transition_item IM)
Hproject_x: foldr equivocators_trace_project_folder
  (Some ([], final_descriptors)) [x] =
Some (projectx, final_descriptors')
HtrX0: foldr equivocators_trace_project_folder
  (Some ([], final_descriptors')) tr =
Some (trX0, initial_descriptors)
HtrXi': equivocator_vlsm_trace_project 
  (IM i)
  (pre_VLSM_projection_finite_trace_project
     (composite_type (equivocator_IM IM))
     (equivocator_IM IM i)
     (composite_project_label
        (equivocator_IM IM) i)
     (λ s : state
              (composite_type
                 (equivocator_IM IM)), 
        s i) tr) (final_descriptors' i) =
Some
  (finite_trace_projection_list IM i trX0,
   initial_descriptors i)
Hproject_xi: equivocator_vlsm_trace_project 
  (IM i)
  (pre_VLSM_projection_finite_trace_project
     (composite_type (equivocator_IM IM))
     (equivocator_IM IM i)
     (composite_project_label
        (equivocator_IM IM) i)
     (λ s : state
              (composite_type
                 (equivocator_IM IM)),
        s i) [x]) eqv_final =
Some
  (finite_trace_projection_list IM i
     projectx, final_descriptors' i)
finite_trace_projection_list IM i (trX0 ++ projectx) =
finite_trace_projection_list IM i trX0 ++
finite_trace_projection_list IM i projectx
    by apply omap_app.
Qed.

A generalization of equivocators_transition_item_project_preserves_zero_descriptors to full (valid) traces.

Lemma equivocators_trace_project_preserves_zero_descriptors
  (is : composite_state (equivocator_IM IM))
  (tr : list (composite_transition_item (equivocator_IM IM)))
  (Htr : finite_constrained_trace_from FreeE is tr)
  (descriptors : equivocator_descriptors IM)
  (idescriptors : equivocator_descriptors IM)
  (trX : list (composite_transition_item IM))
  (HtrX : equivocators_trace_project descriptors tr = Some (trX, idescriptors))
  : forall i, descriptors i = Existing 0 -> idescriptors i = Existing 0.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_constrained_trace_from FreeE is tr
descriptors, idescriptors: equivocator_descriptors IM
trX: list (composite_transition_item IM)
HtrX: equivocators_trace_project descriptors tr =
Some (trX, idescriptors)
∀ i : index,
  descriptors i = Existing 0
  → idescriptors i = Existing 0
Proof.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_constrained_trace_from FreeE is tr
descriptors, idescriptors: equivocator_descriptors IM
trX: list (composite_transition_item IM)
HtrX: equivocators_trace_project descriptors tr =
Some (trX, idescriptors)
∀ i : index,
  descriptors i = Existing 0
  → idescriptors i = Existing 0
  generalize dependent trX.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_constrained_trace_from FreeE is tr
descriptors, idescriptors: equivocator_descriptors IM
∀ trX : list (composite_transition_item IM),
  equivocators_trace_project descriptors tr =
  Some (trX, idescriptors)
  → ∀ i : index,
      descriptors i = Existing 0
      → idescriptors i = Existing 0 generalize dependent descriptors.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_constrained_trace_from FreeE is tr
idescriptors: equivocator_descriptors IM
∀ (descriptors : equivocator_descriptors IM) (trX : 
                                              list
                                                (composite_transition_item
                                                 IM)),
  equivocators_trace_project descriptors tr =
  Some (trX, idescriptors)
  → ∀ i : index,
      descriptors i = Existing 0
      → idescriptors i = Existing 0
  induction Htr using finite_valid_trace_from_rev_ind; [by inversion 1; subst |].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
s: state (preloaded_with_all_messages_vlsm FreeE)
tr: list transition_item
Htr: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) s tr
sf: state (preloaded_with_all_messages_vlsm FreeE)
iom, oom: option message
l: label (preloaded_with_all_messages_vlsm FreeE)
Hx: input_valid_transition
  (preloaded_with_all_messages_vlsm FreeE) l
  (finite_trace_last s tr, iom) (
  sf, oom)
x:= {|
  l := l;
  input := iom;
  destination := sf;
  output := oom
|}: transition_item
idescriptors: equivocator_descriptors IM
IHHtr: ∀ (descriptors : equivocator_descriptors IM) 
  (trX : list (composite_transition_item IM)),
  equivocators_trace_project descriptors tr =
  Some (trX, idescriptors)
  → ∀ i : index,
      descriptors i = Existing 0
      → idescriptors i = Existing 0
∀ (descriptors : equivocator_descriptors IM) (trX : 
                                              list
                                                (composite_transition_item
                                                 IM)),
  equivocators_trace_project descriptors (tr ++ [x]) =
  Some (trX, idescriptors)
  → ∀ i : index,
      descriptors i = Existing 0
      → idescriptors i = Existing 0
  intros descriptors trX HtrX i Hi.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
s: state (preloaded_with_all_messages_vlsm FreeE)
tr: list transition_item
Htr: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) s tr
sf: state (preloaded_with_all_messages_vlsm FreeE)
iom, oom: option message
l: label (preloaded_with_all_messages_vlsm FreeE)
Hx: input_valid_transition
  (preloaded_with_all_messages_vlsm FreeE) l
  (finite_trace_last s tr, iom) (
  sf, oom)
x:= {|
  l := l;
  input := iom;
  destination := sf;
  output := oom
|}: transition_item
idescriptors: equivocator_descriptors IM
IHHtr: ∀ (descriptors : equivocator_descriptors IM) 
  (trX : list (composite_transition_item IM)),
  equivocators_trace_project descriptors tr =
  Some (trX, idescriptors)
  → ∀ i : index,
      descriptors i = Existing 0
      → idescriptors i = Existing 0
descriptors: equivocator_descriptors IM
trX: list (composite_transition_item IM)
HtrX: equivocators_trace_project descriptors
  (tr ++ [x]) = Some (trX, idescriptors)
i: index
Hi: descriptors i = Existing 0
idescriptors i = Existing 0
  apply equivocators_trace_project_app_iff in HtrX
    as (preX & sufX & descriptors' & Hproject_x & Hproject_tr & _).message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
s: state (preloaded_with_all_messages_vlsm FreeE)
tr: list transition_item
Htr: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) s tr
sf: state (preloaded_with_all_messages_vlsm FreeE)
iom, oom: option message
l: label (preloaded_with_all_messages_vlsm FreeE)
Hx: input_valid_transition
  (preloaded_with_all_messages_vlsm FreeE) l
  (finite_trace_last s tr, iom) (
  sf, oom)
x:= {|
  l := l;
  input := iom;
  destination := sf;
  output := oom
|}: transition_item
idescriptors: equivocator_descriptors IM
IHHtr: ∀ (descriptors : equivocator_descriptors IM) 
  (trX : list (composite_transition_item IM)),
  equivocators_trace_project descriptors tr =
  Some (trX, idescriptors)
  → ∀ i : index,
      descriptors i = Existing 0
      → idescriptors i = Existing 0
descriptors: equivocator_descriptors IM
trX, preX, sufX: list (composite_transition_item IM)
descriptors': equivocator_descriptors IM
Hproject_x: equivocators_trace_project descriptors
  [x] = Some (sufX, descriptors')
Hproject_tr: equivocators_trace_project descriptors'
  tr = Some (preX, idescriptors)
i: index
Hi: descriptors i = Existing 0
idescriptors i = Existing 0
  simpl in Hproject_x.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
s: state (preloaded_with_all_messages_vlsm FreeE)
tr: list transition_item
Htr: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) s tr
sf: state (preloaded_with_all_messages_vlsm FreeE)
iom, oom: option message
l: label (preloaded_with_all_messages_vlsm FreeE)
Hx: input_valid_transition
  (preloaded_with_all_messages_vlsm FreeE) l
  (finite_trace_last s tr, iom) (
  sf, oom)
x:= {|
  l := l;
  input := iom;
  destination := sf;
  output := oom
|}: transition_item
idescriptors: equivocator_descriptors IM
IHHtr: ∀ (descriptors : equivocator_descriptors IM) 
  (trX : list (composite_transition_item IM)),
  equivocators_trace_project descriptors tr =
  Some (trX, idescriptors)
  → ∀ i : index,
      descriptors i = Existing 0
      → idescriptors i = Existing 0
descriptors: equivocator_descriptors IM
trX, preX, sufX: list (composite_transition_item IM)
descriptors': equivocator_descriptors IM
Hproject_x: match
  equivocators_transition_item_project
    descriptors x
with
| Some (Some item', odescriptor) =>
    Some ([item'], odescriptor)
| Some (None, odescriptor) =>
    Some ([], odescriptor)
| None => None
end = Some (sufX, descriptors')
Hproject_tr: equivocators_trace_project descriptors'
  tr = Some (preX, idescriptors)
i: index
Hi: descriptors i = Existing 0
idescriptors i = Existing 0
  destruct
    (equivocators_transition_item_project descriptors x)
    as [(oitemx, _descriptors') |] eqn: Hpr_x; [| by congruence].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
s: state (preloaded_with_all_messages_vlsm FreeE)
tr: list transition_item
Htr: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) s tr
sf: state (preloaded_with_all_messages_vlsm FreeE)
iom, oom: option message
l: label (preloaded_with_all_messages_vlsm FreeE)
Hx: input_valid_transition
  (preloaded_with_all_messages_vlsm FreeE) l
  (finite_trace_last s tr, iom) (
  sf, oom)
x:= {|
  l := l;
  input := iom;
  destination := sf;
  output := oom
|}: transition_item
idescriptors: equivocator_descriptors IM
IHHtr: ∀ (descriptors : equivocator_descriptors IM) 
  (trX : list (composite_transition_item IM)),
  equivocators_trace_project descriptors tr =
  Some (trX, idescriptors)
  → ∀ i : index,
      descriptors i = Existing 0
      → idescriptors i = Existing 0
descriptors: equivocator_descriptors IM
trX, preX, sufX: list (composite_transition_item IM)
descriptors': equivocator_descriptors IM
oitemx: option (composite_transition_item IM)
_descriptors': equivocator_descriptors IM
Hpr_x: equivocators_transition_item_project
  descriptors x = Some (oitemx, _descriptors')
Hproject_x: match oitemx with
| Some item' =>
    Some ([item'], _descriptors')
| None => Some ([], _descriptors')
end = Some (sufX, descriptors')
Hproject_tr: equivocators_trace_project descriptors'
  tr = Some (preX, idescriptors)
i: index
Hi: descriptors i = Existing 0
idescriptors i = Existing 0
  assert (_descriptors' = descriptors') as -> by (destruct oitemx; congruence).message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
s: state (preloaded_with_all_messages_vlsm FreeE)
tr: list transition_item
Htr: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) s tr
sf: state (preloaded_with_all_messages_vlsm FreeE)
iom, oom: option message
l: label (preloaded_with_all_messages_vlsm FreeE)
Hx: input_valid_transition
  (preloaded_with_all_messages_vlsm FreeE) l
  (finite_trace_last s tr, iom) (
  sf, oom)
x:= {|
  l := l;
  input := iom;
  destination := sf;
  output := oom
|}: transition_item
idescriptors: equivocator_descriptors IM
IHHtr: ∀ (descriptors : equivocator_descriptors IM) 
  (trX : list (composite_transition_item IM)),
  equivocators_trace_project descriptors tr =
  Some (trX, idescriptors)
  → ∀ i : index,
      descriptors i = Existing 0
      → idescriptors i = Existing 0
descriptors: equivocator_descriptors IM
trX, preX, sufX: list (composite_transition_item IM)
descriptors': equivocator_descriptors IM
oitemx: option (composite_transition_item IM)
Hproject_x: match oitemx with
| Some item' =>
    Some ([item'], descriptors')
| None => Some ([], descriptors')
end = Some (sufX, descriptors')
Hpr_x: equivocators_transition_item_project
  descriptors x = Some (oitemx, descriptors')
Hproject_tr: equivocators_trace_project descriptors'
  tr = Some (preX, idescriptors)
i: index
Hi: descriptors i = Existing 0
idescriptors i = Existing 0
  clear Hproject_x trX sufX.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
s: state (preloaded_with_all_messages_vlsm FreeE)
tr: list transition_item
Htr: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) s tr
sf: state (preloaded_with_all_messages_vlsm FreeE)
iom, oom: option message
l: label (preloaded_with_all_messages_vlsm FreeE)
Hx: input_valid_transition
  (preloaded_with_all_messages_vlsm FreeE) l
  (finite_trace_last s tr, iom) (
  sf, oom)
x:= {|
  l := l;
  input := iom;
  destination := sf;
  output := oom
|}: transition_item
idescriptors: equivocator_descriptors IM
IHHtr: ∀ (descriptors : equivocator_descriptors IM) 
  (trX : list (composite_transition_item IM)),
  equivocators_trace_project descriptors tr =
  Some (trX, idescriptors)
  → ∀ i : index,
      descriptors i = Existing 0
      → idescriptors i = Existing 0
descriptors: equivocator_descriptors IM
preX: list (composite_transition_item IM)
descriptors': equivocator_descriptors IM
oitemx: option (composite_transition_item IM)
Hpr_x: equivocators_transition_item_project
  descriptors x = Some (oitemx, descriptors')
Hproject_tr: equivocators_trace_project descriptors'
  tr = Some (preX, idescriptors)
i: index
Hi: descriptors i = Existing 0
idescriptors i = Existing 0
  destruct Hx as [(_ & _  & Hv) Ht].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
s: state (preloaded_with_all_messages_vlsm FreeE)
tr: list transition_item
Htr: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) s tr
sf: state (preloaded_with_all_messages_vlsm FreeE)
iom, oom: option message
l: label (preloaded_with_all_messages_vlsm FreeE)
Hv: valid l (finite_trace_last s tr, iom)
Ht: transition l (finite_trace_last s tr, iom) =
(sf, oom)
x:= {|
  l := l;
  input := iom;
  destination := sf;
  output := oom
|}: transition_item
idescriptors: equivocator_descriptors IM
IHHtr: ∀ (descriptors : equivocator_descriptors IM) 
  (trX : list (composite_transition_item IM)),
  equivocators_trace_project descriptors tr =
  Some (trX, idescriptors)
  → ∀ i : index,
      descriptors i = Existing 0
      → idescriptors i = Existing 0
descriptors: equivocator_descriptors IM
preX: list (composite_transition_item IM)
descriptors': equivocator_descriptors IM
oitemx: option (composite_transition_item IM)
Hpr_x: equivocators_transition_item_project
  descriptors x = Some (oitemx, descriptors')
Hproject_tr: equivocators_trace_project descriptors'
  tr = Some (preX, idescriptors)
i: index
Hi: descriptors i = Existing 0
idescriptors i = Existing 0
  eapply IHHtr; [done |].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
s: state (preloaded_with_all_messages_vlsm FreeE)
tr: list transition_item
Htr: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) s tr
sf: state (preloaded_with_all_messages_vlsm FreeE)
iom, oom: option message
l: label (preloaded_with_all_messages_vlsm FreeE)
Hv: valid l (finite_trace_last s tr, iom)
Ht: transition l (finite_trace_last s tr, iom) =
(sf, oom)
x:= {|
  l := l;
  input := iom;
  destination := sf;
  output := oom
|}: transition_item
idescriptors: equivocator_descriptors IM
IHHtr: ∀ (descriptors : equivocator_descriptors IM) 
  (trX : list (composite_transition_item IM)),
  equivocators_trace_project descriptors tr =
  Some (trX, idescriptors)
  → ∀ i : index,
      descriptors i = Existing 0
      → idescriptors i = Existing 0
descriptors: equivocator_descriptors IM
preX: list (composite_transition_item IM)
descriptors': equivocator_descriptors IM
oitemx: option (composite_transition_item IM)
Hpr_x: equivocators_transition_item_project
  descriptors x = Some (oitemx, descriptors')
Hproject_tr: equivocators_trace_project descriptors'
  tr = Some (preX, idescriptors)
i: index
Hi: descriptors i = Existing 0
descriptors' i = Existing 0
  by eapply equivocators_transition_item_project_preserves_zero_descriptors
       with (item := x); [| apply Hv | |].
Qed.

Lemma preloaded_equivocators_valid_trace_from_project
  (final_descriptors : equivocator_descriptors IM)
  (is : composite_state (equivocator_IM IM))
  (tr : list (composite_transition_item (equivocator_IM IM)))
  (final_state := finite_trace_last is tr)
  (Hproper : proper_equivocator_descriptors IM final_descriptors final_state)
  (Htr : finite_constrained_trace_from FreeE is tr)
  : exists
    (trX : list (composite_transition_item IM))
    (initial_descriptors : equivocator_descriptors IM),
    equivocators_trace_project final_descriptors tr = Some (trX, initial_descriptors)
    /\ proper_equivocator_descriptors IM initial_descriptors is
    /\ equivocators_state_project IM final_descriptors (finite_trace_last is tr)
     = finite_trace_last (equivocators_state_project IM initial_descriptors is) trX.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
final_descriptors: equivocator_descriptors IM
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
final_state:= finite_trace_last is tr: state
  (composite_type (equivocator_IM IM))
Hproper: proper_equivocator_descriptors IM
  final_descriptors final_state
Htr: finite_constrained_trace_from FreeE is tr
∃ (trX : list (composite_transition_item IM)) (initial_descriptors : 
                                               equivocator_descriptors
                                                 IM),
  equivocators_trace_project final_descriptors tr =
  Some (trX, initial_descriptors)
  ∧ proper_equivocator_descriptors IM
      initial_descriptors is
    ∧ equivocators_state_project IM final_descriptors
        (finite_trace_last is tr) =
      finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX
Proof.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
final_descriptors: equivocator_descriptors IM
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
final_state:= finite_trace_last is tr: state
  (composite_type (equivocator_IM IM))
Hproper: proper_equivocator_descriptors IM
  final_descriptors final_state
Htr: finite_constrained_trace_from FreeE is tr
∃ (trX : list (composite_transition_item IM)) (initial_descriptors : 
                                               equivocator_descriptors
                                                 IM),
  equivocators_trace_project final_descriptors tr =
  Some (trX, initial_descriptors)
  ∧ proper_equivocator_descriptors IM
      initial_descriptors is
    ∧ equivocators_state_project IM final_descriptors
        (finite_trace_last is tr) =
      finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX
  generalize dependent final_descriptors; generalize dependent is.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
tr: list
  (composite_transition_item (equivocator_IM IM))
∀ is : composite_state (equivocator_IM IM),
  let final_state := finite_trace_last is tr in
  finite_constrained_trace_from FreeE is tr
  → ∀ final_descriptors : equivocator_descriptors IM,
      proper_equivocator_descriptors IM
        final_descriptors final_state
      → ∃ (trX : list (composite_transition_item IM)) 
          (initial_descriptors : equivocator_descriptors
                                   IM),
          equivocators_trace_project final_descriptors
            tr = Some (trX, initial_descriptors)
          ∧ proper_equivocator_descriptors IM
              initial_descriptors is
            ∧ equivocators_state_project IM
                final_descriptors
                (finite_trace_last is tr) =
              finite_trace_last
                (equivocators_state_project IM
                   initial_descriptors is) trX
  induction tr using rev_ind; intros; [by exists [], final_descriptors |].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
x: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
IHtr: ∀ is : composite_state (equivocator_IM IM),
  let final_state := finite_trace_last is tr in
  finite_constrained_trace_from FreeE is tr
  → ∀ final_descriptors : 
       equivocator_descriptors IM,
      proper_equivocator_descriptors IM
        final_descriptors final_state
      → ∃ (trX : list
                   (composite_transition_item
                      IM)) 
          (initial_descriptors : 
           equivocator_descriptors IM),
          equivocators_trace_project
            final_descriptors tr =
          Some (trX, initial_descriptors)
          ∧ proper_equivocator_descriptors IM
              initial_descriptors is
            ∧ equivocators_state_project IM
                final_descriptors
                (finite_trace_last is tr) =
              finite_trace_last
                (equivocators_state_project IM
                   initial_descriptors is) trX
is: composite_state (equivocator_IM IM)
final_state:= finite_trace_last is (tr ++ [x]): state
  (composite_type (equivocator_IM IM))
Htr: finite_constrained_trace_from FreeE is
  (tr ++ [x])
final_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  final_descriptors final_state
∃ (trX : list (composite_transition_item IM)) (initial_descriptors : 
                                               equivocator_descriptors
                                                 IM),
  equivocators_trace_project final_descriptors
    (tr ++ [x]) = Some (trX, initial_descriptors)
  ∧ proper_equivocator_descriptors IM
      initial_descriptors is
    ∧ equivocators_state_project IM final_descriptors
        (finite_trace_last is (tr ++ [x])) =
      finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX
  apply finite_valid_trace_from_app_iff in Htr.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
x: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
IHtr: ∀ is : composite_state (equivocator_IM IM),
  let final_state := finite_trace_last is tr in
  finite_constrained_trace_from FreeE is tr
  → ∀ final_descriptors : 
       equivocator_descriptors IM,
      proper_equivocator_descriptors IM
        final_descriptors final_state
      → ∃ (trX : list
                   (composite_transition_item
                      IM)) 
          (initial_descriptors : 
           equivocator_descriptors IM),
          equivocators_trace_project
            final_descriptors tr =
          Some (trX, initial_descriptors)
          ∧ proper_equivocator_descriptors IM
              initial_descriptors is
            ∧ equivocators_state_project IM
                final_descriptors
                (finite_trace_last is tr) =
              finite_trace_last
                (equivocators_state_project IM
                   initial_descriptors is) trX
is: composite_state (equivocator_IM IM)
final_state:= finite_trace_last is (tr ++ [x]): state
  (composite_type (equivocator_IM IM))
Htr: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) is tr
∧ finite_valid_trace_from
    (preloaded_with_all_messages_vlsm FreeE)
    (finite_trace_last is tr) [x]
final_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  final_descriptors final_state
∃ (trX : list (composite_transition_item IM)) (initial_descriptors : 
                                               equivocator_descriptors
                                                 IM),
  equivocators_trace_project final_descriptors
    (tr ++ [x]) = Some (trX, initial_descriptors)
  ∧ proper_equivocator_descriptors IM
      initial_descriptors is
    ∧ equivocators_state_project IM final_descriptors
        (finite_trace_last is (tr ++ [x])) =
      finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX
  destruct Htr as [Htr Hx].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
x: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
IHtr: ∀ is : composite_state (equivocator_IM IM),
  let final_state := finite_trace_last is tr in
  finite_constrained_trace_from FreeE is tr
  → ∀ final_descriptors : 
       equivocator_descriptors IM,
      proper_equivocator_descriptors IM
        final_descriptors final_state
      → ∃ (trX : list
                   (composite_transition_item
                      IM)) 
          (initial_descriptors : 
           equivocator_descriptors IM),
          equivocators_trace_project
            final_descriptors tr =
          Some (trX, initial_descriptors)
          ∧ proper_equivocator_descriptors IM
              initial_descriptors is
            ∧ equivocators_state_project IM
                final_descriptors
                (finite_trace_last is tr) =
              finite_trace_last
                (equivocators_state_project IM
                   initial_descriptors is) trX
is: composite_state (equivocator_IM IM)
final_state:= finite_trace_last is (tr ++ [x]): state
  (composite_type (equivocator_IM IM))
Htr: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) is tr
Hx: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE)
  (finite_trace_last is tr) [x]
final_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  final_descriptors final_state
∃ (trX : list (composite_transition_item IM)) (initial_descriptors : 
                                               equivocator_descriptors
                                                 IM),
  equivocators_trace_project final_descriptors
    (tr ++ [x]) = Some (trX, initial_descriptors)
  ∧ proper_equivocator_descriptors IM
      initial_descriptors is
    ∧ equivocators_state_project IM final_descriptors
        (finite_trace_last is (tr ++ [x])) =
      finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX
  specialize (IHtr _ Htr).message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
x: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
is: composite_state (equivocator_IM IM)
IHtr: ∀ final_descriptors : equivocator_descriptors
                      IM,
  proper_equivocator_descriptors IM
    final_descriptors (finite_trace_last is tr)
  → ∃ (trX : list
               (composite_transition_item IM)) 
      (initial_descriptors : 
       equivocator_descriptors IM),
      equivocators_trace_project
        final_descriptors tr =
      Some (trX, initial_descriptors)
      ∧ proper_equivocator_descriptors IM
          initial_descriptors is
        ∧ equivocators_state_project IM
            final_descriptors
            (finite_trace_last is tr) =
          finite_trace_last
            (equivocators_state_project IM
               initial_descriptors is) trX
final_state:= finite_trace_last is (tr ++ [x]): state
  (composite_type (equivocator_IM IM))
Htr: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) is tr
Hx: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE)
  (finite_trace_last is tr) [x]
final_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  final_descriptors final_state
∃ (trX : list (composite_transition_item IM)) (initial_descriptors : 
                                               equivocator_descriptors
                                                 IM),
  equivocators_trace_project final_descriptors
    (tr ++ [x]) = Some (trX, initial_descriptors)
  ∧ proper_equivocator_descriptors IM
      initial_descriptors is
    ∧ equivocators_state_project IM final_descriptors
        (finite_trace_last is (tr ++ [x])) =
      finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX
  specialize (equivocators_transition_item_project_proper_characterization final_descriptors x)
    as Hproperx.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
x: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
is: composite_state (equivocator_IM IM)
IHtr: ∀ final_descriptors : equivocator_descriptors
                      IM,
  proper_equivocator_descriptors IM
    final_descriptors (finite_trace_last is tr)
  → ∃ (trX : list
               (composite_transition_item IM)) 
      (initial_descriptors : 
       equivocator_descriptors IM),
      equivocators_trace_project
        final_descriptors tr =
      Some (trX, initial_descriptors)
      ∧ proper_equivocator_descriptors IM
          initial_descriptors is
        ∧ equivocators_state_project IM
            final_descriptors
            (finite_trace_last is tr) =
          finite_trace_last
            (equivocators_state_project IM
               initial_descriptors is) trX
final_state:= finite_trace_last is (tr ++ [x]): state
  (composite_type (equivocator_IM IM))
Htr: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) is tr
Hx: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE)
  (finite_trace_last is tr) [x]
final_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  final_descriptors final_state
Hproperx: proper_equivocator_descriptors IM
  final_descriptors (destination x)
→ ∃ (oitem : option
               (composite_transition_item
                  IM)) 
    (eqv_descriptors' : 
     equivocator_descriptors IM),
    equivocators_transition_item_project
      final_descriptors x =
    Some (oitem, eqv_descriptors')
    ∧ match oitem with
      | Some itemx =>
          (∃ Hex : existing_equivocator_label
                     (IM (projT1 (l x)))
                     (projT2 (l x)),
             existT (projT1 (l x))
               (existing_equivocator_label_extract
                  (IM (projT1 (l x)))
                  (projT2 (l x)) Hex) =
             l itemx)
          ∧ input x = input itemx
            ∧ output x = output itemx
              ∧ equivocators_state_project
                  IM final_descriptors
                  (destination x) =
                destination itemx
                ∧ eqv_descriptors'
                    (projT1 (l x)) =
                  equivocator_label_descriptor
                    (projT2 (l x))
      | None => True
      end
      ∧ (∀ s : composite_state
                 (equivocator_IM IM),
           composite_valid
             (equivocator_IM IM) 
             (l x) (s, input x)
           → composite_transition
               (equivocator_IM IM) 
               (l x) (s, input x) =
             (destination x, output x)
             → proper_equivocator_descriptors
                 IM eqv_descriptors' s
               ∧ eqv_descriptors' =
                 equivocator_descriptors_update
                   IM final_descriptors
                   (projT1 (l x))
                   (eqv_descriptors'
                      (projT1 (l x)))
                 ∧ s =
                   state_update
                     (equivocator_IM IM)
                     (destination x)
                     (projT1 (l x))
                     (s (projT1 (l x)))
                   ∧ previous_state_descriptor_prop
                      (IM (projT1 (l x)))
                      (final_descriptors
                      (projT1 (l x)))
                      (s (projT1 (l x)))
                      (eqv_descriptors'
                      (projT1 (l x)))
                     ∧ 
                     match oitem with
                     | Some itemx =>
                      ∀ sx : 
                      composite_state IM,
                      sx =
                      equivocators_state_project
                      IM eqv_descriptors' s
                      → 
                      composite_valid IM
                      (l itemx)
                      (sx, input itemx)
                      ∧ 
                      composite_transition
                      IM 
                      (l itemx)
                      (sx, input itemx) =
                      (
                      destination itemx,
                      output itemx)
                     | None =>
                      equivocators_state_project
                      IM final_descriptors
                      (destination x) =
                      equivocators_state_project
                      IM eqv_descriptors' s
                     end)
∃ (trX : list (composite_transition_item IM)) (initial_descriptors : 
                                               equivocator_descriptors
                                                 IM),
  equivocators_trace_project final_descriptors
    (tr ++ [x]) = Some (trX, initial_descriptors)
  ∧ proper_equivocator_descriptors IM
      initial_descriptors is
    ∧ equivocators_state_project IM final_descriptors
        (finite_trace_last is (tr ++ [x])) =
      finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX
  unfold final_state in Hproper.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
x: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
is: composite_state (equivocator_IM IM)
IHtr: ∀ final_descriptors : equivocator_descriptors
                      IM,
  proper_equivocator_descriptors IM
    final_descriptors (finite_trace_last is tr)
  → ∃ (trX : list
               (composite_transition_item IM)) 
      (initial_descriptors : 
       equivocator_descriptors IM),
      equivocators_trace_project
        final_descriptors tr =
      Some (trX, initial_descriptors)
      ∧ proper_equivocator_descriptors IM
          initial_descriptors is
        ∧ equivocators_state_project IM
            final_descriptors
            (finite_trace_last is tr) =
          finite_trace_last
            (equivocators_state_project IM
               initial_descriptors is) trX
final_state:= finite_trace_last is (tr ++ [x]): state
  (composite_type (equivocator_IM IM))
Htr: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) is tr
Hx: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE)
  (finite_trace_last is tr) [x]
final_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  final_descriptors
  (finite_trace_last is (tr ++ [x]))
Hproperx: proper_equivocator_descriptors IM
  final_descriptors (destination x)
→ ∃ (oitem : option
               (composite_transition_item
                  IM)) 
    (eqv_descriptors' : 
     equivocator_descriptors IM),
    equivocators_transition_item_project
      final_descriptors x =
    Some (oitem, eqv_descriptors')
    ∧ match oitem with
      | Some itemx =>
          (∃ Hex : existing_equivocator_label
                     (IM (projT1 (l x)))
                     (projT2 (l x)),
             existT (projT1 (l x))
               (existing_equivocator_label_extract
                  (IM (projT1 (l x)))
                  (projT2 (l x)) Hex) =
             l itemx)
          ∧ input x = input itemx
            ∧ output x = output itemx
              ∧ equivocators_state_project
                  IM final_descriptors
                  (destination x) =
                destination itemx
                ∧ eqv_descriptors'
                    (projT1 (l x)) =
                  equivocator_label_descriptor
                    (projT2 (l x))
      | None => True
      end
      ∧ (∀ s : composite_state
                 (equivocator_IM IM),
           composite_valid
             (equivocator_IM IM) 
             (l x) (s, input x)
           → composite_transition
               (equivocator_IM IM) 
               (l x) (s, input x) =
             (destination x, output x)
             → proper_equivocator_descriptors
                 IM eqv_descriptors' s
               ∧ eqv_descriptors' =
                 equivocator_descriptors_update
                   IM final_descriptors
                   (projT1 (l x))
                   (eqv_descriptors'
                      (projT1 (l x)))
                 ∧ s =
                   state_update
                     (equivocator_IM IM)
                     (destination x)
                     (projT1 (l x))
                     (s (projT1 (l x)))
                   ∧ previous_state_descriptor_prop
                      (IM (projT1 (l x)))
                      (final_descriptors
                      (projT1 (l x)))
                      (s (projT1 (l x)))
                      (eqv_descriptors'
                      (projT1 (l x)))
                     ∧ 
                     match oitem with
                     | Some itemx =>
                      ∀ sx : 
                      composite_state IM,
                      sx =
                      equivocators_state_project
                      IM eqv_descriptors' s
                      → 
                      composite_valid IM
                      (l itemx)
                      (sx, input itemx)
                      ∧ 
                      composite_transition
                      IM 
                      (l itemx)
                      (sx, input itemx) =
                      (
                      destination itemx,
                      output itemx)
                     | None =>
                      equivocators_state_project
                      IM final_descriptors
                      (destination x) =
                      equivocators_state_project
                      IM eqv_descriptors' s
                     end)
∃ (trX : list (composite_transition_item IM)) (initial_descriptors : 
                                               equivocator_descriptors
                                                 IM),
  equivocators_trace_project final_descriptors
    (tr ++ [x]) = Some (trX, initial_descriptors)
  ∧ proper_equivocator_descriptors IM
      initial_descriptors is
    ∧ equivocators_state_project IM final_descriptors
        (finite_trace_last is (tr ++ [x])) =
      finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX
  rewrite finite_trace_last_is_last in Hproper.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
x: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
is: composite_state (equivocator_IM IM)
IHtr: ∀ final_descriptors : equivocator_descriptors
                      IM,
  proper_equivocator_descriptors IM
    final_descriptors (finite_trace_last is tr)
  → ∃ (trX : list
               (composite_transition_item IM)) 
      (initial_descriptors : 
       equivocator_descriptors IM),
      equivocators_trace_project
        final_descriptors tr =
      Some (trX, initial_descriptors)
      ∧ proper_equivocator_descriptors IM
          initial_descriptors is
        ∧ equivocators_state_project IM
            final_descriptors
            (finite_trace_last is tr) =
          finite_trace_last
            (equivocators_state_project IM
               initial_descriptors is) trX
final_state:= finite_trace_last is (tr ++ [x]): state
  (composite_type (equivocator_IM IM))
Htr: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) is tr
Hx: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE)
  (finite_trace_last is tr) [x]
final_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  final_descriptors (destination x)
Hproperx: proper_equivocator_descriptors IM
  final_descriptors (destination x)
→ ∃ (oitem : option
               (composite_transition_item
                  IM)) 
    (eqv_descriptors' : 
     equivocator_descriptors IM),
    equivocators_transition_item_project
      final_descriptors x =
    Some (oitem, eqv_descriptors')
    ∧ match oitem with
      | Some itemx =>
          (∃ Hex : existing_equivocator_label
                     (IM (projT1 (l x)))
                     (projT2 (l x)),
             existT (projT1 (l x))
               (existing_equivocator_label_extract
                  (IM (projT1 (l x)))
                  (projT2 (l x)) Hex) =
             l itemx)
          ∧ input x = input itemx
            ∧ output x = output itemx
              ∧ equivocators_state_project
                  IM final_descriptors
                  (destination x) =
                destination itemx
                ∧ eqv_descriptors'
                    (projT1 (l x)) =
                  equivocator_label_descriptor
                    (projT2 (l x))
      | None => True
      end
      ∧ (∀ s : composite_state
                 (equivocator_IM IM),
           composite_valid
             (equivocator_IM IM) 
             (l x) (s, input x)
           → composite_transition
               (equivocator_IM IM) 
               (l x) (s, input x) =
             (destination x, output x)
             → proper_equivocator_descriptors
                 IM eqv_descriptors' s
               ∧ eqv_descriptors' =
                 equivocator_descriptors_update
                   IM final_descriptors
                   (projT1 (l x))
                   (eqv_descriptors'
                      (projT1 (l x)))
                 ∧ s =
                   state_update
                     (equivocator_IM IM)
                     (destination x)
                     (projT1 (l x))
                     (s (projT1 (l x)))
                   ∧ previous_state_descriptor_prop
                      (IM (projT1 (l x)))
                      (final_descriptors
                      (projT1 (l x)))
                      (s (projT1 (l x)))
                      (eqv_descriptors'
                      (projT1 (l x)))
                     ∧ 
                     match oitem with
                     | Some itemx =>
                      ∀ sx : 
                      composite_state IM,
                      sx =
                      equivocators_state_project
                      IM eqv_descriptors' s
                      → 
                      composite_valid IM
                      (l itemx)
                      (sx, input itemx)
                      ∧ 
                      composite_transition
                      IM 
                      (l itemx)
                      (sx, input itemx) =
                      (
                      destination itemx,
                      output itemx)
                     | None =>
                      equivocators_state_project
                      IM final_descriptors
                      (destination x) =
                      equivocators_state_project
                      IM eqv_descriptors' s
                     end)
∃ (trX : list (composite_transition_item IM)) (initial_descriptors : 
                                               equivocator_descriptors
                                                 IM),
  equivocators_trace_project final_descriptors
    (tr ++ [x]) = Some (trX, initial_descriptors)
  ∧ proper_equivocator_descriptors IM
      initial_descriptors is
    ∧ equivocators_state_project IM final_descriptors
        (finite_trace_last is (tr ++ [x])) =
      finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX
  specialize (Hproperx Hproper).message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
x: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
is: composite_state (equivocator_IM IM)
IHtr: ∀ final_descriptors : equivocator_descriptors
                      IM,
  proper_equivocator_descriptors IM
    final_descriptors (finite_trace_last is tr)
  → ∃ (trX : list
               (composite_transition_item IM)) 
      (initial_descriptors : 
       equivocator_descriptors IM),
      equivocators_trace_project
        final_descriptors tr =
      Some (trX, initial_descriptors)
      ∧ proper_equivocator_descriptors IM
          initial_descriptors is
        ∧ equivocators_state_project IM
            final_descriptors
            (finite_trace_last is tr) =
          finite_trace_last
            (equivocators_state_project IM
               initial_descriptors is) trX
final_state:= finite_trace_last is (tr ++ [x]): state
  (composite_type (equivocator_IM IM))
Htr: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) is tr
Hx: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE)
  (finite_trace_last is tr) [x]
final_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  final_descriptors (destination x)
Hproperx: ∃ (oitem : option
             (composite_transition_item IM)) 
  (eqv_descriptors' : equivocator_descriptors
                      IM),
  equivocators_transition_item_project
    final_descriptors x =
  Some (oitem, eqv_descriptors')
  ∧ match oitem with
    | Some itemx =>
        (∃ Hex : existing_equivocator_label
                   (IM (projT1 (l x)))
                   (projT2 (l x)),
           existT (projT1 (l x))
             (existing_equivocator_label_extract
                (IM (projT1 (l x)))
                (projT2 (l x)) Hex) =
           l itemx)
        ∧ input x = input itemx
          ∧ output x = output itemx
            ∧ equivocators_state_project IM
                final_descriptors
                (destination x) =
              destination itemx
              ∧ eqv_descriptors'
                  (projT1 (l x)) =
                equivocator_label_descriptor
                  (projT2 (l x))
    | None => True
    end
    ∧ (∀ s : composite_state
               (equivocator_IM IM),
         composite_valid
           (equivocator_IM IM) 
           (l x) (s, input x)
         → composite_transition
             (equivocator_IM IM) 
             (l x) (s, input x) =
           (destination x, output x)
           → proper_equivocator_descriptors
               IM eqv_descriptors' s
             ∧ eqv_descriptors' =
               equivocator_descriptors_update
                 IM final_descriptors
                 (projT1 (l x))
                 (eqv_descriptors'
                    (projT1 (l x)))
               ∧ s =
                 state_update
                   (equivocator_IM IM)
                   (destination x)
                   (projT1 (l x))
                   (s (projT1 (l x)))
                 ∧ previous_state_descriptor_prop
                     (IM (projT1 (l x)))
                     (final_descriptors
                      (projT1 (l x)))
                     (s (projT1 (l x)))
                     (eqv_descriptors'
                      (projT1 (l x)))
                   ∧ match oitem with
                     | Some itemx =>
                      ∀ sx : 
                      composite_state IM,
                      sx =
                      equivocators_state_project
                      IM eqv_descriptors' s
                      → 
                      composite_valid IM
                      (l itemx)
                      (sx, input itemx)
                      ∧ 
                      composite_transition
                      IM 
                      (l itemx)
                      (sx, input itemx) =
                      (
                      destination itemx,
                      output itemx)
                     | None =>
                      equivocators_state_project
                      IM final_descriptors
                      (destination x) =
                      equivocators_state_project
                      IM eqv_descriptors' s
                     end)
∃ (trX : list (composite_transition_item IM)) (initial_descriptors : 
                                               equivocator_descriptors
                                                 IM),
  equivocators_trace_project final_descriptors
    (tr ++ [x]) = Some (trX, initial_descriptors)
  ∧ proper_equivocator_descriptors IM
      initial_descriptors is
    ∧ equivocators_state_project IM final_descriptors
        (finite_trace_last is (tr ++ [x])) =
      finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX
  destruct Hproperx as [oitem [final_descriptors' [Hprojectx [Hitemx Hproperx]]]].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
x: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
is: composite_state (equivocator_IM IM)
IHtr: ∀ final_descriptors : equivocator_descriptors
                      IM,
  proper_equivocator_descriptors IM
    final_descriptors (finite_trace_last is tr)
  → ∃ (trX : list
               (composite_transition_item IM)) 
      (initial_descriptors : 
       equivocator_descriptors IM),
      equivocators_trace_project
        final_descriptors tr =
      Some (trX, initial_descriptors)
      ∧ proper_equivocator_descriptors IM
          initial_descriptors is
        ∧ equivocators_state_project IM
            final_descriptors
            (finite_trace_last is tr) =
          finite_trace_last
            (equivocators_state_project IM
               initial_descriptors is) trX
final_state:= finite_trace_last is (tr ++ [x]): state
  (composite_type (equivocator_IM IM))
Htr: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) is tr
Hx: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE)
  (finite_trace_last is tr) [x]
final_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  final_descriptors (destination x)
oitem: option (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hprojectx: equivocators_transition_item_project
  final_descriptors x =
Some (oitem, final_descriptors')
Hitemx: match oitem with
| Some itemx =>
    (∃ Hex : existing_equivocator_label
               (IM (projT1 (l x)))
               (projT2 (l x)),
       existT (projT1 (l x))
         (existing_equivocator_label_extract
            (IM (projT1 (l x)))
            (projT2 (l x)) Hex) = 
       l itemx)
    ∧ input x = input itemx
      ∧ output x = output itemx
        ∧ equivocators_state_project IM
            final_descriptors 
            (destination x) =
          destination itemx
          ∧ final_descriptors' (projT1 (l x)) =
            equivocator_label_descriptor
              (projT2 (l x))
| None => True
end
Hproperx: ∀ s : composite_state (equivocator_IM IM),
  composite_valid (equivocator_IM IM) 
    (l x) (s, input x)
  → composite_transition
      (equivocator_IM IM) 
      (l x) (s, input x) =
    (destination x, output x)
    → proper_equivocator_descriptors IM
        final_descriptors' s
      ∧ final_descriptors' =
        equivocator_descriptors_update IM
          final_descriptors 
          (projT1 (l x))
          (final_descriptors'
             (projT1 (l x)))
        ∧ s =
          state_update 
            (equivocator_IM IM)
            (destination x) 
            (projT1 (l x))
            (s (projT1 (l x)))
          ∧ previous_state_descriptor_prop
              (IM (projT1 (l x)))
              (final_descriptors
                 (projT1 (l x)))
              (s (projT1 (l x)))
              (final_descriptors'
                 (projT1 (l x)))
            ∧ match oitem with
              | Some itemx =>
                  ∀ sx : composite_state IM,
                    sx =
                    equivocators_state_project
                      IM final_descriptors'
                      s
                    → composite_valid IM
                      (l itemx)
                      (sx, input itemx)
                      ∧ 
                      composite_transition
                      IM 
                      (l itemx)
                      (sx, input itemx) =
                      (
                      destination itemx,
                      output itemx)
              | None =>
                  equivocators_state_project
                    IM final_descriptors
                    (destination x) =
                  equivocators_state_project
                    IM final_descriptors' s
              end
∃ (trX : list (composite_transition_item IM)) (initial_descriptors : 
                                               equivocator_descriptors
                                                 IM),
  equivocators_trace_project final_descriptors
    (tr ++ [x]) = Some (trX, initial_descriptors)
  ∧ proper_equivocator_descriptors IM
      initial_descriptors is
    ∧ equivocators_state_project IM final_descriptors
        (finite_trace_last is (tr ++ [x])) =
      finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX
  specialize (Hproperx (finite_trace_last is tr)).message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
x: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
is: composite_state (equivocator_IM IM)
IHtr: ∀ final_descriptors : equivocator_descriptors
                      IM,
  proper_equivocator_descriptors IM
    final_descriptors (finite_trace_last is tr)
  → ∃ (trX : list
               (composite_transition_item IM)) 
      (initial_descriptors : 
       equivocator_descriptors IM),
      equivocators_trace_project
        final_descriptors tr =
      Some (trX, initial_descriptors)
      ∧ proper_equivocator_descriptors IM
          initial_descriptors is
        ∧ equivocators_state_project IM
            final_descriptors
            (finite_trace_last is tr) =
          finite_trace_last
            (equivocators_state_project IM
               initial_descriptors is) trX
final_state:= finite_trace_last is (tr ++ [x]): state
  (composite_type (equivocator_IM IM))
Htr: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) is tr
Hx: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE)
  (finite_trace_last is tr) [x]
final_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  final_descriptors (destination x)
oitem: option (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hprojectx: equivocators_transition_item_project
  final_descriptors x =
Some (oitem, final_descriptors')
Hitemx: match oitem with
| Some itemx =>
    (∃ Hex : existing_equivocator_label
               (IM (projT1 (l x)))
               (projT2 (l x)),
       existT (projT1 (l x))
         (existing_equivocator_label_extract
            (IM (projT1 (l x)))
            (projT2 (l x)) Hex) = 
       l itemx)
    ∧ input x = input itemx
      ∧ output x = output itemx
        ∧ equivocators_state_project IM
            final_descriptors 
            (destination x) =
          destination itemx
          ∧ final_descriptors' (projT1 (l x)) =
            equivocator_label_descriptor
              (projT2 (l x))
| None => True
end
Hproperx: composite_valid (equivocator_IM IM) 
  (l x) (finite_trace_last is tr, input x)
→ composite_transition 
    (equivocator_IM IM) 
    (l x)
    (finite_trace_last is tr, input x) =
  (destination x, output x)
  → proper_equivocator_descriptors IM
      final_descriptors'
      (finite_trace_last is tr)
    ∧ final_descriptors' =
      equivocator_descriptors_update IM
        final_descriptors 
        (projT1 (l x))
        (final_descriptors' (projT1 (l x)))
      ∧ finite_trace_last is tr =
        state_update (equivocator_IM IM)
          (destination x) 
          (projT1 (l x))
          (finite_trace_last is tr
             (projT1 (l x)))
        ∧ previous_state_descriptor_prop
            (IM (projT1 (l x)))
            (final_descriptors
               (projT1 (l x)))
            (finite_trace_last is tr
               (projT1 (l x)))
            (final_descriptors'
               (projT1 (l x)))
          ∧ match oitem with
            | Some itemx =>
                ∀ sx : composite_state IM,
                  sx =
                  equivocators_state_project
                    IM final_descriptors'
                    (finite_trace_last is
                      tr)
                  → composite_valid IM
                      (l itemx)
                      (
                      sx, input itemx)
                    ∧ composite_transition
                      IM 
                      (l itemx)
                      (sx, input itemx) =
                      (
                      destination itemx,
                      output itemx)
            | None =>
                equivocators_state_project
                  IM final_descriptors
                  (destination x) =
                equivocators_state_project
                  IM final_descriptors'
                  (finite_trace_last is tr)
            end
∃ (trX : list (composite_transition_item IM)) (initial_descriptors : 
                                               equivocator_descriptors
                                                 IM),
  equivocators_trace_project final_descriptors
    (tr ++ [x]) = Some (trX, initial_descriptors)
  ∧ proper_equivocator_descriptors IM
      initial_descriptors is
    ∧ equivocators_state_project IM final_descriptors
        (finite_trace_last is (tr ++ [x])) =
      finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX
  unfold equivocators_trace_project.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
x: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
is: composite_state (equivocator_IM IM)
IHtr: ∀ final_descriptors : equivocator_descriptors
                      IM,
  proper_equivocator_descriptors IM
    final_descriptors (finite_trace_last is tr)
  → ∃ (trX : list
               (composite_transition_item IM)) 
      (initial_descriptors : 
       equivocator_descriptors IM),
      equivocators_trace_project
        final_descriptors tr =
      Some (trX, initial_descriptors)
      ∧ proper_equivocator_descriptors IM
          initial_descriptors is
        ∧ equivocators_state_project IM
            final_descriptors
            (finite_trace_last is tr) =
          finite_trace_last
            (equivocators_state_project IM
               initial_descriptors is) trX
final_state:= finite_trace_last is (tr ++ [x]): state
  (composite_type (equivocator_IM IM))
Htr: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) is tr
Hx: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE)
  (finite_trace_last is tr) [x]
final_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  final_descriptors (destination x)
oitem: option (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hprojectx: equivocators_transition_item_project
  final_descriptors x =
Some (oitem, final_descriptors')
Hitemx: match oitem with
| Some itemx =>
    (∃ Hex : existing_equivocator_label
               (IM (projT1 (l x)))
               (projT2 (l x)),
       existT (projT1 (l x))
         (existing_equivocator_label_extract
            (IM (projT1 (l x)))
            (projT2 (l x)) Hex) = 
       l itemx)
    ∧ input x = input itemx
      ∧ output x = output itemx
        ∧ equivocators_state_project IM
            final_descriptors 
            (destination x) =
          destination itemx
          ∧ final_descriptors' (projT1 (l x)) =
            equivocator_label_descriptor
              (projT2 (l x))
| None => True
end
Hproperx: composite_valid (equivocator_IM IM) 
  (l x) (finite_trace_last is tr, input x)
→ composite_transition 
    (equivocator_IM IM) 
    (l x)
    (finite_trace_last is tr, input x) =
  (destination x, output x)
  → proper_equivocator_descriptors IM
      final_descriptors'
      (finite_trace_last is tr)
    ∧ final_descriptors' =
      equivocator_descriptors_update IM
        final_descriptors 
        (projT1 (l x))
        (final_descriptors' (projT1 (l x)))
      ∧ finite_trace_last is tr =
        state_update (equivocator_IM IM)
          (destination x) 
          (projT1 (l x))
          (finite_trace_last is tr
             (projT1 (l x)))
        ∧ previous_state_descriptor_prop
            (IM (projT1 (l x)))
            (final_descriptors
               (projT1 (l x)))
            (finite_trace_last is tr
               (projT1 (l x)))
            (final_descriptors'
               (projT1 (l x)))
          ∧ match oitem with
            | Some itemx =>
                ∀ sx : composite_state IM,
                  sx =
                  equivocators_state_project
                    IM final_descriptors'
                    (finite_trace_last is
                      tr)
                  → composite_valid IM
                      (l itemx)
                      (
                      sx, input itemx)
                    ∧ composite_transition
                      IM 
                      (l itemx)
                      (sx, input itemx) =
                      (
                      destination itemx,
                      output itemx)
            | None =>
                equivocators_state_project
                  IM final_descriptors
                  (destination x) =
                equivocators_state_project
                  IM final_descriptors'
                  (finite_trace_last is tr)
            end
∃ (trX : list (composite_transition_item IM)) (initial_descriptors : 
                                               equivocator_descriptors
                                                 IM),
  foldr equivocators_trace_project_folder
    (Some ([], final_descriptors)) (tr ++ [x]) =
  Some (trX, initial_descriptors)
  ∧ proper_equivocator_descriptors IM
      initial_descriptors is
    ∧ equivocators_state_project IM final_descriptors
        (finite_trace_last is (tr ++ [x])) =
      finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX
  rewrite fold_right_app.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
x: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
is: composite_state (equivocator_IM IM)
IHtr: ∀ final_descriptors : equivocator_descriptors
                      IM,
  proper_equivocator_descriptors IM
    final_descriptors (finite_trace_last is tr)
  → ∃ (trX : list
               (composite_transition_item IM)) 
      (initial_descriptors : 
       equivocator_descriptors IM),
      equivocators_trace_project
        final_descriptors tr =
      Some (trX, initial_descriptors)
      ∧ proper_equivocator_descriptors IM
          initial_descriptors is
        ∧ equivocators_state_project IM
            final_descriptors
            (finite_trace_last is tr) =
          finite_trace_last
            (equivocators_state_project IM
               initial_descriptors is) trX
final_state:= finite_trace_last is (tr ++ [x]): state
  (composite_type (equivocator_IM IM))
Htr: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) is tr
Hx: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE)
  (finite_trace_last is tr) [x]
final_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  final_descriptors (destination x)
oitem: option (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hprojectx: equivocators_transition_item_project
  final_descriptors x =
Some (oitem, final_descriptors')
Hitemx: match oitem with
| Some itemx =>
    (∃ Hex : existing_equivocator_label
               (IM (projT1 (l x)))
               (projT2 (l x)),
       existT (projT1 (l x))
         (existing_equivocator_label_extract
            (IM (projT1 (l x)))
            (projT2 (l x)) Hex) = 
       l itemx)
    ∧ input x = input itemx
      ∧ output x = output itemx
        ∧ equivocators_state_project IM
            final_descriptors 
            (destination x) =
          destination itemx
          ∧ final_descriptors' (projT1 (l x)) =
            equivocator_label_descriptor
              (projT2 (l x))
| None => True
end
Hproperx: composite_valid (equivocator_IM IM) 
  (l x) (finite_trace_last is tr, input x)
→ composite_transition 
    (equivocator_IM IM) 
    (l x)
    (finite_trace_last is tr, input x) =
  (destination x, output x)
  → proper_equivocator_descriptors IM
      final_descriptors'
      (finite_trace_last is tr)
    ∧ final_descriptors' =
      equivocator_descriptors_update IM
        final_descriptors 
        (projT1 (l x))
        (final_descriptors' (projT1 (l x)))
      ∧ finite_trace_last is tr =
        state_update (equivocator_IM IM)
          (destination x) 
          (projT1 (l x))
          (finite_trace_last is tr
             (projT1 (l x)))
        ∧ previous_state_descriptor_prop
            (IM (projT1 (l x)))
            (final_descriptors
               (projT1 (l x)))
            (finite_trace_last is tr
               (projT1 (l x)))
            (final_descriptors'
               (projT1 (l x)))
          ∧ match oitem with
            | Some itemx =>
                ∀ sx : composite_state IM,
                  sx =
                  equivocators_state_project
                    IM final_descriptors'
                    (finite_trace_last is
                      tr)
                  → composite_valid IM
                      (l itemx)
                      (
                      sx, input itemx)
                    ∧ composite_transition
                      IM 
                      (l itemx)
                      (sx, input itemx) =
                      (
                      destination itemx,
                      output itemx)
            | None =>
                equivocators_state_project
                  IM final_descriptors
                  (destination x) =
                equivocators_state_project
                  IM final_descriptors'
                  (finite_trace_last is tr)
            end
∃ (trX : list (composite_transition_item IM)) (initial_descriptors : 
                                               equivocator_descriptors
                                                 IM),
  foldr equivocators_trace_project_folder
    (foldr equivocators_trace_project_folder
       (Some ([], final_descriptors)) [x]) tr =
  Some (trX, initial_descriptors)
  ∧ proper_equivocator_descriptors IM
      initial_descriptors is
    ∧ equivocators_state_project IM final_descriptors
        (finite_trace_last is (tr ++ [x])) =
      finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX
  match goal with
  |- context [fold_right _ ?fld _] => remember fld as foldx
  end.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
x: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
is: composite_state (equivocator_IM IM)
IHtr: ∀ final_descriptors : equivocator_descriptors
                      IM,
  proper_equivocator_descriptors IM
    final_descriptors (finite_trace_last is tr)
  → ∃ (trX : list
               (composite_transition_item IM)) 
      (initial_descriptors : 
       equivocator_descriptors IM),
      equivocators_trace_project
        final_descriptors tr =
      Some (trX, initial_descriptors)
      ∧ proper_equivocator_descriptors IM
          initial_descriptors is
        ∧ equivocators_state_project IM
            final_descriptors
            (finite_trace_last is tr) =
          finite_trace_last
            (equivocators_state_project IM
               initial_descriptors is) trX
final_state:= finite_trace_last is (tr ++ [x]): state
  (composite_type (equivocator_IM IM))
Htr: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) is tr
Hx: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE)
  (finite_trace_last is tr) [x]
final_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  final_descriptors (destination x)
oitem: option (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hprojectx: equivocators_transition_item_project
  final_descriptors x =
Some (oitem, final_descriptors')
Hitemx: match oitem with
| Some itemx =>
    (∃ Hex : existing_equivocator_label
               (IM (projT1 (l x)))
               (projT2 (l x)),
       existT (projT1 (l x))
         (existing_equivocator_label_extract
            (IM (projT1 (l x)))
            (projT2 (l x)) Hex) = 
       l itemx)
    ∧ input x = input itemx
      ∧ output x = output itemx
        ∧ equivocators_state_project IM
            final_descriptors 
            (destination x) =
          destination itemx
          ∧ final_descriptors' (projT1 (l x)) =
            equivocator_label_descriptor
              (projT2 (l x))
| None => True
end
Hproperx: composite_valid (equivocator_IM IM) 
  (l x) (finite_trace_last is tr, input x)
→ composite_transition 
    (equivocator_IM IM) 
    (l x)
    (finite_trace_last is tr, input x) =
  (destination x, output x)
  → proper_equivocator_descriptors IM
      final_descriptors'
      (finite_trace_last is tr)
    ∧ final_descriptors' =
      equivocator_descriptors_update IM
        final_descriptors 
        (projT1 (l x))
        (final_descriptors' (projT1 (l x)))
      ∧ finite_trace_last is tr =
        state_update (equivocator_IM IM)
          (destination x) 
          (projT1 (l x))
          (finite_trace_last is tr
             (projT1 (l x)))
        ∧ previous_state_descriptor_prop
            (IM (projT1 (l x)))
            (final_descriptors
               (projT1 (l x)))
            (finite_trace_last is tr
               (projT1 (l x)))
            (final_descriptors'
               (projT1 (l x)))
          ∧ match oitem with
            | Some itemx =>
                ∀ sx : composite_state IM,
                  sx =
                  equivocators_state_project
                    IM final_descriptors'
                    (finite_trace_last is
                      tr)
                  → composite_valid IM
                      (l itemx)
                      (
                      sx, input itemx)
                    ∧ composite_transition
                      IM 
                      (l itemx)
                      (sx, input itemx) =
                      (
                      destination itemx,
                      output itemx)
            | None =>
                equivocators_state_project
                  IM final_descriptors
                  (destination x) =
                equivocators_state_project
                  IM final_descriptors'
                  (finite_trace_last is tr)
            end
foldx: option
  (list (composite_transition_item IM) *
   equivocator_descriptors IM)
Heqfoldx: foldx =
foldr equivocators_trace_project_folder
  (Some ([], final_descriptors)) [x]
∃ (trX : list (composite_transition_item IM)) (initial_descriptors : 
                                               equivocator_descriptors
                                                 IM),
  foldr equivocators_trace_project_folder foldx tr =
  Some (trX, initial_descriptors)
  ∧ proper_equivocator_descriptors IM
      initial_descriptors is
    ∧ equivocators_state_project IM final_descriptors
        (finite_trace_last is (tr ++ [x])) =
      finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX
  simpl in Heqfoldx.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
x: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
is: composite_state (equivocator_IM IM)
IHtr: ∀ final_descriptors : equivocator_descriptors
                      IM,
  proper_equivocator_descriptors IM
    final_descriptors (finite_trace_last is tr)
  → ∃ (trX : list
               (composite_transition_item IM)) 
      (initial_descriptors : 
       equivocator_descriptors IM),
      equivocators_trace_project
        final_descriptors tr =
      Some (trX, initial_descriptors)
      ∧ proper_equivocator_descriptors IM
          initial_descriptors is
        ∧ equivocators_state_project IM
            final_descriptors
            (finite_trace_last is tr) =
          finite_trace_last
            (equivocators_state_project IM
               initial_descriptors is) trX
final_state:= finite_trace_last is (tr ++ [x]): state
  (composite_type (equivocator_IM IM))
Htr: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) is tr
Hx: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE)
  (finite_trace_last is tr) [x]
final_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  final_descriptors (destination x)
oitem: option (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hprojectx: equivocators_transition_item_project
  final_descriptors x =
Some (oitem, final_descriptors')
Hitemx: match oitem with
| Some itemx =>
    (∃ Hex : existing_equivocator_label
               (IM (projT1 (l x)))
               (projT2 (l x)),
       existT (projT1 (l x))
         (existing_equivocator_label_extract
            (IM (projT1 (l x)))
            (projT2 (l x)) Hex) = 
       l itemx)
    ∧ input x = input itemx
      ∧ output x = output itemx
        ∧ equivocators_state_project IM
            final_descriptors 
            (destination x) =
          destination itemx
          ∧ final_descriptors' (projT1 (l x)) =
            equivocator_label_descriptor
              (projT2 (l x))
| None => True
end
Hproperx: composite_valid (equivocator_IM IM) 
  (l x) (finite_trace_last is tr, input x)
→ composite_transition 
    (equivocator_IM IM) 
    (l x)
    (finite_trace_last is tr, input x) =
  (destination x, output x)
  → proper_equivocator_descriptors IM
      final_descriptors'
      (finite_trace_last is tr)
    ∧ final_descriptors' =
      equivocator_descriptors_update IM
        final_descriptors 
        (projT1 (l x))
        (final_descriptors' (projT1 (l x)))
      ∧ finite_trace_last is tr =
        state_update (equivocator_IM IM)
          (destination x) 
          (projT1 (l x))
          (finite_trace_last is tr
             (projT1 (l x)))
        ∧ previous_state_descriptor_prop
            (IM (projT1 (l x)))
            (final_descriptors
               (projT1 (l x)))
            (finite_trace_last is tr
               (projT1 (l x)))
            (final_descriptors'
               (projT1 (l x)))
          ∧ match oitem with
            | Some itemx =>
                ∀ sx : composite_state IM,
                  sx =
                  equivocators_state_project
                    IM final_descriptors'
                    (finite_trace_last is
                      tr)
                  → composite_valid IM
                      (l itemx)
                      (
                      sx, input itemx)
                    ∧ composite_transition
                      IM 
                      (l itemx)
                      (sx, input itemx) =
                      (
                      destination itemx,
                      output itemx)
            | None =>
                equivocators_state_project
                  IM final_descriptors
                  (destination x) =
                equivocators_state_project
                  IM final_descriptors'
                  (finite_trace_last is tr)
            end
foldx: option
  (list (composite_transition_item IM) *
   equivocator_descriptors IM)
Heqfoldx: foldx =
match
  equivocators_transition_item_project
    final_descriptors x
with
| Some (Some item', odescriptor) =>
    Some ([item'], odescriptor)
| Some (None, odescriptor) =>
    Some ([], odescriptor)
| None => None
end
∃ (trX : list (composite_transition_item IM)) (initial_descriptors : 
                                               equivocator_descriptors
                                                 IM),
  foldr equivocators_trace_project_folder foldx tr =
  Some (trX, initial_descriptors)
  ∧ proper_equivocator_descriptors IM
      initial_descriptors is
    ∧ equivocators_state_project IM final_descriptors
        (finite_trace_last is (tr ++ [x])) =
      finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX
  rewrite Hprojectx in Heqfoldx.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
x: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
is: composite_state (equivocator_IM IM)
IHtr: ∀ final_descriptors : equivocator_descriptors
                      IM,
  proper_equivocator_descriptors IM
    final_descriptors (finite_trace_last is tr)
  → ∃ (trX : list
               (composite_transition_item IM)) 
      (initial_descriptors : 
       equivocator_descriptors IM),
      equivocators_trace_project
        final_descriptors tr =
      Some (trX, initial_descriptors)
      ∧ proper_equivocator_descriptors IM
          initial_descriptors is
        ∧ equivocators_state_project IM
            final_descriptors
            (finite_trace_last is tr) =
          finite_trace_last
            (equivocators_state_project IM
               initial_descriptors is) trX
final_state:= finite_trace_last is (tr ++ [x]): state
  (composite_type (equivocator_IM IM))
Htr: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) is tr
Hx: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE)
  (finite_trace_last is tr) [x]
final_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  final_descriptors (destination x)
oitem: option (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hprojectx: equivocators_transition_item_project
  final_descriptors x =
Some (oitem, final_descriptors')
Hitemx: match oitem with
| Some itemx =>
    (∃ Hex : existing_equivocator_label
               (IM (projT1 (l x)))
               (projT2 (l x)),
       existT (projT1 (l x))
         (existing_equivocator_label_extract
            (IM (projT1 (l x)))
            (projT2 (l x)) Hex) = 
       l itemx)
    ∧ input x = input itemx
      ∧ output x = output itemx
        ∧ equivocators_state_project IM
            final_descriptors 
            (destination x) =
          destination itemx
          ∧ final_descriptors' (projT1 (l x)) =
            equivocator_label_descriptor
              (projT2 (l x))
| None => True
end
Hproperx: composite_valid (equivocator_IM IM) 
  (l x) (finite_trace_last is tr, input x)
→ composite_transition 
    (equivocator_IM IM) 
    (l x)
    (finite_trace_last is tr, input x) =
  (destination x, output x)
  → proper_equivocator_descriptors IM
      final_descriptors'
      (finite_trace_last is tr)
    ∧ final_descriptors' =
      equivocator_descriptors_update IM
        final_descriptors 
        (projT1 (l x))
        (final_descriptors' (projT1 (l x)))
      ∧ finite_trace_last is tr =
        state_update (equivocator_IM IM)
          (destination x) 
          (projT1 (l x))
          (finite_trace_last is tr
             (projT1 (l x)))
        ∧ previous_state_descriptor_prop
            (IM (projT1 (l x)))
            (final_descriptors
               (projT1 (l x)))
            (finite_trace_last is tr
               (projT1 (l x)))
            (final_descriptors'
               (projT1 (l x)))
          ∧ match oitem with
            | Some itemx =>
                ∀ sx : composite_state IM,
                  sx =
                  equivocators_state_project
                    IM final_descriptors'
                    (finite_trace_last is
                      tr)
                  → composite_valid IM
                      (l itemx)
                      (
                      sx, input itemx)
                    ∧ composite_transition
                      IM 
                      (l itemx)
                      (sx, input itemx) =
                      (
                      destination itemx,
                      output itemx)
            | None =>
                equivocators_state_project
                  IM final_descriptors
                  (destination x) =
                equivocators_state_project
                  IM final_descriptors'
                  (finite_trace_last is tr)
            end
foldx: option
  (list (composite_transition_item IM) *
   equivocator_descriptors IM)
Heqfoldx: foldx =
match oitem with
| Some item' =>
    Some ([item'], final_descriptors')
| None => Some ([], final_descriptors')
end
∃ (trX : list (composite_transition_item IM)) (initial_descriptors : 
                                               equivocator_descriptors
                                                 IM),
  foldr equivocators_trace_project_folder foldx tr =
  Some (trX, initial_descriptors)
  ∧ proper_equivocator_descriptors IM
      initial_descriptors is
    ∧ equivocators_state_project IM final_descriptors
        (finite_trace_last is (tr ++ [x])) =
      finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX
  inversion Hx.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
x: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
is: composite_state (equivocator_IM IM)
IHtr: ∀ final_descriptors : equivocator_descriptors
                      IM,
  proper_equivocator_descriptors IM
    final_descriptors (finite_trace_last is tr)
  → ∃ (trX : list
               (composite_transition_item IM)) 
      (initial_descriptors : 
       equivocator_descriptors IM),
      equivocators_trace_project
        final_descriptors tr =
      Some (trX, initial_descriptors)
      ∧ proper_equivocator_descriptors IM
          initial_descriptors is
        ∧ equivocators_state_project IM
            final_descriptors
            (finite_trace_last is tr) =
          finite_trace_last
            (equivocators_state_project IM
               initial_descriptors is) trX
final_state:= finite_trace_last is (tr ++ [x]): state
  (composite_type (equivocator_IM IM))
Htr: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) is tr
Hx: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE)
  (finite_trace_last is tr) [x]
final_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  final_descriptors (destination x)
oitem: option (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hprojectx: equivocators_transition_item_project
  final_descriptors x =
Some (oitem, final_descriptors')
Hitemx: match oitem with
| Some itemx =>
    (∃ Hex : existing_equivocator_label
               (IM (projT1 (VLSM.l x)))
               (projT2 (VLSM.l x)),
       existT (projT1 (VLSM.l x))
         (existing_equivocator_label_extract
            (IM (projT1 (VLSM.l x)))
            (projT2 (VLSM.l x)) Hex) =
       VLSM.l itemx)
    ∧ input x = input itemx
      ∧ output x = output itemx
        ∧ equivocators_state_project IM
            final_descriptors 
            (destination x) =
          destination itemx
          ∧ final_descriptors'
              (projT1 (VLSM.l x)) =
            equivocator_label_descriptor
              (projT2 (VLSM.l x))
| None => True
end
Hproperx: composite_valid (equivocator_IM IM)
  (VLSM.l x)
  (finite_trace_last is tr, input x)
→ composite_transition 
    (equivocator_IM IM) 
    (VLSM.l x)
    (finite_trace_last is tr, input x) =
  (destination x, output x)
  → proper_equivocator_descriptors IM
      final_descriptors'
      (finite_trace_last is tr)
    ∧ final_descriptors' =
      equivocator_descriptors_update IM
        final_descriptors
        (projT1 (VLSM.l x))
        (final_descriptors'
           (projT1 (VLSM.l x)))
      ∧ finite_trace_last is tr =
        state_update (equivocator_IM IM)
          (destination x)
          (projT1 (VLSM.l x))
          (finite_trace_last is tr
             (projT1 (VLSM.l x)))
        ∧ previous_state_descriptor_prop
            (IM (projT1 (VLSM.l x)))
            (final_descriptors
               (projT1 (VLSM.l x)))
            (finite_trace_last is tr
               (projT1 (VLSM.l x)))
            (final_descriptors'
               (projT1 (VLSM.l x)))
          ∧ match oitem with
            | Some itemx =>
                ∀ sx : composite_state IM,
                  sx =
                  equivocators_state_project
                    IM final_descriptors'
                    (finite_trace_last is
                      tr)
                  → composite_valid IM
                      (VLSM.l itemx)
                      (
                      sx, input itemx)
                    ∧ composite_transition
                      IM 
                      (VLSM.l itemx)
                      (sx, input itemx) =
                      (
                      destination itemx,
                      output itemx)
            | None =>
                equivocators_state_project
                  IM final_descriptors
                  (destination x) =
                equivocators_state_project
                  IM final_descriptors'
                  (finite_trace_last is tr)
            end
foldx: option
  (list (composite_transition_item IM) *
   equivocator_descriptors IM)
Heqfoldx: foldx =
match oitem with
| Some item' =>
    Some ([item'], final_descriptors')
| None => Some ([], final_descriptors')
end
s: state (preloaded_with_all_messages_vlsm FreeE)
tl: list transition_item
s': state (preloaded_with_all_messages_vlsm FreeE)
iom, oom: option message
l: label (preloaded_with_all_messages_vlsm FreeE)
Htl: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) s []
Ht: input_valid_transition
  (preloaded_with_all_messages_vlsm FreeE) l
  (finite_trace_last is tr, iom) (
  s, oom)
H1: s' = finite_trace_last is tr
H3: {|
  l := l;
  input := iom;
  destination := s;
  output := oom
|} = x
H4: tl = []
∃ (trX : list (composite_transition_item IM)) (initial_descriptors : 
                                               equivocator_descriptors
                                                 IM),
  foldr equivocators_trace_project_folder foldx tr =
  Some (trX, initial_descriptors)
  ∧ proper_equivocator_descriptors IM
      initial_descriptors is
    ∧ equivocators_state_project IM final_descriptors
        (finite_trace_last is
           (tr ++
            [{|
               l := l;
               input := iom;
               destination := s;
               output := oom
             |}])) =
      finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX subst tl s' x.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
tr: list
  (composite_transition_item (equivocator_IM IM))
is: composite_state (equivocator_IM IM)
IHtr: ∀ final_descriptors : equivocator_descriptors
                      IM,
  proper_equivocator_descriptors IM
    final_descriptors (finite_trace_last is tr)
  → ∃ (trX : list
               (composite_transition_item IM)) 
      (initial_descriptors : 
       equivocator_descriptors IM),
      equivocators_trace_project
        final_descriptors tr =
      Some (trX, initial_descriptors)
      ∧ proper_equivocator_descriptors IM
          initial_descriptors is
        ∧ equivocators_state_project IM
            final_descriptors
            (finite_trace_last is tr) =
          finite_trace_last
            (equivocators_state_project IM
               initial_descriptors is) trX
s: state (preloaded_with_all_messages_vlsm FreeE)
iom, oom: option message
l: label (preloaded_with_all_messages_vlsm FreeE)
final_state:= finite_trace_last is
  (tr ++
   [{|
      l := l;
      input := iom;
      destination := s;
      output := oom
    |}]): state
  (composite_type (equivocator_IM IM))
Htr: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) is tr
Hx: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE)
  (finite_trace_last is tr)
  [{|
     l := l;
     input := iom;
     destination := s;
     output := oom
   |}]
final_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  final_descriptors
  (destination
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |})
oitem: option (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hproperx: composite_valid (equivocator_IM IM)
  (VLSM.l
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |})
  (finite_trace_last is tr,
   input
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |})
→ composite_transition 
    (equivocator_IM IM)
    (VLSM.l
       {|
         l := l;
         input := iom;
         destination := s;
         output := oom
       |})
    (finite_trace_last is tr,
     input
       {|
         l := l;
         input := iom;
         destination := s;
         output := oom
       |}) =
  (destination
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |},
   output
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |})
  → proper_equivocator_descriptors IM
      final_descriptors'
      (finite_trace_last is tr)
    ∧ final_descriptors' =
      equivocator_descriptors_update IM
        final_descriptors
        (projT1
           (VLSM.l
              {|
                l := l;
                input := iom;
                destination := s;
                output := oom
              |}))
        (final_descriptors'
           (projT1
              (VLSM.l
                 {|
                   l := l;
                   input := iom;
                   destination := s;
                   output := oom
                 |})))
      ∧ finite_trace_last is tr =
        state_update (equivocator_IM IM)
          (destination
             {|
               l := l;
               input := iom;
               destination := s;
               output := oom
             |})
          (projT1
             (VLSM.l
                {|
                  l := l;
                  input := iom;
                  destination := s;
                  output := oom
                |}))
          (finite_trace_last is tr
             (projT1
                (VLSM.l
                   {|
                     l := l;
                     input := iom;
                     destination := s;
                     output := oom
                   |})))
        ∧ previous_state_descriptor_prop
            (IM
               (projT1
                  (VLSM.l
                     {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                     |})))
            (final_descriptors
               (projT1
                  (VLSM.l
                     {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                     |})))
            (finite_trace_last is tr
               (projT1
                  (VLSM.l
                     {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                     |})))
            (final_descriptors'
               (projT1
                  (VLSM.l
                     {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                     |})))
          ∧ match oitem with
            | Some itemx =>
                ∀ sx : composite_state IM,
                  sx =
                  equivocators_state_project
                    IM final_descriptors'
                    (finite_trace_last is
                      tr)
                  → composite_valid IM
                      (VLSM.l itemx)
                      (
                      sx, input itemx)
                    ∧ composite_transition
                      IM 
                      (VLSM.l itemx)
                      (sx, input itemx) =
                      (
                      destination itemx,
                      output itemx)
            | None =>
                equivocators_state_project
                  IM final_descriptors
                  (destination
                     {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                     |}) =
                equivocators_state_project
                  IM final_descriptors'
                  (finite_trace_last is tr)
            end
Hitemx: match oitem with
| Some itemx =>
    (∃ Hex : existing_equivocator_label
               (IM
                  (projT1
                     (VLSM.l
                      {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                      |})))
               (projT2
                  (VLSM.l
                     {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                     |})),
       existT
         (projT1
            (VLSM.l
               {|
                 l := l;
                 input := iom;
                 destination := s;
                 output := oom
               |}))
         (existing_equivocator_label_extract
            (IM
               (projT1
                  (VLSM.l
                     {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                     |})))
            (projT2
               (VLSM.l
                  {|
                    l := l;
                    input := iom;
                    destination := s;
                    output := oom
                  |})) Hex) = 
       VLSM.l itemx)
    ∧ input
        {|
          l := l;
          input := iom;
          destination := s;
          output := oom
        |} = input itemx
      ∧ output
          {|
            l := l;
            input := iom;
            destination := s;
            output := oom
          |} = output itemx
        ∧ equivocators_state_project IM
            final_descriptors
            (destination
               {|
                 l := l;
                 input := iom;
                 destination := s;
                 output := oom
               |}) = destination itemx
          ∧ final_descriptors'
              (projT1
                 (VLSM.l
                    {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                    |})) =
            equivocator_label_descriptor
              (projT2
                 (VLSM.l
                    {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                    |}))
| None => True
end
Hprojectx: equivocators_transition_item_project
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some (oitem, final_descriptors')
foldx: option
  (list (composite_transition_item IM) *
   equivocator_descriptors IM)
Heqfoldx: foldx =
match oitem with
| Some item' =>
    Some ([item'], final_descriptors')
| None => Some ([], final_descriptors')
end
Htl: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) s []
Ht: input_valid_transition
  (preloaded_with_all_messages_vlsm FreeE) l
  (finite_trace_last is tr, iom) (
  s, oom)
∃ (trX : list (composite_transition_item IM)) (initial_descriptors : 
                                               equivocator_descriptors
                                                 IM),
  foldr equivocators_trace_project_folder foldx tr =
  Some (trX, initial_descriptors)
  ∧ proper_equivocator_descriptors IM
      initial_descriptors is
    ∧ equivocators_state_project IM final_descriptors
        (finite_trace_last is
           (tr ++
            [{|
               l := l;
               input := iom;
               destination := s;
               output := oom
             |}])) =
      finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX clear Hx.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
tr: list
  (composite_transition_item (equivocator_IM IM))
is: composite_state (equivocator_IM IM)
IHtr: ∀ final_descriptors : equivocator_descriptors
                      IM,
  proper_equivocator_descriptors IM
    final_descriptors (finite_trace_last is tr)
  → ∃ (trX : list
               (composite_transition_item IM)) 
      (initial_descriptors : 
       equivocator_descriptors IM),
      equivocators_trace_project
        final_descriptors tr =
      Some (trX, initial_descriptors)
      ∧ proper_equivocator_descriptors IM
          initial_descriptors is
        ∧ equivocators_state_project IM
            final_descriptors
            (finite_trace_last is tr) =
          finite_trace_last
            (equivocators_state_project IM
               initial_descriptors is) trX
s: state (preloaded_with_all_messages_vlsm FreeE)
iom, oom: option message
l: label (preloaded_with_all_messages_vlsm FreeE)
final_state:= finite_trace_last is
  (tr ++
   [{|
      l := l;
      input := iom;
      destination := s;
      output := oom
    |}]): state
  (composite_type (equivocator_IM IM))
Htr: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) is tr
final_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  final_descriptors
  (destination
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |})
oitem: option (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hproperx: composite_valid (equivocator_IM IM)
  (VLSM.l
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |})
  (finite_trace_last is tr,
   input
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |})
→ composite_transition 
    (equivocator_IM IM)
    (VLSM.l
       {|
         l := l;
         input := iom;
         destination := s;
         output := oom
       |})
    (finite_trace_last is tr,
     input
       {|
         l := l;
         input := iom;
         destination := s;
         output := oom
       |}) =
  (destination
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |},
   output
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |})
  → proper_equivocator_descriptors IM
      final_descriptors'
      (finite_trace_last is tr)
    ∧ final_descriptors' =
      equivocator_descriptors_update IM
        final_descriptors
        (projT1
           (VLSM.l
              {|
                l := l;
                input := iom;
                destination := s;
                output := oom
              |}))
        (final_descriptors'
           (projT1
              (VLSM.l
                 {|
                   l := l;
                   input := iom;
                   destination := s;
                   output := oom
                 |})))
      ∧ finite_trace_last is tr =
        state_update (equivocator_IM IM)
          (destination
             {|
               l := l;
               input := iom;
               destination := s;
               output := oom
             |})
          (projT1
             (VLSM.l
                {|
                  l := l;
                  input := iom;
                  destination := s;
                  output := oom
                |}))
          (finite_trace_last is tr
             (projT1
                (VLSM.l
                   {|
                     l := l;
                     input := iom;
                     destination := s;
                     output := oom
                   |})))
        ∧ previous_state_descriptor_prop
            (IM
               (projT1
                  (VLSM.l
                     {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                     |})))
            (final_descriptors
               (projT1
                  (VLSM.l
                     {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                     |})))
            (finite_trace_last is tr
               (projT1
                  (VLSM.l
                     {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                     |})))
            (final_descriptors'
               (projT1
                  (VLSM.l
                     {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                     |})))
          ∧ match oitem with
            | Some itemx =>
                ∀ sx : composite_state IM,
                  sx =
                  equivocators_state_project
                    IM final_descriptors'
                    (finite_trace_last is
                      tr)
                  → composite_valid IM
                      (VLSM.l itemx)
                      (
                      sx, input itemx)
                    ∧ composite_transition
                      IM 
                      (VLSM.l itemx)
                      (sx, input itemx) =
                      (
                      destination itemx,
                      output itemx)
            | None =>
                equivocators_state_project
                  IM final_descriptors
                  (destination
                     {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                     |}) =
                equivocators_state_project
                  IM final_descriptors'
                  (finite_trace_last is tr)
            end
Hitemx: match oitem with
| Some itemx =>
    (∃ Hex : existing_equivocator_label
               (IM
                  (projT1
                     (VLSM.l
                      {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                      |})))
               (projT2
                  (VLSM.l
                     {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                     |})),
       existT
         (projT1
            (VLSM.l
               {|
                 l := l;
                 input := iom;
                 destination := s;
                 output := oom
               |}))
         (existing_equivocator_label_extract
            (IM
               (projT1
                  (VLSM.l
                     {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                     |})))
            (projT2
               (VLSM.l
                  {|
                    l := l;
                    input := iom;
                    destination := s;
                    output := oom
                  |})) Hex) = 
       VLSM.l itemx)
    ∧ input
        {|
          l := l;
          input := iom;
          destination := s;
          output := oom
        |} = input itemx
      ∧ output
          {|
            l := l;
            input := iom;
            destination := s;
            output := oom
          |} = output itemx
        ∧ equivocators_state_project IM
            final_descriptors
            (destination
               {|
                 l := l;
                 input := iom;
                 destination := s;
                 output := oom
               |}) = destination itemx
          ∧ final_descriptors'
              (projT1
                 (VLSM.l
                    {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                    |})) =
            equivocator_label_descriptor
              (projT2
                 (VLSM.l
                    {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                    |}))
| None => True
end
Hprojectx: equivocators_transition_item_project
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some (oitem, final_descriptors')
foldx: option
  (list (composite_transition_item IM) *
   equivocator_descriptors IM)
Heqfoldx: foldx =
match oitem with
| Some item' =>
    Some ([item'], final_descriptors')
| None => Some ([], final_descriptors')
end
Htl: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) s []
Ht: input_valid_transition
  (preloaded_with_all_messages_vlsm FreeE) l
  (finite_trace_last is tr, iom) (
  s, oom)
∃ (trX : list (composite_transition_item IM)) (initial_descriptors : 
                                               equivocator_descriptors
                                                 IM),
  foldr equivocators_trace_project_folder foldx tr =
  Some (trX, initial_descriptors)
  ∧ proper_equivocator_descriptors IM
      initial_descriptors is
    ∧ equivocators_state_project IM final_descriptors
        (finite_trace_last is
           (tr ++
            [{|
               l := l;
               input := iom;
               destination := s;
               output := oom
             |}])) =
      finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX
  destruct Ht as [[_ [_ Hv]] Ht].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
tr: list
  (composite_transition_item (equivocator_IM IM))
is: composite_state (equivocator_IM IM)
IHtr: ∀ final_descriptors : equivocator_descriptors
                      IM,
  proper_equivocator_descriptors IM
    final_descriptors (finite_trace_last is tr)
  → ∃ (trX : list
               (composite_transition_item IM)) 
      (initial_descriptors : 
       equivocator_descriptors IM),
      equivocators_trace_project
        final_descriptors tr =
      Some (trX, initial_descriptors)
      ∧ proper_equivocator_descriptors IM
          initial_descriptors is
        ∧ equivocators_state_project IM
            final_descriptors
            (finite_trace_last is tr) =
          finite_trace_last
            (equivocators_state_project IM
               initial_descriptors is) trX
s: state (preloaded_with_all_messages_vlsm FreeE)
iom, oom: option message
l: label (preloaded_with_all_messages_vlsm FreeE)
final_state:= finite_trace_last is
  (tr ++
   [{|
      l := l;
      input := iom;
      destination := s;
      output := oom
    |}]): state
  (composite_type (equivocator_IM IM))
Htr: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) is tr
final_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  final_descriptors
  (destination
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |})
oitem: option (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hproperx: composite_valid (equivocator_IM IM)
  (VLSM.l
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |})
  (finite_trace_last is tr,
   input
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |})
→ composite_transition 
    (equivocator_IM IM)
    (VLSM.l
       {|
         l := l;
         input := iom;
         destination := s;
         output := oom
       |})
    (finite_trace_last is tr,
     input
       {|
         l := l;
         input := iom;
         destination := s;
         output := oom
       |}) =
  (destination
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |},
   output
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |})
  → proper_equivocator_descriptors IM
      final_descriptors'
      (finite_trace_last is tr)
    ∧ final_descriptors' =
      equivocator_descriptors_update IM
        final_descriptors
        (projT1
           (VLSM.l
              {|
                l := l;
                input := iom;
                destination := s;
                output := oom
              |}))
        (final_descriptors'
           (projT1
              (VLSM.l
                 {|
                   l := l;
                   input := iom;
                   destination := s;
                   output := oom
                 |})))
      ∧ finite_trace_last is tr =
        state_update (equivocator_IM IM)
          (destination
             {|
               l := l;
               input := iom;
               destination := s;
               output := oom
             |})
          (projT1
             (VLSM.l
                {|
                  l := l;
                  input := iom;
                  destination := s;
                  output := oom
                |}))
          (finite_trace_last is tr
             (projT1
                (VLSM.l
                   {|
                     l := l;
                     input := iom;
                     destination := s;
                     output := oom
                   |})))
        ∧ previous_state_descriptor_prop
            (IM
               (projT1
                  (VLSM.l
                     {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                     |})))
            (final_descriptors
               (projT1
                  (VLSM.l
                     {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                     |})))
            (finite_trace_last is tr
               (projT1
                  (VLSM.l
                     {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                     |})))
            (final_descriptors'
               (projT1
                  (VLSM.l
                     {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                     |})))
          ∧ match oitem with
            | Some itemx =>
                ∀ sx : composite_state IM,
                  sx =
                  equivocators_state_project
                    IM final_descriptors'
                    (finite_trace_last is
                      tr)
                  → composite_valid IM
                      (VLSM.l itemx)
                      (
                      sx, input itemx)
                    ∧ composite_transition
                      IM 
                      (VLSM.l itemx)
                      (sx, input itemx) =
                      (
                      destination itemx,
                      output itemx)
            | None =>
                equivocators_state_project
                  IM final_descriptors
                  (destination
                     {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                     |}) =
                equivocators_state_project
                  IM final_descriptors'
                  (finite_trace_last is tr)
            end
Hitemx: match oitem with
| Some itemx =>
    (∃ Hex : existing_equivocator_label
               (IM
                  (projT1
                     (VLSM.l
                      {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                      |})))
               (projT2
                  (VLSM.l
                     {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                     |})),
       existT
         (projT1
            (VLSM.l
               {|
                 l := l;
                 input := iom;
                 destination := s;
                 output := oom
               |}))
         (existing_equivocator_label_extract
            (IM
               (projT1
                  (VLSM.l
                     {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                     |})))
            (projT2
               (VLSM.l
                  {|
                    l := l;
                    input := iom;
                    destination := s;
                    output := oom
                  |})) Hex) = 
       VLSM.l itemx)
    ∧ input
        {|
          l := l;
          input := iom;
          destination := s;
          output := oom
        |} = input itemx
      ∧ output
          {|
            l := l;
            input := iom;
            destination := s;
            output := oom
          |} = output itemx
        ∧ equivocators_state_project IM
            final_descriptors
            (destination
               {|
                 l := l;
                 input := iom;
                 destination := s;
                 output := oom
               |}) = destination itemx
          ∧ final_descriptors'
              (projT1
                 (VLSM.l
                    {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                    |})) =
            equivocator_label_descriptor
              (projT2
                 (VLSM.l
                    {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                    |}))
| None => True
end
Hprojectx: equivocators_transition_item_project
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some (oitem, final_descriptors')
foldx: option
  (list (composite_transition_item IM) *
   equivocator_descriptors IM)
Heqfoldx: foldx =
match oitem with
| Some item' =>
    Some ([item'], final_descriptors')
| None => Some ([], final_descriptors')
end
Htl: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) s []
Hv: valid l (finite_trace_last is tr, iom)
Ht: transition l (finite_trace_last is tr, iom) =
(s, oom)
∃ (trX : list (composite_transition_item IM)) (initial_descriptors : 
                                               equivocator_descriptors
                                                 IM),
  foldr equivocators_trace_project_folder foldx tr =
  Some (trX, initial_descriptors)
  ∧ proper_equivocator_descriptors IM
      initial_descriptors is
    ∧ equivocators_state_project IM final_descriptors
        (finite_trace_last is
           (tr ++
            [{|
               l := l;
               input := iom;
               destination := s;
               output := oom
             |}])) =
      finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX
  specialize (Hproperx Hv Ht).message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
tr: list
  (composite_transition_item (equivocator_IM IM))
is: composite_state (equivocator_IM IM)
IHtr: ∀ final_descriptors : equivocator_descriptors
                      IM,
  proper_equivocator_descriptors IM
    final_descriptors (finite_trace_last is tr)
  → ∃ (trX : list
               (composite_transition_item IM)) 
      (initial_descriptors : 
       equivocator_descriptors IM),
      equivocators_trace_project
        final_descriptors tr =
      Some (trX, initial_descriptors)
      ∧ proper_equivocator_descriptors IM
          initial_descriptors is
        ∧ equivocators_state_project IM
            final_descriptors
            (finite_trace_last is tr) =
          finite_trace_last
            (equivocators_state_project IM
               initial_descriptors is) trX
s: state (preloaded_with_all_messages_vlsm FreeE)
iom, oom: option message
l: label (preloaded_with_all_messages_vlsm FreeE)
final_state:= finite_trace_last is
  (tr ++
   [{|
      l := l;
      input := iom;
      destination := s;
      output := oom
    |}]): state
  (composite_type (equivocator_IM IM))
Htr: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) is tr
final_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  final_descriptors
  (destination
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |})
oitem: option (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hproperx: proper_equivocator_descriptors IM
  final_descriptors'
  (finite_trace_last is tr)
∧ final_descriptors' =
  equivocator_descriptors_update IM
    final_descriptors
    (projT1
       (VLSM.l
          {|
            l := l;
            input := iom;
            destination := s;
            output := oom
          |}))
    (final_descriptors'
       (projT1
          (VLSM.l
             {|
               l := l;
               input := iom;
               destination := s;
               output := oom
             |})))
  ∧ finite_trace_last is tr =
    state_update (equivocator_IM IM)
      (destination
         {|
           l := l;
           input := iom;
           destination := s;
           output := oom
         |})
      (projT1
         (VLSM.l
            {|
              l := l;
              input := iom;
              destination := s;
              output := oom
            |}))
      (finite_trace_last is tr
         (projT1
            (VLSM.l
               {|
                 l := l;
                 input := iom;
                 destination := s;
                 output := oom
               |})))
    ∧ previous_state_descriptor_prop
        (IM
           (projT1
              (VLSM.l
                 {|
                   l := l;
                   input := iom;
                   destination := s;
                   output := oom
                 |})))
        (final_descriptors
           (projT1
              (VLSM.l
                 {|
                   l := l;
                   input := iom;
                   destination := s;
                   output := oom
                 |})))
        (finite_trace_last is tr
           (projT1
              (VLSM.l
                 {|
                   l := l;
                   input := iom;
                   destination := s;
                   output := oom
                 |})))
        (final_descriptors'
           (projT1
              (VLSM.l
                 {|
                   l := l;
                   input := iom;
                   destination := s;
                   output := oom
                 |})))
      ∧ match oitem with
        | Some itemx =>
            ∀ sx : composite_state IM,
              sx =
              equivocators_state_project IM
                final_descriptors'
                (finite_trace_last is tr)
              → composite_valid IM
                  (VLSM.l itemx)
                  (sx, input itemx)
                ∧ composite_transition IM
                    (VLSM.l itemx)
                    (sx, input itemx) =
                  (destination itemx,
                   output itemx)
        | None =>
            equivocators_state_project IM
              final_descriptors
              (destination
                 {|
                   l := l;
                   input := iom;
                   destination := s;
                   output := oom
                 |}) =
            equivocators_state_project IM
              final_descriptors'
              (finite_trace_last is tr)
        end
Hitemx: match oitem with
| Some itemx =>
    (∃ Hex : existing_equivocator_label
               (IM
                  (projT1
                     (VLSM.l
                      {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                      |})))
               (projT2
                  (VLSM.l
                     {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                     |})),
       existT
         (projT1
            (VLSM.l
               {|
                 l := l;
                 input := iom;
                 destination := s;
                 output := oom
               |}))
         (existing_equivocator_label_extract
            (IM
               (projT1
                  (VLSM.l
                     {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                     |})))
            (projT2
               (VLSM.l
                  {|
                    l := l;
                    input := iom;
                    destination := s;
                    output := oom
                  |})) Hex) = 
       VLSM.l itemx)
    ∧ input
        {|
          l := l;
          input := iom;
          destination := s;
          output := oom
        |} = input itemx
      ∧ output
          {|
            l := l;
            input := iom;
            destination := s;
            output := oom
          |} = output itemx
        ∧ equivocators_state_project IM
            final_descriptors
            (destination
               {|
                 l := l;
                 input := iom;
                 destination := s;
                 output := oom
               |}) = destination itemx
          ∧ final_descriptors'
              (projT1
                 (VLSM.l
                    {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                    |})) =
            equivocator_label_descriptor
              (projT2
                 (VLSM.l
                    {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                    |}))
| None => True
end
Hprojectx: equivocators_transition_item_project
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some (oitem, final_descriptors')
foldx: option
  (list (composite_transition_item IM) *
   equivocator_descriptors IM)
Heqfoldx: foldx =
match oitem with
| Some item' =>
    Some ([item'], final_descriptors')
| None => Some ([], final_descriptors')
end
Htl: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) s []
Hv: valid l (finite_trace_last is tr, iom)
Ht: transition l (finite_trace_last is tr, iom) =
(s, oom)
∃ (trX : list (composite_transition_item IM)) (initial_descriptors : 
                                               equivocator_descriptors
                                                 IM),
  foldr equivocators_trace_project_folder foldx tr =
  Some (trX, initial_descriptors)
  ∧ proper_equivocator_descriptors IM
      initial_descriptors is
    ∧ equivocators_state_project IM final_descriptors
        (finite_trace_last is
           (tr ++
            [{|
               l := l;
               input := iom;
               destination := s;
               output := oom
             |}])) =
      finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX
  destruct Hproperx as [Hproper' [Heq_final_descriptors' [Heq_ltr [Hex_new Hx]]]].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
tr: list
  (composite_transition_item (equivocator_IM IM))
is: composite_state (equivocator_IM IM)
IHtr: ∀ final_descriptors : equivocator_descriptors
                      IM,
  proper_equivocator_descriptors IM
    final_descriptors (finite_trace_last is tr)
  → ∃ (trX : list
               (composite_transition_item IM)) 
      (initial_descriptors : 
       equivocator_descriptors IM),
      equivocators_trace_project
        final_descriptors tr =
      Some (trX, initial_descriptors)
      ∧ proper_equivocator_descriptors IM
          initial_descriptors is
        ∧ equivocators_state_project IM
            final_descriptors
            (finite_trace_last is tr) =
          finite_trace_last
            (equivocators_state_project IM
               initial_descriptors is) trX
s: state (preloaded_with_all_messages_vlsm FreeE)
iom, oom: option message
l: label (preloaded_with_all_messages_vlsm FreeE)
final_state:= finite_trace_last is
  (tr ++
   [{|
      l := l;
      input := iom;
      destination := s;
      output := oom
    |}]): state
  (composite_type (equivocator_IM IM))
Htr: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) is tr
final_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  final_descriptors
  (destination
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |})
oitem: option (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hproper': proper_equivocator_descriptors IM
  final_descriptors'
  (finite_trace_last is tr)
Heq_final_descriptors': final_descriptors' =
equivocator_descriptors_update
  IM final_descriptors
  (projT1
     (VLSM.l
        {|
          l := l;
          input := iom;
          destination := s;
          output := oom
        |}))
  (final_descriptors'
     (projT1
        (VLSM.l
           {|
             l := l;
             input := iom;
             destination := s;
             output := oom
           |})))
Heq_ltr: finite_trace_last is tr =
state_update (equivocator_IM IM)
  (destination
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |})
  (projT1
     (VLSM.l
        {|
          l := l;
          input := iom;
          destination := s;
          output := oom
        |}))
  (finite_trace_last is tr
     (projT1
        (VLSM.l
           {|
             l := l;
             input := iom;
             destination := s;
             output := oom
           |})))
Hex_new: previous_state_descriptor_prop
  (IM
     (projT1
        (VLSM.l
           {|
             l := l;
             input := iom;
             destination := s;
             output := oom
           |})))
  (final_descriptors
     (projT1
        (VLSM.l
           {|
             l := l;
             input := iom;
             destination := s;
             output := oom
           |})))
  (finite_trace_last is tr
     (projT1
        (VLSM.l
           {|
             l := l;
             input := iom;
             destination := s;
             output := oom
           |})))
  (final_descriptors'
     (projT1
        (VLSM.l
           {|
             l := l;
             input := iom;
             destination := s;
             output := oom
           |})))
Hx: match oitem with
| Some itemx =>
    ∀ sx : composite_state IM,
      sx =
      equivocators_state_project IM
        final_descriptors'
        (finite_trace_last is tr)
      → composite_valid IM 
          (VLSM.l itemx) (
          sx, input itemx)
        ∧ composite_transition IM 
            (VLSM.l itemx) (
            sx, input itemx) =
          (destination itemx, output itemx)
| None =>
    equivocators_state_project IM
      final_descriptors
      (destination
         {|
           l := l;
           input := iom;
           destination := s;
           output := oom
         |}) =
    equivocators_state_project IM
      final_descriptors'
      (finite_trace_last is tr)
end
Hitemx: match oitem with
| Some itemx =>
    (∃ Hex : existing_equivocator_label
               (IM
                  (projT1
                     (VLSM.l
                      {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                      |})))
               (projT2
                  (VLSM.l
                     {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                     |})),
       existT
         (projT1
            (VLSM.l
               {|
                 l := l;
                 input := iom;
                 destination := s;
                 output := oom
               |}))
         (existing_equivocator_label_extract
            (IM
               (projT1
                  (VLSM.l
                     {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                     |})))
            (projT2
               (VLSM.l
                  {|
                    l := l;
                    input := iom;
                    destination := s;
                    output := oom
                  |})) Hex) = 
       VLSM.l itemx)
    ∧ input
        {|
          l := l;
          input := iom;
          destination := s;
          output := oom
        |} = input itemx
      ∧ output
          {|
            l := l;
            input := iom;
            destination := s;
            output := oom
          |} = output itemx
        ∧ equivocators_state_project IM
            final_descriptors
            (destination
               {|
                 l := l;
                 input := iom;
                 destination := s;
                 output := oom
               |}) = destination itemx
          ∧ final_descriptors'
              (projT1
                 (VLSM.l
                    {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                    |})) =
            equivocator_label_descriptor
              (projT2
                 (VLSM.l
                    {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                    |}))
| None => True
end
Hprojectx: equivocators_transition_item_project
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some (oitem, final_descriptors')
foldx: option
  (list (composite_transition_item IM) *
   equivocator_descriptors IM)
Heqfoldx: foldx =
match oitem with
| Some item' =>
    Some ([item'], final_descriptors')
| None => Some ([], final_descriptors')
end
Htl: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) s []
Hv: valid l (finite_trace_last is tr, iom)
Ht: transition l (finite_trace_last is tr, iom) =
(s, oom)
∃ (trX : list (composite_transition_item IM)) (initial_descriptors : 
                                               equivocator_descriptors
                                                 IM),
  foldr equivocators_trace_project_folder foldx tr =
  Some (trX, initial_descriptors)
  ∧ proper_equivocator_descriptors IM
      initial_descriptors is
    ∧ equivocators_state_project IM final_descriptors
        (finite_trace_last is
           (tr ++
            [{|
               l := l;
               input := iom;
               destination := s;
               output := oom
             |}])) =
      finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX
  specialize (IHtr _ Hproper').message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
tr: list
  (composite_transition_item (equivocator_IM IM))
is: composite_state (equivocator_IM IM)
final_descriptors': equivocator_descriptors IM
IHtr: ∃ (trX : list (composite_transition_item IM)) 
  (initial_descriptors : 
   equivocator_descriptors IM),
  equivocators_trace_project final_descriptors'
    tr = Some (trX, initial_descriptors)
  ∧ proper_equivocator_descriptors IM
      initial_descriptors is
    ∧ equivocators_state_project IM
        final_descriptors'
        (finite_trace_last is tr) =
      finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX
s: state (preloaded_with_all_messages_vlsm FreeE)
iom, oom: option message
l: label (preloaded_with_all_messages_vlsm FreeE)
final_state:= finite_trace_last is
  (tr ++
   [{|
      l := l;
      input := iom;
      destination := s;
      output := oom
    |}]): state
  (composite_type (equivocator_IM IM))
Htr: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) is tr
final_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  final_descriptors
  (destination
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |})
oitem: option (composite_transition_item IM)
Hproper': proper_equivocator_descriptors IM
  final_descriptors'
  (finite_trace_last is tr)
Heq_final_descriptors': final_descriptors' =
equivocator_descriptors_update
  IM final_descriptors
  (projT1
     (VLSM.l
        {|
          l := l;
          input := iom;
          destination := s;
          output := oom
        |}))
  (final_descriptors'
     (projT1
        (VLSM.l
           {|
             l := l;
             input := iom;
             destination := s;
             output := oom
           |})))
Heq_ltr: finite_trace_last is tr =
state_update (equivocator_IM IM)
  (destination
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |})
  (projT1
     (VLSM.l
        {|
          l := l;
          input := iom;
          destination := s;
          output := oom
        |}))
  (finite_trace_last is tr
     (projT1
        (VLSM.l
           {|
             l := l;
             input := iom;
             destination := s;
             output := oom
           |})))
Hex_new: previous_state_descriptor_prop
  (IM
     (projT1
        (VLSM.l
           {|
             l := l;
             input := iom;
             destination := s;
             output := oom
           |})))
  (final_descriptors
     (projT1
        (VLSM.l
           {|
             l := l;
             input := iom;
             destination := s;
             output := oom
           |})))
  (finite_trace_last is tr
     (projT1
        (VLSM.l
           {|
             l := l;
             input := iom;
             destination := s;
             output := oom
           |})))
  (final_descriptors'
     (projT1
        (VLSM.l
           {|
             l := l;
             input := iom;
             destination := s;
             output := oom
           |})))
Hx: match oitem with
| Some itemx =>
    ∀ sx : composite_state IM,
      sx =
      equivocators_state_project IM
        final_descriptors'
        (finite_trace_last is tr)
      → composite_valid IM 
          (VLSM.l itemx) (
          sx, input itemx)
        ∧ composite_transition IM 
            (VLSM.l itemx) (
            sx, input itemx) =
          (destination itemx, output itemx)
| None =>
    equivocators_state_project IM
      final_descriptors
      (destination
         {|
           l := l;
           input := iom;
           destination := s;
           output := oom
         |}) =
    equivocators_state_project IM
      final_descriptors'
      (finite_trace_last is tr)
end
Hitemx: match oitem with
| Some itemx =>
    (∃ Hex : existing_equivocator_label
               (IM
                  (projT1
                     (VLSM.l
                      {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                      |})))
               (projT2
                  (VLSM.l
                     {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                     |})),
       existT
         (projT1
            (VLSM.l
               {|
                 l := l;
                 input := iom;
                 destination := s;
                 output := oom
               |}))
         (existing_equivocator_label_extract
            (IM
               (projT1
                  (VLSM.l
                     {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                     |})))
            (projT2
               (VLSM.l
                  {|
                    l := l;
                    input := iom;
                    destination := s;
                    output := oom
                  |})) Hex) = 
       VLSM.l itemx)
    ∧ input
        {|
          l := l;
          input := iom;
          destination := s;
          output := oom
        |} = input itemx
      ∧ output
          {|
            l := l;
            input := iom;
            destination := s;
            output := oom
          |} = output itemx
        ∧ equivocators_state_project IM
            final_descriptors
            (destination
               {|
                 l := l;
                 input := iom;
                 destination := s;
                 output := oom
               |}) = destination itemx
          ∧ final_descriptors'
              (projT1
                 (VLSM.l
                    {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                    |})) =
            equivocator_label_descriptor
              (projT2
                 (VLSM.l
                    {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                    |}))
| None => True
end
Hprojectx: equivocators_transition_item_project
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some (oitem, final_descriptors')
foldx: option
  (list (composite_transition_item IM) *
   equivocator_descriptors IM)
Heqfoldx: foldx =
match oitem with
| Some item' =>
    Some ([item'], final_descriptors')
| None => Some ([], final_descriptors')
end
Htl: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) s []
Hv: valid l (finite_trace_last is tr, iom)
Ht: transition l (finite_trace_last is tr, iom) =
(s, oom)
∃ (trX : list (composite_transition_item IM)) (initial_descriptors : 
                                               equivocator_descriptors
                                                 IM),
  foldr equivocators_trace_project_folder foldx tr =
  Some (trX, initial_descriptors)
  ∧ proper_equivocator_descriptors IM
      initial_descriptors is
    ∧ equivocators_state_project IM final_descriptors
        (finite_trace_last is
           (tr ++
            [{|
               l := l;
               input := iom;
               destination := s;
               output := oom
             |}])) =
      finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX
  destruct IHtr as [trX' [initial_descriptors [Htr_project [Hproper_initial Hlst]]]].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
tr: list
  (composite_transition_item (equivocator_IM IM))
is: composite_state (equivocator_IM IM)
final_descriptors': equivocator_descriptors IM
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project
  final_descriptors' tr =
Some (trX', initial_descriptors)
Hproper_initial: proper_equivocator_descriptors IM
  initial_descriptors is
Hlst: equivocators_state_project IM
  final_descriptors' (finite_trace_last is tr) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
s: state (preloaded_with_all_messages_vlsm FreeE)
iom, oom: option message
l: label (preloaded_with_all_messages_vlsm FreeE)
final_state:= finite_trace_last is
  (tr ++
   [{|
      l := l;
      input := iom;
      destination := s;
      output := oom
    |}]): state
  (composite_type (equivocator_IM IM))
Htr: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) is tr
final_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  final_descriptors
  (destination
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |})
oitem: option (composite_transition_item IM)
Hproper': proper_equivocator_descriptors IM
  final_descriptors'
  (finite_trace_last is tr)
Heq_final_descriptors': final_descriptors' =
equivocator_descriptors_update
  IM final_descriptors
  (projT1
     (VLSM.l
        {|
          l := l;
          input := iom;
          destination := s;
          output := oom
        |}))
  (final_descriptors'
     (projT1
        (VLSM.l
           {|
             l := l;
             input := iom;
             destination := s;
             output := oom
           |})))
Heq_ltr: finite_trace_last is tr =
state_update (equivocator_IM IM)
  (destination
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |})
  (projT1
     (VLSM.l
        {|
          l := l;
          input := iom;
          destination := s;
          output := oom
        |}))
  (finite_trace_last is tr
     (projT1
        (VLSM.l
           {|
             l := l;
             input := iom;
             destination := s;
             output := oom
           |})))
Hex_new: previous_state_descriptor_prop
  (IM
     (projT1
        (VLSM.l
           {|
             l := l;
             input := iom;
             destination := s;
             output := oom
           |})))
  (final_descriptors
     (projT1
        (VLSM.l
           {|
             l := l;
             input := iom;
             destination := s;
             output := oom
           |})))
  (finite_trace_last is tr
     (projT1
        (VLSM.l
           {|
             l := l;
             input := iom;
             destination := s;
             output := oom
           |})))
  (final_descriptors'
     (projT1
        (VLSM.l
           {|
             l := l;
             input := iom;
             destination := s;
             output := oom
           |})))
Hx: match oitem with
| Some itemx =>
    ∀ sx : composite_state IM,
      sx =
      equivocators_state_project IM
        final_descriptors'
        (finite_trace_last is tr)
      → composite_valid IM 
          (VLSM.l itemx) (
          sx, input itemx)
        ∧ composite_transition IM 
            (VLSM.l itemx) (
            sx, input itemx) =
          (destination itemx, output itemx)
| None =>
    equivocators_state_project IM
      final_descriptors
      (destination
         {|
           l := l;
           input := iom;
           destination := s;
           output := oom
         |}) =
    equivocators_state_project IM
      final_descriptors'
      (finite_trace_last is tr)
end
Hitemx: match oitem with
| Some itemx =>
    (∃ Hex : existing_equivocator_label
               (IM
                  (projT1
                     (VLSM.l
                      {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                      |})))
               (projT2
                  (VLSM.l
                     {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                     |})),
       existT
         (projT1
            (VLSM.l
               {|
                 l := l;
                 input := iom;
                 destination := s;
                 output := oom
               |}))
         (existing_equivocator_label_extract
            (IM
               (projT1
                  (VLSM.l
                     {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                     |})))
            (projT2
               (VLSM.l
                  {|
                    l := l;
                    input := iom;
                    destination := s;
                    output := oom
                  |})) Hex) = 
       VLSM.l itemx)
    ∧ input
        {|
          l := l;
          input := iom;
          destination := s;
          output := oom
        |} = input itemx
      ∧ output
          {|
            l := l;
            input := iom;
            destination := s;
            output := oom
          |} = output itemx
        ∧ equivocators_state_project IM
            final_descriptors
            (destination
               {|
                 l := l;
                 input := iom;
                 destination := s;
                 output := oom
               |}) = destination itemx
          ∧ final_descriptors'
              (projT1
                 (VLSM.l
                    {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                    |})) =
            equivocator_label_descriptor
              (projT2
                 (VLSM.l
                    {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                    |}))
| None => True
end
Hprojectx: equivocators_transition_item_project
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some (oitem, final_descriptors')
foldx: option
  (list (composite_transition_item IM) *
   equivocator_descriptors IM)
Heqfoldx: foldx =
match oitem with
| Some item' =>
    Some ([item'], final_descriptors')
| None => Some ([], final_descriptors')
end
Htl: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) s []
Hv: valid l (finite_trace_last is tr, iom)
Ht: transition l (finite_trace_last is tr, iom) =
(s, oom)
∃ (trX : list (composite_transition_item IM)) (initial_descriptors : 
                                               equivocator_descriptors
                                                 IM),
  foldr equivocators_trace_project_folder foldx tr =
  Some (trX, initial_descriptors)
  ∧ proper_equivocator_descriptors IM
      initial_descriptors is
    ∧ equivocators_state_project IM final_descriptors
        (finite_trace_last is
           (tr ++
            [{|
               l := l;
               input := iom;
               destination := s;
               output := oom
             |}])) =
      finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX
  destruct oitem as [item |].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
tr: list
  (composite_transition_item (equivocator_IM IM))
is: composite_state (equivocator_IM IM)
final_descriptors': equivocator_descriptors IM
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project
  final_descriptors' tr =
Some (trX', initial_descriptors)
Hproper_initial: proper_equivocator_descriptors IM
  initial_descriptors is
Hlst: equivocators_state_project IM
  final_descriptors' (finite_trace_last is tr) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
s: state (preloaded_with_all_messages_vlsm FreeE)
iom, oom: option message
l: label (preloaded_with_all_messages_vlsm FreeE)
final_state:= finite_trace_last is
  (tr ++
   [{|
      l := l;
      input := iom;
      destination := s;
      output := oom
    |}]): state
  (composite_type (equivocator_IM IM))
Htr: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) is tr
final_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  final_descriptors
  (destination
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |})
item: composite_transition_item IM
Hproper': proper_equivocator_descriptors IM
  final_descriptors'
  (finite_trace_last is tr)
Heq_final_descriptors': final_descriptors' =
equivocator_descriptors_update
  IM final_descriptors
  (projT1
     (VLSM.l
        {|
          l := l;
          input := iom;
          destination := s;
          output := oom
        |}))
  (final_descriptors'
     (projT1
        (VLSM.l
           {|
             l := l;
             input := iom;
             destination := s;
             output := oom
           |})))
Heq_ltr: finite_trace_last is tr =
state_update (equivocator_IM IM)
  (destination
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |})
  (projT1
     (VLSM.l
        {|
          l := l;
          input := iom;
          destination := s;
          output := oom
        |}))
  (finite_trace_last is tr
     (projT1
        (VLSM.l
           {|
             l := l;
             input := iom;
             destination := s;
             output := oom
           |})))
Hex_new: previous_state_descriptor_prop
  (IM
     (projT1
        (VLSM.l
           {|
             l := l;
             input := iom;
             destination := s;
             output := oom
           |})))
  (final_descriptors
     (projT1
        (VLSM.l
           {|
             l := l;
             input := iom;
             destination := s;
             output := oom
           |})))
  (finite_trace_last is tr
     (projT1
        (VLSM.l
           {|
             l := l;
             input := iom;
             destination := s;
             output := oom
           |})))
  (final_descriptors'
     (projT1
        (VLSM.l
           {|
             l := l;
             input := iom;
             destination := s;
             output := oom
           |})))
Hx: ∀ sx : composite_state IM,
  sx =
  equivocators_state_project IM
    final_descriptors' 
    (finite_trace_last is tr)
  → composite_valid IM 
      (VLSM.l item) (sx, input item)
    ∧ composite_transition IM 
        (VLSM.l item) (
        sx, input item) =
      (destination item, output item)
Hitemx: (∃ Hex : existing_equivocator_label
           (IM
              (projT1
                 (VLSM.l
                    {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                    |})))
           (projT2
              (VLSM.l
                 {|
                   l := l;
                   input := iom;
                   destination := s;
                   output := oom
                 |})),
   existT
     (projT1
        (VLSM.l
           {|
             l := l;
             input := iom;
             destination := s;
             output := oom
           |}))
     (existing_equivocator_label_extract
        (IM
           (projT1
              (VLSM.l
                 {|
                   l := l;
                   input := iom;
                   destination := s;
                   output := oom
                 |})))
        (projT2
           (VLSM.l
              {|
                l := l;
                input := iom;
                destination := s;
                output := oom
              |})) Hex) = 
   VLSM.l item)
∧ input
    {|
      l := l;
      input := iom;
      destination := s;
      output := oom
    |} = input item
  ∧ output
      {|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |} = output item
    ∧ equivocators_state_project IM
        final_descriptors
        (destination
           {|
             l := l;
             input := iom;
             destination := s;
             output := oom
           |}) = destination item
      ∧ final_descriptors'
          (projT1
             (VLSM.l
                {|
                  l := l;
                  input := iom;
                  destination := s;
                  output := oom
                |})) =
        equivocator_label_descriptor
          (projT2
             (VLSM.l
                {|
                  l := l;
                  input := iom;
                  destination := s;
                  output := oom
                |}))
Hprojectx: equivocators_transition_item_project
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} =
Some (Some item, final_descriptors')
foldx: option
  (list (composite_transition_item IM) *
   equivocator_descriptors IM)
Heqfoldx: foldx = Some ([item], final_descriptors')
Htl: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) s []
Hv: valid l (finite_trace_last is tr, iom)
Ht: transition l (finite_trace_last is tr, iom) =
(s, oom)
∃ (trX : list (composite_transition_item IM)) (initial_descriptors : 
                                               equivocator_descriptors
                                                 IM),
  foldr equivocators_trace_project_folder foldx tr =
  Some (trX, initial_descriptors)
  ∧ proper_equivocator_descriptors IM
      initial_descriptors is
    ∧ equivocators_state_project IM final_descriptors
        (finite_trace_last is
           (tr ++
            [{|
               l := l;
               input := iom;
               destination := s;
               output := oom
             |}])) =
      finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX
message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
tr: list
  (composite_transition_item (equivocator_IM IM))
is: composite_state (equivocator_IM IM)
final_descriptors': equivocator_descriptors IM
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project
  final_descriptors' tr =
Some (trX', initial_descriptors)
Hproper_initial: proper_equivocator_descriptors IM
  initial_descriptors is
Hlst: equivocators_state_project IM
  final_descriptors' (finite_trace_last is tr) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
s: state (preloaded_with_all_messages_vlsm FreeE)
iom, oom: option message
l: label (preloaded_with_all_messages_vlsm FreeE)
final_state:= finite_trace_last is
  (tr ++
   [{|
      l := l;
      input := iom;
      destination := s;
      output := oom
    |}]): state
  (composite_type (equivocator_IM IM))
Htr: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) is tr
final_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  final_descriptors
  (destination
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |})
Hproper': proper_equivocator_descriptors IM
  final_descriptors'
  (finite_trace_last is tr)
Heq_final_descriptors': final_descriptors' =
equivocator_descriptors_update
  IM final_descriptors
  (projT1
     (VLSM.l
        {|
          l := l;
          input := iom;
          destination := s;
          output := oom
        |}))
  (final_descriptors'
     (projT1
        (VLSM.l
           {|
             l := l;
             input := iom;
             destination := s;
             output := oom
           |})))
Heq_ltr: finite_trace_last is tr =
state_update (equivocator_IM IM)
  (destination
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |})
  (projT1
     (VLSM.l
        {|
          l := l;
          input := iom;
          destination := s;
          output := oom
        |}))
  (finite_trace_last is tr
     (projT1
        (VLSM.l
           {|
             l := l;
             input := iom;
             destination := s;
             output := oom
           |})))
Hex_new: previous_state_descriptor_prop
  (IM
     (projT1
        (VLSM.l
           {|
             l := l;
             input := iom;
             destination := s;
             output := oom
           |})))
  (final_descriptors
     (projT1
        (VLSM.l
           {|
             l := l;
             input := iom;
             destination := s;
             output := oom
           |})))
  (finite_trace_last is tr
     (projT1
        (VLSM.l
           {|
             l := l;
             input := iom;
             destination := s;
             output := oom
           |})))
  (final_descriptors'
     (projT1
        (VLSM.l
           {|
             l := l;
             input := iom;
             destination := s;
             output := oom
           |})))
Hx: equivocators_state_project IM final_descriptors
  (destination
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |}) =
equivocators_state_project IM final_descriptors'
  (finite_trace_last is tr)
Hitemx: True
Hprojectx: equivocators_transition_item_project
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some (None, final_descriptors')
foldx: option
  (list (composite_transition_item IM) *
   equivocator_descriptors IM)
Heqfoldx: foldx = Some ([], final_descriptors')
Htl: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) s []
Hv: valid l (finite_trace_last is tr, iom)
Ht: transition l (finite_trace_last is tr, iom) =
(s, oom)
∃ (trX : list (composite_transition_item IM)) 
  (initial_descriptors : 
   equivocator_descriptors IM),
  foldr equivocators_trace_project_folder foldx tr =
  Some (trX, initial_descriptors)
  ∧ proper_equivocator_descriptors IM
      initial_descriptors is
    ∧ equivocators_state_project IM final_descriptors
        (finite_trace_last is
           (tr ++
            [{|
               l := l;
               input := iom;
               destination := s;
               output := oom
             |}])) =
      finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX
  -message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
tr: list
  (composite_transition_item (equivocator_IM IM))
is: composite_state (equivocator_IM IM)
final_descriptors': equivocator_descriptors IM
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project
  final_descriptors' tr =
Some (trX', initial_descriptors)
Hproper_initial: proper_equivocator_descriptors IM
  initial_descriptors is
Hlst: equivocators_state_project IM
  final_descriptors' (finite_trace_last is tr) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
s: state (preloaded_with_all_messages_vlsm FreeE)
iom, oom: option message
l: label (preloaded_with_all_messages_vlsm FreeE)
final_state:= finite_trace_last is
  (tr ++
   [{|
      l := l;
      input := iom;
      destination := s;
      output := oom
    |}]): state
  (composite_type (equivocator_IM IM))
Htr: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) is tr
final_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  final_descriptors
  (destination
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |})
item: composite_transition_item IM
Hproper': proper_equivocator_descriptors IM
  final_descriptors'
  (finite_trace_last is tr)
Heq_final_descriptors': final_descriptors' =
equivocator_descriptors_update
  IM final_descriptors
  (projT1
     (VLSM.l
        {|
          l := l;
          input := iom;
          destination := s;
          output := oom
        |}))
  (final_descriptors'
     (projT1
        (VLSM.l
           {|
             l := l;
             input := iom;
             destination := s;
             output := oom
           |})))
Heq_ltr: finite_trace_last is tr =
state_update (equivocator_IM IM)
  (destination
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |})
  (projT1
     (VLSM.l
        {|
          l := l;
          input := iom;
          destination := s;
          output := oom
        |}))
  (finite_trace_last is tr
     (projT1
        (VLSM.l
           {|
             l := l;
             input := iom;
             destination := s;
             output := oom
           |})))
Hex_new: previous_state_descriptor_prop
  (IM
     (projT1
        (VLSM.l
           {|
             l := l;
             input := iom;
             destination := s;
             output := oom
           |})))
  (final_descriptors
     (projT1
        (VLSM.l
           {|
             l := l;
             input := iom;
             destination := s;
             output := oom
           |})))
  (finite_trace_last is tr
     (projT1
        (VLSM.l
           {|
             l := l;
             input := iom;
             destination := s;
             output := oom
           |})))
  (final_descriptors'
     (projT1
        (VLSM.l
           {|
             l := l;
             input := iom;
             destination := s;
             output := oom
           |})))
Hx: ∀ sx : composite_state IM,
  sx =
  equivocators_state_project IM
    final_descriptors' 
    (finite_trace_last is tr)
  → composite_valid IM 
      (VLSM.l item) (sx, input item)
    ∧ composite_transition IM 
        (VLSM.l item) (
        sx, input item) =
      (destination item, output item)
Hitemx: (∃ Hex : existing_equivocator_label
           (IM
              (projT1
                 (VLSM.l
                    {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                    |})))
           (projT2
              (VLSM.l
                 {|
                   l := l;
                   input := iom;
                   destination := s;
                   output := oom
                 |})),
   existT
     (projT1
        (VLSM.l
           {|
             l := l;
             input := iom;
             destination := s;
             output := oom
           |}))
     (existing_equivocator_label_extract
        (IM
           (projT1
              (VLSM.l
                 {|
                   l := l;
                   input := iom;
                   destination := s;
                   output := oom
                 |})))
        (projT2
           (VLSM.l
              {|
                l := l;
                input := iom;
                destination := s;
                output := oom
              |})) Hex) = 
   VLSM.l item)
∧ input
    {|
      l := l;
      input := iom;
      destination := s;
      output := oom
    |} = input item
  ∧ output
      {|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |} = output item
    ∧ equivocators_state_project IM
        final_descriptors
        (destination
           {|
             l := l;
             input := iom;
             destination := s;
             output := oom
           |}) = destination item
      ∧ final_descriptors'
          (projT1
             (VLSM.l
                {|
                  l := l;
                  input := iom;
                  destination := s;
                  output := oom
                |})) =
        equivocator_label_descriptor
          (projT2
             (VLSM.l
                {|
                  l := l;
                  input := iom;
                  destination := s;
                  output := oom
                |}))
Hprojectx: equivocators_transition_item_project
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} =
Some (Some item, final_descriptors')
foldx: option
  (list (composite_transition_item IM) *
   equivocator_descriptors IM)
Heqfoldx: foldx = Some ([item], final_descriptors')
Htl: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) s []
Hv: valid l (finite_trace_last is tr, iom)
Ht: transition l (finite_trace_last is tr, iom) =
(s, oom)
∃ (trX : list (composite_transition_item IM)) (initial_descriptors : 
                                               equivocator_descriptors
                                                 IM),
  foldr equivocators_trace_project_folder foldx tr =
  Some (trX, initial_descriptors)
  ∧ proper_equivocator_descriptors IM
      initial_descriptors is
    ∧ equivocators_state_project IM final_descriptors
        (finite_trace_last is
           (tr ++
            [{|
               l := l;
               input := iom;
               destination := s;
               output := oom
             |}])) =
      finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX simpl in Hitemx.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
tr: list
  (composite_transition_item (equivocator_IM IM))
is: composite_state (equivocator_IM IM)
final_descriptors': equivocator_descriptors IM
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project
  final_descriptors' tr =
Some (trX', initial_descriptors)
Hproper_initial: proper_equivocator_descriptors IM
  initial_descriptors is
Hlst: equivocators_state_project IM
  final_descriptors' (finite_trace_last is tr) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
s: state (preloaded_with_all_messages_vlsm FreeE)
iom, oom: option message
l: label (preloaded_with_all_messages_vlsm FreeE)
final_state:= finite_trace_last is
  (tr ++
   [{|
      l := l;
      input := iom;
      destination := s;
      output := oom
    |}]): state
  (composite_type (equivocator_IM IM))
Htr: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) is tr
final_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  final_descriptors
  (destination
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |})
item: composite_transition_item IM
Hproper': proper_equivocator_descriptors IM
  final_descriptors'
  (finite_trace_last is tr)
Heq_final_descriptors': final_descriptors' =
equivocator_descriptors_update
  IM final_descriptors
  (projT1
     (VLSM.l
        {|
          l := l;
          input := iom;
          destination := s;
          output := oom
        |}))
  (final_descriptors'
     (projT1
        (VLSM.l
           {|
             l := l;
             input := iom;
             destination := s;
             output := oom
           |})))
Heq_ltr: finite_trace_last is tr =
state_update (equivocator_IM IM)
  (destination
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |})
  (projT1
     (VLSM.l
        {|
          l := l;
          input := iom;
          destination := s;
          output := oom
        |}))
  (finite_trace_last is tr
     (projT1
        (VLSM.l
           {|
             l := l;
             input := iom;
             destination := s;
             output := oom
           |})))
Hex_new: previous_state_descriptor_prop
  (IM
     (projT1
        (VLSM.l
           {|
             l := l;
             input := iom;
             destination := s;
             output := oom
           |})))
  (final_descriptors
     (projT1
        (VLSM.l
           {|
             l := l;
             input := iom;
             destination := s;
             output := oom
           |})))
  (finite_trace_last is tr
     (projT1
        (VLSM.l
           {|
             l := l;
             input := iom;
             destination := s;
             output := oom
           |})))
  (final_descriptors'
     (projT1
        (VLSM.l
           {|
             l := l;
             input := iom;
             destination := s;
             output := oom
           |})))
Hx: ∀ sx : composite_state IM,
  sx =
  equivocators_state_project IM
    final_descriptors' 
    (finite_trace_last is tr)
  → composite_valid IM 
      (VLSM.l item) (sx, input item)
    ∧ composite_transition IM 
        (VLSM.l item) (
        sx, input item) =
      (destination item, output item)
Hitemx: (∃ Hex : existing_equivocator_label
           (IM (projT1 l)) 
           (projT2 l),
   existT (projT1 l)
     (existing_equivocator_label_extract
        (IM (projT1 l)) 
        (projT2 l) Hex) = 
   VLSM.l item)
∧ iom = input item
  ∧ oom = output item
    ∧ equivocators_state_project IM
        final_descriptors s =
      destination item
      ∧ final_descriptors' (projT1 l) =
        equivocator_label_descriptor
          (projT2 l)
Hprojectx: equivocators_transition_item_project
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} =
Some (Some item, final_descriptors')
foldx: option
  (list (composite_transition_item IM) *
   equivocator_descriptors IM)
Heqfoldx: foldx = Some ([item], final_descriptors')
Htl: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) s []
Hv: valid l (finite_trace_last is tr, iom)
Ht: transition l (finite_trace_last is tr, iom) =
(s, oom)
∃ (trX : list (composite_transition_item IM)) (initial_descriptors : 
                                               equivocator_descriptors
                                                 IM),
  foldr equivocators_trace_project_folder foldx tr =
  Some (trX, initial_descriptors)
  ∧ proper_equivocator_descriptors IM
      initial_descriptors is
    ∧ equivocators_state_project IM final_descriptors
        (finite_trace_last is
           (tr ++
            [{|
               l := l;
               input := iom;
               destination := s;
               output := oom
             |}])) =
      finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX destruct Hitemx as [Hl [Hinput [Houtput [Hdestination _]]]].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
tr: list
  (composite_transition_item (equivocator_IM IM))
is: composite_state (equivocator_IM IM)
final_descriptors': equivocator_descriptors IM
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project
  final_descriptors' tr =
Some (trX', initial_descriptors)
Hproper_initial: proper_equivocator_descriptors IM
  initial_descriptors is
Hlst: equivocators_state_project IM
  final_descriptors' (finite_trace_last is tr) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
s: state (preloaded_with_all_messages_vlsm FreeE)
iom, oom: option message
l: label (preloaded_with_all_messages_vlsm FreeE)
final_state:= finite_trace_last is
  (tr ++
   [{|
      l := l;
      input := iom;
      destination := s;
      output := oom
    |}]): state
  (composite_type (equivocator_IM IM))
Htr: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) is tr
final_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  final_descriptors
  (destination
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |})
item: composite_transition_item IM
Hproper': proper_equivocator_descriptors IM
  final_descriptors'
  (finite_trace_last is tr)
Heq_final_descriptors': final_descriptors' =
equivocator_descriptors_update
  IM final_descriptors
  (projT1
     (VLSM.l
        {|
          l := l;
          input := iom;
          destination := s;
          output := oom
        |}))
  (final_descriptors'
     (projT1
        (VLSM.l
           {|
             l := l;
             input := iom;
             destination := s;
             output := oom
           |})))
Heq_ltr: finite_trace_last is tr =
state_update (equivocator_IM IM)
  (destination
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |})
  (projT1
     (VLSM.l
        {|
          l := l;
          input := iom;
          destination := s;
          output := oom
        |}))
  (finite_trace_last is tr
     (projT1
        (VLSM.l
           {|
             l := l;
             input := iom;
             destination := s;
             output := oom
           |})))
Hex_new: previous_state_descriptor_prop
  (IM
     (projT1
        (VLSM.l
           {|
             l := l;
             input := iom;
             destination := s;
             output := oom
           |})))
  (final_descriptors
     (projT1
        (VLSM.l
           {|
             l := l;
             input := iom;
             destination := s;
             output := oom
           |})))
  (finite_trace_last is tr
     (projT1
        (VLSM.l
           {|
             l := l;
             input := iom;
             destination := s;
             output := oom
           |})))
  (final_descriptors'
     (projT1
        (VLSM.l
           {|
             l := l;
             input := iom;
             destination := s;
             output := oom
           |})))
Hx: ∀ sx : composite_state IM,
  sx =
  equivocators_state_project IM
    final_descriptors' 
    (finite_trace_last is tr)
  → composite_valid IM 
      (VLSM.l item) (sx, input item)
    ∧ composite_transition IM 
        (VLSM.l item) (
        sx, input item) =
      (destination item, output item)
Hl: ∃ Hex : existing_equivocator_label
          (IM (projT1 l)) 
          (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM (projT1 l)) 
       (projT2 l) Hex) = 
  VLSM.l item
Hinput: iom = input item
Houtput: oom = output item
Hdestination: equivocators_state_project IM
  final_descriptors s =
destination item
Hprojectx: equivocators_transition_item_project
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} =
Some (Some item, final_descriptors')
foldx: option
  (list (composite_transition_item IM) *
   equivocator_descriptors IM)
Heqfoldx: foldx = Some ([item], final_descriptors')
Htl: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) s []
Hv: valid l (finite_trace_last is tr, iom)
Ht: transition l (finite_trace_last is tr, iom) =
(s, oom)
∃ (trX : list (composite_transition_item IM)) (initial_descriptors : 
                                               equivocator_descriptors
                                                 IM),
  foldr equivocators_trace_project_folder foldx tr =
  Some (trX, initial_descriptors)
  ∧ proper_equivocator_descriptors IM
      initial_descriptors is
    ∧ equivocators_state_project IM final_descriptors
        (finite_trace_last is
           (tr ++
            [{|
               l := l;
               input := iom;
               destination := s;
               output := oom
             |}])) =
      finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX
    specialize (Hx _ eq_refl).message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
tr: list
  (composite_transition_item (equivocator_IM IM))
is: composite_state (equivocator_IM IM)
final_descriptors': equivocator_descriptors IM
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project
  final_descriptors' tr =
Some (trX', initial_descriptors)
Hproper_initial: proper_equivocator_descriptors IM
  initial_descriptors is
Hlst: equivocators_state_project IM
  final_descriptors' (finite_trace_last is tr) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
s: state (preloaded_with_all_messages_vlsm FreeE)
iom, oom: option message
l: label (preloaded_with_all_messages_vlsm FreeE)
final_state:= finite_trace_last is
  (tr ++
   [{|
      l := l;
      input := iom;
      destination := s;
      output := oom
    |}]): state
  (composite_type (equivocator_IM IM))
Htr: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) is tr
final_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  final_descriptors
  (destination
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |})
item: composite_transition_item IM
Hproper': proper_equivocator_descriptors IM
  final_descriptors'
  (finite_trace_last is tr)
Heq_final_descriptors': final_descriptors' =
equivocator_descriptors_update
  IM final_descriptors
  (projT1
     (VLSM.l
        {|
          l := l;
          input := iom;
          destination := s;
          output := oom
        |}))
  (final_descriptors'
     (projT1
        (VLSM.l
           {|
             l := l;
             input := iom;
             destination := s;
             output := oom
           |})))
Heq_ltr: finite_trace_last is tr =
state_update (equivocator_IM IM)
  (destination
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |})
  (projT1
     (VLSM.l
        {|
          l := l;
          input := iom;
          destination := s;
          output := oom
        |}))
  (finite_trace_last is tr
     (projT1
        (VLSM.l
           {|
             l := l;
             input := iom;
             destination := s;
             output := oom
           |})))
Hex_new: previous_state_descriptor_prop
  (IM
     (projT1
        (VLSM.l
           {|
             l := l;
             input := iom;
             destination := s;
             output := oom
           |})))
  (final_descriptors
     (projT1
        (VLSM.l
           {|
             l := l;
             input := iom;
             destination := s;
             output := oom
           |})))
  (finite_trace_last is tr
     (projT1
        (VLSM.l
           {|
             l := l;
             input := iom;
             destination := s;
             output := oom
           |})))
  (final_descriptors'
     (projT1
        (VLSM.l
           {|
             l := l;
             input := iom;
             destination := s;
             output := oom
           |})))
Hx: composite_valid IM (VLSM.l item)
  (equivocators_state_project IM
     final_descriptors' 
     (finite_trace_last is tr), 
   input item)
∧ composite_transition IM 
    (VLSM.l item)
    (equivocators_state_project IM
       final_descriptors'
       (finite_trace_last is tr), 
     input item) =
  (destination item, output item)
Hl: ∃ Hex : existing_equivocator_label
          (IM (projT1 l)) 
          (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM (projT1 l)) 
       (projT2 l) Hex) = 
  VLSM.l item
Hinput: iom = input item
Houtput: oom = output item
Hdestination: equivocators_state_project IM
  final_descriptors s =
destination item
Hprojectx: equivocators_transition_item_project
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} =
Some (Some item, final_descriptors')
foldx: option
  (list (composite_transition_item IM) *
   equivocator_descriptors IM)
Heqfoldx: foldx = Some ([item], final_descriptors')
Htl: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) s []
Hv: valid l (finite_trace_last is tr, iom)
Ht: transition l (finite_trace_last is tr, iom) =
(s, oom)
∃ (trX : list (composite_transition_item IM)) (initial_descriptors : 
                                               equivocator_descriptors
                                                 IM),
  foldr equivocators_trace_project_folder foldx tr =
  Some (trX, initial_descriptors)
  ∧ proper_equivocator_descriptors IM
      initial_descriptors is
    ∧ equivocators_state_project IM final_descriptors
        (finite_trace_last is
           (tr ++
            [{|
               l := l;
               input := iom;
               destination := s;
               output := oom
             |}])) =
      finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX
    destruct Hx as [Hvx Htx].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
tr: list
  (composite_transition_item (equivocator_IM IM))
is: composite_state (equivocator_IM IM)
final_descriptors': equivocator_descriptors IM
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project
  final_descriptors' tr =
Some (trX', initial_descriptors)
Hproper_initial: proper_equivocator_descriptors IM
  initial_descriptors is
Hlst: equivocators_state_project IM
  final_descriptors' (finite_trace_last is tr) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
s: state (preloaded_with_all_messages_vlsm FreeE)
iom, oom: option message
l: label (preloaded_with_all_messages_vlsm FreeE)
final_state:= finite_trace_last is
  (tr ++
   [{|
      l := l;
      input := iom;
      destination := s;
      output := oom
    |}]): state
  (composite_type (equivocator_IM IM))
Htr: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) is tr
final_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  final_descriptors
  (destination
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |})
item: composite_transition_item IM
Hproper': proper_equivocator_descriptors IM
  final_descriptors'
  (finite_trace_last is tr)
Heq_final_descriptors': final_descriptors' =
equivocator_descriptors_update
  IM final_descriptors
  (projT1
     (VLSM.l
        {|
          l := l;
          input := iom;
          destination := s;
          output := oom
        |}))
  (final_descriptors'
     (projT1
        (VLSM.l
           {|
             l := l;
             input := iom;
             destination := s;
             output := oom
           |})))
Heq_ltr: finite_trace_last is tr =
state_update (equivocator_IM IM)
  (destination
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |})
  (projT1
     (VLSM.l
        {|
          l := l;
          input := iom;
          destination := s;
          output := oom
        |}))
  (finite_trace_last is tr
     (projT1
        (VLSM.l
           {|
             l := l;
             input := iom;
             destination := s;
             output := oom
           |})))
Hex_new: previous_state_descriptor_prop
  (IM
     (projT1
        (VLSM.l
           {|
             l := l;
             input := iom;
             destination := s;
             output := oom
           |})))
  (final_descriptors
     (projT1
        (VLSM.l
           {|
             l := l;
             input := iom;
             destination := s;
             output := oom
           |})))
  (finite_trace_last is tr
     (projT1
        (VLSM.l
           {|
             l := l;
             input := iom;
             destination := s;
             output := oom
           |})))
  (final_descriptors'
     (projT1
        (VLSM.l
           {|
             l := l;
             input := iom;
             destination := s;
             output := oom
           |})))
Hvx: composite_valid IM (VLSM.l item)
  (equivocators_state_project IM
     final_descriptors'
     (finite_trace_last is tr), 
   input item)
Htx: composite_transition IM 
  (VLSM.l item)
  (equivocators_state_project IM
     final_descriptors'
     (finite_trace_last is tr), 
   input item) = (destination item, output item)
Hl: ∃ Hex : existing_equivocator_label
          (IM (projT1 l)) 
          (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM (projT1 l)) 
       (projT2 l) Hex) = 
  VLSM.l item
Hinput: iom = input item
Houtput: oom = output item
Hdestination: equivocators_state_project IM
  final_descriptors s =
destination item
Hprojectx: equivocators_transition_item_project
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} =
Some (Some item, final_descriptors')
foldx: option
  (list (composite_transition_item IM) *
   equivocator_descriptors IM)
Heqfoldx: foldx = Some ([item], final_descriptors')
Htl: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) s []
Hv: valid l (finite_trace_last is tr, iom)
Ht: transition l (finite_trace_last is tr, iom) =
(s, oom)
∃ (trX : list (composite_transition_item IM)) (initial_descriptors : 
                                               equivocator_descriptors
                                                 IM),
  foldr equivocators_trace_project_folder foldx tr =
  Some (trX, initial_descriptors)
  ∧ proper_equivocator_descriptors IM
      initial_descriptors is
    ∧ equivocators_state_project IM final_descriptors
        (finite_trace_last is
           (tr ++
            [{|
               l := l;
               input := iom;
               destination := s;
               output := oom
             |}])) =
      finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX
    exists (trX' ++ [item]), initial_descriptors.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
tr: list
  (composite_transition_item (equivocator_IM IM))
is: composite_state (equivocator_IM IM)
final_descriptors': equivocator_descriptors IM
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project
  final_descriptors' tr =
Some (trX', initial_descriptors)
Hproper_initial: proper_equivocator_descriptors IM
  initial_descriptors is
Hlst: equivocators_state_project IM
  final_descriptors' (finite_trace_last is tr) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
s: state (preloaded_with_all_messages_vlsm FreeE)
iom, oom: option message
l: label (preloaded_with_all_messages_vlsm FreeE)
final_state:= finite_trace_last is
  (tr ++
   [{|
      l := l;
      input := iom;
      destination := s;
      output := oom
    |}]): state
  (composite_type (equivocator_IM IM))
Htr: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) is tr
final_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  final_descriptors
  (destination
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |})
item: composite_transition_item IM
Hproper': proper_equivocator_descriptors IM
  final_descriptors'
  (finite_trace_last is tr)
Heq_final_descriptors': final_descriptors' =
equivocator_descriptors_update
  IM final_descriptors
  (projT1
     (VLSM.l
        {|
          l := l;
          input := iom;
          destination := s;
          output := oom
        |}))
  (final_descriptors'
     (projT1
        (VLSM.l
           {|
             l := l;
             input := iom;
             destination := s;
             output := oom
           |})))
Heq_ltr: finite_trace_last is tr =
state_update (equivocator_IM IM)
  (destination
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |})
  (projT1
     (VLSM.l
        {|
          l := l;
          input := iom;
          destination := s;
          output := oom
        |}))
  (finite_trace_last is tr
     (projT1
        (VLSM.l
           {|
             l := l;
             input := iom;
             destination := s;
             output := oom
           |})))
Hex_new: previous_state_descriptor_prop
  (IM
     (projT1
        (VLSM.l
           {|
             l := l;
             input := iom;
             destination := s;
             output := oom
           |})))
  (final_descriptors
     (projT1
        (VLSM.l
           {|
             l := l;
             input := iom;
             destination := s;
             output := oom
           |})))
  (finite_trace_last is tr
     (projT1
        (VLSM.l
           {|
             l := l;
             input := iom;
             destination := s;
             output := oom
           |})))
  (final_descriptors'
     (projT1
        (VLSM.l
           {|
             l := l;
             input := iom;
             destination := s;
             output := oom
           |})))
Hvx: composite_valid IM (VLSM.l item)
  (equivocators_state_project IM
     final_descriptors'
     (finite_trace_last is tr), 
   input item)
Htx: composite_transition IM 
  (VLSM.l item)
  (equivocators_state_project IM
     final_descriptors'
     (finite_trace_last is tr), 
   input item) = (destination item, output item)
Hl: ∃ Hex : existing_equivocator_label
          (IM (projT1 l)) 
          (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM (projT1 l)) 
       (projT2 l) Hex) = 
  VLSM.l item
Hinput: iom = input item
Houtput: oom = output item
Hdestination: equivocators_state_project IM
  final_descriptors s =
destination item
Hprojectx: equivocators_transition_item_project
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} =
Some (Some item, final_descriptors')
foldx: option
  (list (composite_transition_item IM) *
   equivocator_descriptors IM)
Heqfoldx: foldx = Some ([item], final_descriptors')
Htl: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) s []
Hv: valid l (finite_trace_last is tr, iom)
Ht: transition l (finite_trace_last is tr, iom) =
(s, oom)
foldr equivocators_trace_project_folder foldx tr =
Some (trX' ++ [item], initial_descriptors)
∧ proper_equivocator_descriptors IM
    initial_descriptors is
  ∧ equivocators_state_project IM final_descriptors
      (finite_trace_last is
         (tr ++
          [{|
             l := l;
             input := iom;
             destination := s;
             output := oom
           |}])) =
    finite_trace_last
      (equivocators_state_project IM
         initial_descriptors is) (trX' ++ [item]) subst foldx.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
tr: list
  (composite_transition_item (equivocator_IM IM))
is: composite_state (equivocator_IM IM)
final_descriptors': equivocator_descriptors IM
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project
  final_descriptors' tr =
Some (trX', initial_descriptors)
Hproper_initial: proper_equivocator_descriptors IM
  initial_descriptors is
Hlst: equivocators_state_project IM
  final_descriptors' (finite_trace_last is tr) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
s: state (preloaded_with_all_messages_vlsm FreeE)
iom, oom: option message
l: label (preloaded_with_all_messages_vlsm FreeE)
final_state:= finite_trace_last is
  (tr ++
   [{|
      l := l;
      input := iom;
      destination := s;
      output := oom
    |}]): state
  (composite_type (equivocator_IM IM))
Htr: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) is tr
final_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  final_descriptors
  (destination
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |})
item: composite_transition_item IM
Hproper': proper_equivocator_descriptors IM
  final_descriptors'
  (finite_trace_last is tr)
Heq_final_descriptors': final_descriptors' =
equivocator_descriptors_update
  IM final_descriptors
  (projT1
     (VLSM.l
        {|
          l := l;
          input := iom;
          destination := s;
          output := oom
        |}))
  (final_descriptors'
     (projT1
        (VLSM.l
           {|
             l := l;
             input := iom;
             destination := s;
             output := oom
           |})))
Heq_ltr: finite_trace_last is tr =
state_update (equivocator_IM IM)
  (destination
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |})
  (projT1
     (VLSM.l
        {|
          l := l;
          input := iom;
          destination := s;
          output := oom
        |}))
  (finite_trace_last is tr
     (projT1
        (VLSM.l
           {|
             l := l;
             input := iom;
             destination := s;
             output := oom
           |})))
Hex_new: previous_state_descriptor_prop
  (IM
     (projT1
        (VLSM.l
           {|
             l := l;
             input := iom;
             destination := s;
             output := oom
           |})))
  (final_descriptors
     (projT1
        (VLSM.l
           {|
             l := l;
             input := iom;
             destination := s;
             output := oom
           |})))
  (finite_trace_last is tr
     (projT1
        (VLSM.l
           {|
             l := l;
             input := iom;
             destination := s;
             output := oom
           |})))
  (final_descriptors'
     (projT1
        (VLSM.l
           {|
             l := l;
             input := iom;
             destination := s;
             output := oom
           |})))
Hvx: composite_valid IM (VLSM.l item)
  (equivocators_state_project IM
     final_descriptors'
     (finite_trace_last is tr), 
   input item)
Htx: composite_transition IM 
  (VLSM.l item)
  (equivocators_state_project IM
     final_descriptors'
     (finite_trace_last is tr), 
   input item) = (destination item, output item)
Hl: ∃ Hex : existing_equivocator_label
          (IM (projT1 l)) 
          (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM (projT1 l)) 
       (projT2 l) Hex) = 
  VLSM.l item
Hinput: iom = input item
Houtput: oom = output item
Hdestination: equivocators_state_project IM
  final_descriptors s =
destination item
Hprojectx: equivocators_transition_item_project
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} =
Some (Some item, final_descriptors')
Htl: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) s []
Hv: valid l (finite_trace_last is tr, iom)
Ht: transition l (finite_trace_last is tr, iom) =
(s, oom)
foldr equivocators_trace_project_folder
  (Some ([item], final_descriptors')) tr =
Some (trX' ++ [item], initial_descriptors)
∧ proper_equivocator_descriptors IM
    initial_descriptors is
  ∧ equivocators_state_project IM final_descriptors
      (finite_trace_last is
         (tr ++
          [{|
             l := l;
             input := iom;
             destination := s;
             output := oom
           |}])) =
    finite_trace_last
      (equivocators_state_project IM
         initial_descriptors is) (trX' ++ [item])
    by erewrite equivocators_trace_project_folder_additive, !finite_trace_last_is_last.
  -message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
tr: list
  (composite_transition_item (equivocator_IM IM))
is: composite_state (equivocator_IM IM)
final_descriptors': equivocator_descriptors IM
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project
  final_descriptors' tr =
Some (trX', initial_descriptors)
Hproper_initial: proper_equivocator_descriptors IM
  initial_descriptors is
Hlst: equivocators_state_project IM
  final_descriptors' (finite_trace_last is tr) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
s: state (preloaded_with_all_messages_vlsm FreeE)
iom, oom: option message
l: label (preloaded_with_all_messages_vlsm FreeE)
final_state:= finite_trace_last is
  (tr ++
   [{|
      l := l;
      input := iom;
      destination := s;
      output := oom
    |}]): state
  (composite_type (equivocator_IM IM))
Htr: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) is tr
final_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  final_descriptors
  (destination
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |})
Hproper': proper_equivocator_descriptors IM
  final_descriptors'
  (finite_trace_last is tr)
Heq_final_descriptors': final_descriptors' =
equivocator_descriptors_update
  IM final_descriptors
  (projT1
     (VLSM.l
        {|
          l := l;
          input := iom;
          destination := s;
          output := oom
        |}))
  (final_descriptors'
     (projT1
        (VLSM.l
           {|
             l := l;
             input := iom;
             destination := s;
             output := oom
           |})))
Heq_ltr: finite_trace_last is tr =
state_update (equivocator_IM IM)
  (destination
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |})
  (projT1
     (VLSM.l
        {|
          l := l;
          input := iom;
          destination := s;
          output := oom
        |}))
  (finite_trace_last is tr
     (projT1
        (VLSM.l
           {|
             l := l;
             input := iom;
             destination := s;
             output := oom
           |})))
Hex_new: previous_state_descriptor_prop
  (IM
     (projT1
        (VLSM.l
           {|
             l := l;
             input := iom;
             destination := s;
             output := oom
           |})))
  (final_descriptors
     (projT1
        (VLSM.l
           {|
             l := l;
             input := iom;
             destination := s;
             output := oom
           |})))
  (finite_trace_last is tr
     (projT1
        (VLSM.l
           {|
             l := l;
             input := iom;
             destination := s;
             output := oom
           |})))
  (final_descriptors'
     (projT1
        (VLSM.l
           {|
             l := l;
             input := iom;
             destination := s;
             output := oom
           |})))
Hx: equivocators_state_project IM final_descriptors
  (destination
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |}) =
equivocators_state_project IM final_descriptors'
  (finite_trace_last is tr)
Hitemx: True
Hprojectx: equivocators_transition_item_project
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some (None, final_descriptors')
foldx: option
  (list (composite_transition_item IM) *
   equivocator_descriptors IM)
Heqfoldx: foldx = Some ([], final_descriptors')
Htl: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) s []
Hv: valid l (finite_trace_last is tr, iom)
Ht: transition l (finite_trace_last is tr, iom) =
(s, oom)
∃ (trX : list (composite_transition_item IM)) (initial_descriptors : 
                                               equivocator_descriptors
                                                 IM),
  foldr equivocators_trace_project_folder foldx tr =
  Some (trX, initial_descriptors)
  ∧ proper_equivocator_descriptors IM
      initial_descriptors is
    ∧ equivocators_state_project IM final_descriptors
        (finite_trace_last is
           (tr ++
            [{|
               l := l;
               input := iom;
               destination := s;
               output := oom
             |}])) =
      finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX exists trX', initial_descriptors.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
tr: list
  (composite_transition_item (equivocator_IM IM))
is: composite_state (equivocator_IM IM)
final_descriptors': equivocator_descriptors IM
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project
  final_descriptors' tr =
Some (trX', initial_descriptors)
Hproper_initial: proper_equivocator_descriptors IM
  initial_descriptors is
Hlst: equivocators_state_project IM
  final_descriptors' (finite_trace_last is tr) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
s: state (preloaded_with_all_messages_vlsm FreeE)
iom, oom: option message
l: label (preloaded_with_all_messages_vlsm FreeE)
final_state:= finite_trace_last is
  (tr ++
   [{|
      l := l;
      input := iom;
      destination := s;
      output := oom
    |}]): state
  (composite_type (equivocator_IM IM))
Htr: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) is tr
final_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  final_descriptors
  (destination
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |})
Hproper': proper_equivocator_descriptors IM
  final_descriptors'
  (finite_trace_last is tr)
Heq_final_descriptors': final_descriptors' =
equivocator_descriptors_update
  IM final_descriptors
  (projT1
     (VLSM.l
        {|
          l := l;
          input := iom;
          destination := s;
          output := oom
        |}))
  (final_descriptors'
     (projT1
        (VLSM.l
           {|
             l := l;
             input := iom;
             destination := s;
             output := oom
           |})))
Heq_ltr: finite_trace_last is tr =
state_update (equivocator_IM IM)
  (destination
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |})
  (projT1
     (VLSM.l
        {|
          l := l;
          input := iom;
          destination := s;
          output := oom
        |}))
  (finite_trace_last is tr
     (projT1
        (VLSM.l
           {|
             l := l;
             input := iom;
             destination := s;
             output := oom
           |})))
Hex_new: previous_state_descriptor_prop
  (IM
     (projT1
        (VLSM.l
           {|
             l := l;
             input := iom;
             destination := s;
             output := oom
           |})))
  (final_descriptors
     (projT1
        (VLSM.l
           {|
             l := l;
             input := iom;
             destination := s;
             output := oom
           |})))
  (finite_trace_last is tr
     (projT1
        (VLSM.l
           {|
             l := l;
             input := iom;
             destination := s;
             output := oom
           |})))
  (final_descriptors'
     (projT1
        (VLSM.l
           {|
             l := l;
             input := iom;
             destination := s;
             output := oom
           |})))
Hx: equivocators_state_project IM final_descriptors
  (destination
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |}) =
equivocators_state_project IM final_descriptors'
  (finite_trace_last is tr)
Hitemx: True
Hprojectx: equivocators_transition_item_project
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some (None, final_descriptors')
foldx: option
  (list (composite_transition_item IM) *
   equivocator_descriptors IM)
Heqfoldx: foldx = Some ([], final_descriptors')
Htl: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) s []
Hv: valid l (finite_trace_last is tr, iom)
Ht: transition l (finite_trace_last is tr, iom) =
(s, oom)
foldr equivocators_trace_project_folder foldx tr =
Some (trX', initial_descriptors)
∧ proper_equivocator_descriptors IM
    initial_descriptors is
  ∧ equivocators_state_project IM final_descriptors
      (finite_trace_last is
         (tr ++
          [{|
             l := l;
             input := iom;
             destination := s;
             output := oom
           |}])) =
    finite_trace_last
      (equivocators_state_project IM
         initial_descriptors is) trX'
    subst; split_and!; [done .. |].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
tr: list
  (composite_transition_item (equivocator_IM IM))
is: composite_state (equivocator_IM IM)
final_descriptors': equivocator_descriptors IM
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project
  final_descriptors' tr =
Some (trX', initial_descriptors)
Hproper_initial: proper_equivocator_descriptors IM
  initial_descriptors is
Hlst: equivocators_state_project IM
  final_descriptors' (finite_trace_last is tr) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
s: state (preloaded_with_all_messages_vlsm FreeE)
iom, oom: option message
l: label (preloaded_with_all_messages_vlsm FreeE)
final_state:= finite_trace_last is
  (tr ++
   [{|
      l := l;
      input := iom;
      destination := s;
      output := oom
    |}]): state
  (composite_type (equivocator_IM IM))
Htr: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) is tr
final_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  final_descriptors
  (destination
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |})
Hproper': proper_equivocator_descriptors IM
  final_descriptors'
  (finite_trace_last is tr)
Heq_final_descriptors': final_descriptors' =
equivocator_descriptors_update
  IM final_descriptors
  (projT1
     (VLSM.l
        {|
          l := l;
          input := iom;
          destination := s;
          output := oom
        |}))
  (final_descriptors'
     (projT1
        (VLSM.l
           {|
             l := l;
             input := iom;
             destination := s;
             output := oom
           |})))
Heq_ltr: finite_trace_last is tr =
state_update (equivocator_IM IM)
  (destination
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |})
  (projT1
     (VLSM.l
        {|
          l := l;
          input := iom;
          destination := s;
          output := oom
        |}))
  (finite_trace_last is tr
     (projT1
        (VLSM.l
           {|
             l := l;
             input := iom;
             destination := s;
             output := oom
           |})))
Hex_new: previous_state_descriptor_prop
  (IM
     (projT1
        (VLSM.l
           {|
             l := l;
             input := iom;
             destination := s;
             output := oom
           |})))
  (final_descriptors
     (projT1
        (VLSM.l
           {|
             l := l;
             input := iom;
             destination := s;
             output := oom
           |})))
  (finite_trace_last is tr
     (projT1
        (VLSM.l
           {|
             l := l;
             input := iom;
             destination := s;
             output := oom
           |})))
  (final_descriptors'
     (projT1
        (VLSM.l
           {|
             l := l;
             input := iom;
             destination := s;
             output := oom
           |})))
Hx: equivocators_state_project IM final_descriptors
  (destination
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |}) =
equivocators_state_project IM final_descriptors'
  (finite_trace_last is tr)
Hitemx: True
Hprojectx: equivocators_transition_item_project
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some (None, final_descriptors')
Htl: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) s []
Hv: valid l (finite_trace_last is tr, iom)
Ht: transition l (finite_trace_last is tr, iom) =
(s, oom)
equivocators_state_project IM final_descriptors
  (finite_trace_last is
     (tr ++
      [{|
         l := l;
         input := iom;
         destination := s;
         output := oom
       |}])) =
finite_trace_last
  (equivocators_state_project IM initial_descriptors
     is) trX'
    by rewrite finite_trace_last_is_last; congruence.
Qed.

Lemma equivocators_trace_project_zero_descriptors
  (is : composite_state (equivocator_IM IM))
  (tr : list (composite_transition_item (equivocator_IM IM)))
  (Htr : finite_constrained_trace_from FreeE is tr)
  : exists (trX : list (composite_transition_item IM)),
    equivocators_trace_project (zero_descriptor IM) tr = Some (trX, (zero_descriptor IM)).message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_constrained_trace_from FreeE is tr
∃ trX : list (composite_transition_item IM),
  equivocators_trace_project (zero_descriptor IM) tr =
  Some (trX, zero_descriptor IM)
Proof.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_constrained_trace_from FreeE is tr
∃ trX : list (composite_transition_item IM),
  equivocators_trace_project (zero_descriptor IM) tr =
  Some (trX, zero_descriptor IM)
  specialize
    (preloaded_equivocators_valid_trace_from_project
      (zero_descriptor IM) is tr)
    as Hproject.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_constrained_trace_from FreeE is tr
Hproject: let final_state := finite_trace_last is tr
  in
proper_equivocator_descriptors IM
  (zero_descriptor IM) final_state
→ finite_constrained_trace_from FreeE is tr
  → ∃ (trX : list
               (composite_transition_item
                  IM)) 
      (initial_descriptors : 
       equivocator_descriptors IM),
      equivocators_trace_project
        (zero_descriptor IM) tr =
      Some (trX, initial_descriptors)
      ∧ proper_equivocator_descriptors IM
          initial_descriptors is
        ∧ equivocators_state_project IM
            (zero_descriptor IM)
            (finite_trace_last is tr) =
          finite_trace_last
            (equivocators_state_project IM
               initial_descriptors is) trX
∃ trX : list (composite_transition_item IM),
  equivocators_trace_project (zero_descriptor IM) tr =
  Some (trX, zero_descriptor IM)
  simpl in Hproject.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_constrained_trace_from FreeE is tr
Hproject: proper_equivocator_descriptors IM
  (zero_descriptor IM)
  (finite_trace_last is tr)
→ finite_constrained_trace_from FreeE is tr
  → ∃ (trX : list
               (composite_transition_item
                  IM)) 
      (initial_descriptors : 
       equivocator_descriptors IM),
      equivocators_trace_project
        (zero_descriptor IM) tr =
      Some (trX, initial_descriptors)
      ∧ proper_equivocator_descriptors IM
          initial_descriptors is
        ∧ equivocators_state_project IM
            (zero_descriptor IM)
            (finite_trace_last is tr) =
          finite_trace_last
            (equivocators_state_project IM
               initial_descriptors is) trX
∃ trX : list (composite_transition_item IM),
  equivocators_trace_project (zero_descriptor IM) tr =
  Some (trX, zero_descriptor IM) spec Hproject; [by apply zero_descriptor_proper |].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_constrained_trace_from FreeE is tr
Hproject: finite_constrained_trace_from FreeE is tr
→ ∃ (trX : list
             (composite_transition_item IM)) 
    (initial_descriptors : 
     equivocator_descriptors IM),
    equivocators_trace_project
      (zero_descriptor IM) tr =
    Some (trX, initial_descriptors)
    ∧ proper_equivocator_descriptors IM
        initial_descriptors is
      ∧ equivocators_state_project IM
          (zero_descriptor IM)
          (finite_trace_last is tr) =
        finite_trace_last
          (equivocators_state_project IM
             initial_descriptors is) trX
∃ trX : list (composite_transition_item IM),
  equivocators_trace_project (zero_descriptor IM) tr =
  Some (trX, zero_descriptor IM)
  specialize (Hproject Htr).message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_constrained_trace_from FreeE is tr
Hproject: ∃ (trX : list
           (composite_transition_item IM)) 
  (initial_descriptors : 
   equivocator_descriptors IM),
  equivocators_trace_project
    (zero_descriptor IM) tr =
  Some (trX, initial_descriptors)
  ∧ proper_equivocator_descriptors IM
      initial_descriptors is
    ∧ equivocators_state_project IM
        (zero_descriptor IM)
        (finite_trace_last is tr) =
      finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX
∃ trX : list (composite_transition_item IM),
  equivocators_trace_project (zero_descriptor IM) tr =
  Some (trX, zero_descriptor IM)
  destruct Hproject as [trX [initial_descriptors [Hproject _]]].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_constrained_trace_from FreeE is tr
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hproject: equivocators_trace_project
  (zero_descriptor IM) tr =
Some (trX, initial_descriptors)
∃ trX : list (composite_transition_item IM),
  equivocators_trace_project (zero_descriptor IM) tr =
  Some (trX, zero_descriptor IM)
  exists trX.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_constrained_trace_from FreeE is tr
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hproject: equivocators_trace_project
  (zero_descriptor IM) tr =
Some (trX, initial_descriptors)
equivocators_trace_project (zero_descriptor IM) tr =
Some (trX, zero_descriptor IM)
  replace initial_descriptors with (zero_descriptor IM) in Hproject; [done |].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_constrained_trace_from FreeE is tr
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hproject: equivocators_trace_project
  (zero_descriptor IM) tr =
Some (trX, initial_descriptors)
zero_descriptor IM = initial_descriptors
  apply functional_extensionality_dep.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_constrained_trace_from FreeE is tr
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hproject: equivocators_trace_project
  (zero_descriptor IM) tr =
Some (trX, initial_descriptors)
∀ x : index,
  zero_descriptor IM x = initial_descriptors x intros i.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_constrained_trace_from FreeE is tr
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hproject: equivocators_trace_project
  (zero_descriptor IM) tr =
Some (trX, initial_descriptors)
i: index
zero_descriptor IM i = initial_descriptors i symmetry.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_constrained_trace_from FreeE is tr
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hproject: equivocators_trace_project
  (zero_descriptor IM) tr =
Some (trX, initial_descriptors)
i: index
initial_descriptors i = zero_descriptor IM i
  by apply (equivocators_trace_project_preserves_zero_descriptors _ _ Htr _ _ _ Hproject).
Qed.

Lemma preloaded_equivocators_valid_trace_project_inv
  (final_descriptors : equivocator_descriptors IM)
  (is : composite_state (equivocator_IM IM))
  (tr : list (composite_transition_item (equivocator_IM IM)))
  (final_state := finite_trace_last is tr)
  (Htr : finite_constrained_trace FreeE is tr)
  (trX : list (composite_transition_item IM))
  (initial_descriptors : equivocator_descriptors IM)
  (Hproject : equivocators_trace_project final_descriptors tr = Some (trX, initial_descriptors))
  (Hproper : proper_equivocator_descriptors IM initial_descriptors is)
  : proper_equivocator_descriptors IM final_descriptors final_state.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
final_descriptors: equivocator_descriptors IM
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
final_state:= finite_trace_last is tr: state
  (composite_type (equivocator_IM IM))
Htr: finite_constrained_trace FreeE is tr
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hproject: equivocators_trace_project
  final_descriptors tr =
Some (trX, initial_descriptors)
Hproper: proper_equivocator_descriptors IM
  initial_descriptors is
proper_equivocator_descriptors IM final_descriptors
  final_state
Proof.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
final_descriptors: equivocator_descriptors IM
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
final_state:= finite_trace_last is tr: state
  (composite_type (equivocator_IM IM))
Htr: finite_constrained_trace FreeE is tr
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hproject: equivocators_trace_project
  final_descriptors tr =
Some (trX, initial_descriptors)
Hproper: proper_equivocator_descriptors IM
  initial_descriptors is
proper_equivocator_descriptors IM final_descriptors
  final_state
  revert Hproject.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
final_descriptors: equivocator_descriptors IM
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
final_state:= finite_trace_last is tr: state
  (composite_type (equivocator_IM IM))
Htr: finite_constrained_trace FreeE is tr
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  initial_descriptors is
equivocators_trace_project final_descriptors tr =
Some (trX, initial_descriptors)
→ proper_equivocator_descriptors IM final_descriptors
    final_state revert trX Htr final_descriptors.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
final_state:= finite_trace_last is tr: state
  (composite_type (equivocator_IM IM))
initial_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  initial_descriptors is
∀ trX : list (composite_transition_item IM),
  finite_constrained_trace FreeE is tr
  → ∀ final_descriptors : equivocator_descriptors IM,
      equivocators_trace_project final_descriptors tr =
      Some (trX, initial_descriptors)
      → proper_equivocator_descriptors IM
          final_descriptors final_state
  induction tr using rev_ind; intros; [by inversion Hproject |].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
x: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
final_state:= finite_trace_last is (tr ++ [x]): state
  (composite_type (equivocator_IM IM))
initial_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  initial_descriptors is
IHtr: ∀ (final_state := finite_trace_last is tr) 
  (trX : list (composite_transition_item IM)),
  finite_constrained_trace FreeE is tr
  → ∀ final_descriptors : 
       equivocator_descriptors IM,
      equivocators_trace_project
        final_descriptors tr =
      Some (trX, initial_descriptors)
      → proper_equivocator_descriptors IM
          final_descriptors final_state
trX: list (composite_transition_item IM)
Htr: finite_constrained_trace FreeE is (tr ++ [x])
final_descriptors: equivocator_descriptors IM
Hproject: equivocators_trace_project
  final_descriptors (tr ++ [x]) =
Some (trX, initial_descriptors)
proper_equivocator_descriptors IM final_descriptors
  final_state
  destruct Htr as [Htr Hinit].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
x: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
final_state:= finite_trace_last is (tr ++ [x]): state
  (composite_type (equivocator_IM IM))
initial_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  initial_descriptors is
IHtr: ∀ (final_state := finite_trace_last is tr) 
  (trX : list (composite_transition_item IM)),
  finite_constrained_trace FreeE is tr
  → ∀ final_descriptors : 
       equivocator_descriptors IM,
      equivocators_trace_project
        final_descriptors tr =
      Some (trX, initial_descriptors)
      → proper_equivocator_descriptors IM
          final_descriptors final_state
trX: list (composite_transition_item IM)
Htr: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) is
  (tr ++ [x])
Hinit: initial_state_prop is
final_descriptors: equivocator_descriptors IM
Hproject: equivocators_trace_project
  final_descriptors (tr ++ [x]) =
Some (trX, initial_descriptors)
proper_equivocator_descriptors IM final_descriptors
  final_state
  apply finite_valid_trace_from_app_iff in Htr.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
x: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
final_state:= finite_trace_last is (tr ++ [x]): state
  (composite_type (equivocator_IM IM))
initial_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  initial_descriptors is
IHtr: ∀ (final_state := finite_trace_last is tr) 
  (trX : list (composite_transition_item IM)),
  finite_constrained_trace FreeE is tr
  → ∀ final_descriptors : 
       equivocator_descriptors IM,
      equivocators_trace_project
        final_descriptors tr =
      Some (trX, initial_descriptors)
      → proper_equivocator_descriptors IM
          final_descriptors final_state
trX: list (composite_transition_item IM)
Htr: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) is tr
∧ finite_valid_trace_from
    (preloaded_with_all_messages_vlsm FreeE)
    (finite_trace_last is tr) [x]
Hinit: initial_state_prop is
final_descriptors: equivocator_descriptors IM
Hproject: equivocators_trace_project
  final_descriptors (tr ++ [x]) =
Some (trX, initial_descriptors)
proper_equivocator_descriptors IM final_descriptors
  final_state
  destruct Htr as [Htr Hx].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
x: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
final_state:= finite_trace_last is (tr ++ [x]): state
  (composite_type (equivocator_IM IM))
initial_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  initial_descriptors is
IHtr: ∀ (final_state := finite_trace_last is tr) 
  (trX : list (composite_transition_item IM)),
  finite_constrained_trace FreeE is tr
  → ∀ final_descriptors : 
       equivocator_descriptors IM,
      equivocators_trace_project
        final_descriptors tr =
      Some (trX, initial_descriptors)
      → proper_equivocator_descriptors IM
          final_descriptors final_state
trX: list (composite_transition_item IM)
Htr: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) is tr
Hx: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE)
  (finite_trace_last is tr) [x]
Hinit: initial_state_prop is
final_descriptors: equivocator_descriptors IM
Hproject: equivocators_trace_project
  final_descriptors (tr ++ [x]) =
Some (trX, initial_descriptors)
proper_equivocator_descriptors IM final_descriptors
  final_state
  unfold equivocators_trace_project in Hproject.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
x: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
final_state:= finite_trace_last is (tr ++ [x]): state
  (composite_type (equivocator_IM IM))
initial_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  initial_descriptors is
IHtr: ∀ (final_state := finite_trace_last is tr) 
  (trX : list (composite_transition_item IM)),
  finite_constrained_trace FreeE is tr
  → ∀ final_descriptors : 
       equivocator_descriptors IM,
      equivocators_trace_project
        final_descriptors tr =
      Some (trX, initial_descriptors)
      → proper_equivocator_descriptors IM
          final_descriptors final_state
trX: list (composite_transition_item IM)
Htr: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) is tr
Hx: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE)
  (finite_trace_last is tr) [x]
Hinit: initial_state_prop is
final_descriptors: equivocator_descriptors IM
Hproject: foldr equivocators_trace_project_folder
  (Some ([], final_descriptors))
  (tr ++ [x]) =
Some (trX, initial_descriptors)
proper_equivocator_descriptors IM final_descriptors
  final_state
  rewrite fold_right_app in Hproject.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
x: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
final_state:= finite_trace_last is (tr ++ [x]): state
  (composite_type (equivocator_IM IM))
initial_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  initial_descriptors is
IHtr: ∀ (final_state := finite_trace_last is tr) 
  (trX : list (composite_transition_item IM)),
  finite_constrained_trace FreeE is tr
  → ∀ final_descriptors : 
       equivocator_descriptors IM,
      equivocators_trace_project
        final_descriptors tr =
      Some (trX, initial_descriptors)
      → proper_equivocator_descriptors IM
          final_descriptors final_state
trX: list (composite_transition_item IM)
Htr: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) is tr
Hx: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE)
  (finite_trace_last is tr) [x]
Hinit: initial_state_prop is
final_descriptors: equivocator_descriptors IM
Hproject: foldr equivocators_trace_project_folder
  (foldr equivocators_trace_project_folder
     (Some ([], final_descriptors)) [x]) tr =
Some (trX, initial_descriptors)
proper_equivocator_descriptors IM final_descriptors
  final_state
  match type of Hproject with
  | fold_right _ ?f _ = _ => remember f as project_x
  end.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
x: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
final_state:= finite_trace_last is (tr ++ [x]): state
  (composite_type (equivocator_IM IM))
initial_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  initial_descriptors is
IHtr: ∀ (final_state := finite_trace_last is tr) 
  (trX : list (composite_transition_item IM)),
  finite_constrained_trace FreeE is tr
  → ∀ final_descriptors : 
       equivocator_descriptors IM,
      equivocators_trace_project
        final_descriptors tr =
      Some (trX, initial_descriptors)
      → proper_equivocator_descriptors IM
          final_descriptors final_state
trX: list (composite_transition_item IM)
Htr: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) is tr
Hx: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE)
  (finite_trace_last is tr) [x]
Hinit: initial_state_prop is
final_descriptors: equivocator_descriptors IM
project_x: option
  (list (composite_transition_item IM) *
   equivocator_descriptors IM)
Heqproject_x: project_x =
foldr equivocators_trace_project_folder
  (Some ([], final_descriptors)) [x]
Hproject: foldr equivocators_trace_project_folder
  project_x tr =
Some (trX, initial_descriptors)
proper_equivocator_descriptors IM final_descriptors
  final_state
  simpl in Heqproject_x.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
x: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
final_state:= finite_trace_last is (tr ++ [x]): state
  (composite_type (equivocator_IM IM))
initial_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  initial_descriptors is
IHtr: ∀ (final_state := finite_trace_last is tr) 
  (trX : list (composite_transition_item IM)),
  finite_constrained_trace FreeE is tr
  → ∀ final_descriptors : 
       equivocator_descriptors IM,
      equivocators_trace_project
        final_descriptors tr =
      Some (trX, initial_descriptors)
      → proper_equivocator_descriptors IM
          final_descriptors final_state
trX: list (composite_transition_item IM)
Htr: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) is tr
Hx: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE)
  (finite_trace_last is tr) [x]
Hinit: initial_state_prop is
final_descriptors: equivocator_descriptors IM
project_x: option
  (list (composite_transition_item IM) *
   equivocator_descriptors IM)
Heqproject_x: project_x =
match
  equivocators_transition_item_project
    final_descriptors x
with
| Some (Some item', odescriptor) =>
    Some ([item'], odescriptor)
| Some (None, odescriptor) =>
    Some ([], odescriptor)
| None => None
end
Hproject: foldr equivocators_trace_project_folder
  project_x tr =
Some (trX, initial_descriptors)
proper_equivocator_descriptors IM final_descriptors
  final_state
  destruct project_x as [(x', x_descriptors) |]
  ; [| by rewrite equivocators_trace_project_fold_None in Hproject; congruence].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
x: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
final_state:= finite_trace_last is (tr ++ [x]): state
  (composite_type (equivocator_IM IM))
initial_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  initial_descriptors is
IHtr: ∀ (final_state := finite_trace_last is tr) 
  (trX : list (composite_transition_item IM)),
  finite_constrained_trace FreeE is tr
  → ∀ final_descriptors : 
       equivocator_descriptors IM,
      equivocators_trace_project
        final_descriptors tr =
      Some (trX, initial_descriptors)
      → proper_equivocator_descriptors IM
          final_descriptors final_state
trX: list (composite_transition_item IM)
Htr: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) is tr
Hx: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE)
  (finite_trace_last is tr) [x]
Hinit: initial_state_prop is
final_descriptors: equivocator_descriptors IM
x': list (composite_transition_item IM)
x_descriptors: equivocator_descriptors IM
Heqproject_x: Some (x', x_descriptors) =
match
  equivocators_transition_item_project
    final_descriptors x
with
| Some (Some item', odescriptor) =>
    Some ([item'], odescriptor)
| Some (None, odescriptor) =>
    Some ([], odescriptor)
| None => None
end
Hproject: foldr equivocators_trace_project_folder
  (Some (x', x_descriptors)) tr =
Some (trX, initial_descriptors)
proper_equivocator_descriptors IM final_descriptors
  final_state
  destruct (equivocators_transition_item_project final_descriptors x) as [(oitem', ditem') |]
    eqn: Hproject_x
  ; [| by congruence].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
x: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
final_state:= finite_trace_last is (tr ++ [x]): state
  (composite_type (equivocator_IM IM))
initial_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  initial_descriptors is
IHtr: ∀ (final_state := finite_trace_last is tr) 
  (trX : list (composite_transition_item IM)),
  finite_constrained_trace FreeE is tr
  → ∀ final_descriptors : 
       equivocator_descriptors IM,
      equivocators_trace_project
        final_descriptors tr =
      Some (trX, initial_descriptors)
      → proper_equivocator_descriptors IM
          final_descriptors final_state
trX: list (composite_transition_item IM)
Htr: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) is tr
Hx: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE)
  (finite_trace_last is tr) [x]
Hinit: initial_state_prop is
final_descriptors: equivocator_descriptors IM
x': list (composite_transition_item IM)
x_descriptors: equivocator_descriptors IM
oitem': option (composite_transition_item IM)
ditem': equivocator_descriptors IM
Hproject_x: equivocators_transition_item_project
  final_descriptors x =
Some (oitem', ditem')
Heqproject_x: Some (x', x_descriptors) =
match oitem' with
| Some item' => Some ([item'], ditem')
| None => Some ([], ditem')
end
Hproject: foldr equivocators_trace_project_folder
  (Some (x', x_descriptors)) tr =
Some (trX, initial_descriptors)
proper_equivocator_descriptors IM final_descriptors
  final_state
  apply (equivocators_trace_project_folder_additive_iff tr x' x_descriptors initial_descriptors trX)
  in Hproject.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
x: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
final_state:= finite_trace_last is (tr ++ [x]): state
  (composite_type (equivocator_IM IM))
initial_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  initial_descriptors is
IHtr: ∀ (final_state := finite_trace_last is tr) 
  (trX : list (composite_transition_item IM)),
  finite_constrained_trace FreeE is tr
  → ∀ final_descriptors : 
       equivocator_descriptors IM,
      equivocators_trace_project
        final_descriptors tr =
      Some (trX, initial_descriptors)
      → proper_equivocator_descriptors IM
          final_descriptors final_state
trX: list (composite_transition_item IM)
Htr: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) is tr
Hx: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE)
  (finite_trace_last is tr) [x]
Hinit: initial_state_prop is
final_descriptors: equivocator_descriptors IM
x': list (composite_transition_item IM)
x_descriptors: equivocator_descriptors IM
oitem': option (composite_transition_item IM)
ditem': equivocator_descriptors IM
Hproject_x: equivocators_transition_item_project
  final_descriptors x =
Some (oitem', ditem')
Heqproject_x: Some (x', x_descriptors) =
match oitem' with
| Some item' => Some ([item'], ditem')
| None => Some ([], ditem')
end
Hproject: ∃ trX0 : list
           (composite_transition_item IM),
  foldr equivocators_trace_project_folder
    (Some ([], x_descriptors)) tr =
  Some (trX0, initial_descriptors)
  ∧ trX = trX0 ++ x'
proper_equivocator_descriptors IM final_descriptors
  final_state
  destruct Hproject as [trX' [Hproject_x' HeqtrX]].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
x: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
final_state:= finite_trace_last is (tr ++ [x]): state
  (composite_type (equivocator_IM IM))
initial_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  initial_descriptors is
IHtr: ∀ (final_state := finite_trace_last is tr) 
  (trX : list (composite_transition_item IM)),
  finite_constrained_trace FreeE is tr
  → ∀ final_descriptors : 
       equivocator_descriptors IM,
      equivocators_trace_project
        final_descriptors tr =
      Some (trX, initial_descriptors)
      → proper_equivocator_descriptors IM
          final_descriptors final_state
trX: list (composite_transition_item IM)
Htr: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) is tr
Hx: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE)
  (finite_trace_last is tr) [x]
Hinit: initial_state_prop is
final_descriptors: equivocator_descriptors IM
x': list (composite_transition_item IM)
x_descriptors: equivocator_descriptors IM
oitem': option (composite_transition_item IM)
ditem': equivocator_descriptors IM
Hproject_x: equivocators_transition_item_project
  final_descriptors x =
Some (oitem', ditem')
Heqproject_x: Some (x', x_descriptors) =
match oitem' with
| Some item' => Some ([item'], ditem')
| None => Some ([], ditem')
end
trX': list (composite_transition_item IM)
Hproject_x': foldr equivocators_trace_project_folder
  (Some ([], x_descriptors)) tr =
Some (trX', initial_descriptors)
HeqtrX: trX = trX' ++ x'
proper_equivocator_descriptors IM final_descriptors
  final_state
  specialize (IHtr trX' (conj Htr Hinit) _ Hproject_x').message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
x: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
final_state:= finite_trace_last is (tr ++ [x]): state
  (composite_type (equivocator_IM IM))
initial_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  initial_descriptors is
x_descriptors: equivocator_descriptors IM
IHtr: proper_equivocator_descriptors IM x_descriptors
  (finite_trace_last is tr)
trX: list (composite_transition_item IM)
Htr: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) is tr
Hx: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE)
  (finite_trace_last is tr) [x]
Hinit: initial_state_prop is
final_descriptors: equivocator_descriptors IM
x': list (composite_transition_item IM)
oitem': option (composite_transition_item IM)
ditem': equivocator_descriptors IM
Hproject_x: equivocators_transition_item_project
  final_descriptors x =
Some (oitem', ditem')
Heqproject_x: Some (x', x_descriptors) =
match oitem' with
| Some item' => Some ([item'], ditem')
| None => Some ([], ditem')
end
trX': list (composite_transition_item IM)
Hproject_x': foldr equivocators_trace_project_folder
  (Some ([], x_descriptors)) tr =
Some (trX', initial_descriptors)
HeqtrX: trX = trX' ++ x'
proper_equivocator_descriptors IM final_descriptors
  final_state
  inversion Hx.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
x: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
final_state:= finite_trace_last is (tr ++ [x]): state
  (composite_type (equivocator_IM IM))
initial_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  initial_descriptors is
x_descriptors: equivocator_descriptors IM
IHtr: proper_equivocator_descriptors IM x_descriptors
  (finite_trace_last is tr)
trX: list (composite_transition_item IM)
Htr: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) is tr
Hx: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE)
  (finite_trace_last is tr) [x]
Hinit: initial_state_prop is
final_descriptors: equivocator_descriptors IM
x': list (composite_transition_item IM)
oitem': option (composite_transition_item IM)
ditem': equivocator_descriptors IM
Hproject_x: equivocators_transition_item_project
  final_descriptors x =
Some (oitem', ditem')
Heqproject_x: Some (x', x_descriptors) =
match oitem' with
| Some item' => Some ([item'], ditem')
| None => Some ([], ditem')
end
trX': list (composite_transition_item IM)
Hproject_x': foldr equivocators_trace_project_folder
  (Some ([], x_descriptors)) tr =
Some (trX', initial_descriptors)
HeqtrX: trX = trX' ++ x'
s: state (preloaded_with_all_messages_vlsm FreeE)
tl: list transition_item
s': state (preloaded_with_all_messages_vlsm FreeE)
iom, oom: option message
l: label (preloaded_with_all_messages_vlsm FreeE)
Htl: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) s []
Ht: input_valid_transition
  (preloaded_with_all_messages_vlsm FreeE) l
  (finite_trace_last is tr, iom) (
  s, oom)
H1: s' = finite_trace_last is tr
H3: {|
  l := l;
  input := iom;
  destination := s;
  output := oom
|} = x
H4: tl = []
proper_equivocator_descriptors IM final_descriptors
  final_state subst.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
s: state (preloaded_with_all_messages_vlsm FreeE)
iom, oom: option message
l: label (preloaded_with_all_messages_vlsm FreeE)
final_state:= finite_trace_last is
  (tr ++
   [{|
      l := l;
      input := iom;
      destination := s;
      output := oom
    |}]): state
  (composite_type (equivocator_IM IM))
initial_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  initial_descriptors is
x_descriptors: equivocator_descriptors IM
IHtr: proper_equivocator_descriptors IM x_descriptors
  (finite_trace_last is tr)
Htr: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) is tr
Hx: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE)
  (finite_trace_last is tr)
  [{|
     l := l;
     input := iom;
     destination := s;
     output := oom
   |}]
Hinit: initial_state_prop is
final_descriptors: equivocator_descriptors IM
x': list (composite_transition_item IM)
oitem': option (composite_transition_item IM)
ditem': equivocator_descriptors IM
Hproject_x: equivocators_transition_item_project
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some (oitem', ditem')
Heqproject_x: Some (x', x_descriptors) =
match oitem' with
| Some item' => Some ([item'], ditem')
| None => Some ([], ditem')
end
trX': list (composite_transition_item IM)
Hproject_x': foldr equivocators_trace_project_folder
  (Some ([], x_descriptors)) tr =
Some (trX', initial_descriptors)
Htl: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) s []
Ht: input_valid_transition
  (preloaded_with_all_messages_vlsm FreeE) l
  (finite_trace_last is tr, iom) (
  s, oom)
proper_equivocator_descriptors IM final_descriptors
  final_state clear Hx.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
s: state (preloaded_with_all_messages_vlsm FreeE)
iom, oom: option message
l: label (preloaded_with_all_messages_vlsm FreeE)
final_state:= finite_trace_last is
  (tr ++
   [{|
      l := l;
      input := iom;
      destination := s;
      output := oom
    |}]): state
  (composite_type (equivocator_IM IM))
initial_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  initial_descriptors is
x_descriptors: equivocator_descriptors IM
IHtr: proper_equivocator_descriptors IM x_descriptors
  (finite_trace_last is tr)
Htr: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) is tr
Hinit: initial_state_prop is
final_descriptors: equivocator_descriptors IM
x': list (composite_transition_item IM)
oitem': option (composite_transition_item IM)
ditem': equivocator_descriptors IM
Hproject_x: equivocators_transition_item_project
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some (oitem', ditem')
Heqproject_x: Some (x', x_descriptors) =
match oitem' with
| Some item' => Some ([item'], ditem')
| None => Some ([], ditem')
end
trX': list (composite_transition_item IM)
Hproject_x': foldr equivocators_trace_project_folder
  (Some ([], x_descriptors)) tr =
Some (trX', initial_descriptors)
Htl: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) s []
Ht: input_valid_transition
  (preloaded_with_all_messages_vlsm FreeE) l
  (finite_trace_last is tr, iom) (
  s, oom)
proper_equivocator_descriptors IM final_descriptors
  final_state
  unfold equivocators_transition_item_project in Hproject_x; simpl in Hproject_x.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
s: state (preloaded_with_all_messages_vlsm FreeE)
iom, oom: option message
l: label (preloaded_with_all_messages_vlsm FreeE)
final_state:= finite_trace_last is
  (tr ++
   [{|
      l := l;
      input := iom;
      destination := s;
      output := oom
    |}]): state
  (composite_type (equivocator_IM IM))
initial_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  initial_descriptors is
x_descriptors: equivocator_descriptors IM
IHtr: proper_equivocator_descriptors IM x_descriptors
  (finite_trace_last is tr)
Htr: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) is tr
Hinit: initial_state_prop is
final_descriptors: equivocator_descriptors IM
x': list (composite_transition_item IM)
oitem': option (composite_transition_item IM)
ditem': equivocator_descriptors IM
Hproject_x: match
  equivocator_vlsm_transition_item_project
    (IM (projT1 l))
    (composite_transition_item_projection
       (equivocator_IM IM)
       {|
         l := l;
         input := iom;
         destination := s;
         output := oom
       |}) (final_descriptors (projT1 l))
with
| Some (Some item', deqv') =>
    Some
      (Some
         {|
           l :=
             existT (projT1 l)
               (VLSM.l item');
           input := iom;
           destination :=
             equivocators_state_project
               IM final_descriptors s;
           output := oom
         |},
       equivocator_descriptors_update IM
         final_descriptors 
         (projT1 l) deqv')
| Some (None, deqv') =>
    Some
      (None,
       equivocator_descriptors_update IM
         final_descriptors 
         (projT1 l) deqv')
| None => None
end = Some (oitem', ditem')
Heqproject_x: Some (x', x_descriptors) =
match oitem' with
| Some item' => Some ([item'], ditem')
| None => Some ([], ditem')
end
trX': list (composite_transition_item IM)
Hproject_x': foldr equivocators_trace_project_folder
  (Some ([], x_descriptors)) tr =
Some (trX', initial_descriptors)
Htl: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) s []
Ht: input_valid_transition
  (preloaded_with_all_messages_vlsm FreeE) l
  (finite_trace_last is tr, iom) (
  s, oom)
proper_equivocator_descriptors IM final_descriptors
  final_state
  unfold composite_transition_item_projection in Hproject_x; simpl in Hproject_x.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
s: state (preloaded_with_all_messages_vlsm FreeE)
iom, oom: option message
l: label (preloaded_with_all_messages_vlsm FreeE)
final_state:= finite_trace_last is
  (tr ++
   [{|
      l := l;
      input := iom;
      destination := s;
      output := oom
    |}]): state
  (composite_type (equivocator_IM IM))
initial_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  initial_descriptors is
x_descriptors: equivocator_descriptors IM
IHtr: proper_equivocator_descriptors IM x_descriptors
  (finite_trace_last is tr)
Htr: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) is tr
Hinit: initial_state_prop is
final_descriptors: equivocator_descriptors IM
x': list (composite_transition_item IM)
oitem': option (composite_transition_item IM)
ditem': equivocator_descriptors IM
Hproject_x: match
  equivocator_vlsm_transition_item_project
    (IM (projT1 l))
    (composite_transition_item_projection_from_eq
       (equivocator_IM IM)
       {|
         l := l;
         input := iom;
         destination := s;
         output := oom
       |} (projT1 l) eq_refl)
    (final_descriptors (projT1 l))
with
| Some (Some item', deqv') =>
    Some
      (Some
         {|
           l :=
             existT (projT1 l)
               (VLSM.l item');
           input := iom;
           destination :=
             equivocators_state_project
               IM final_descriptors s;
           output := oom
         |},
       equivocator_descriptors_update IM
         final_descriptors 
         (projT1 l) deqv')
| Some (None, deqv') =>
    Some
      (None,
       equivocator_descriptors_update IM
         final_descriptors 
         (projT1 l) deqv')
| None => None
end = Some (oitem', ditem')
Heqproject_x: Some (x', x_descriptors) =
match oitem' with
| Some item' => Some ([item'], ditem')
| None => Some ([], ditem')
end
trX': list (composite_transition_item IM)
Hproject_x': foldr equivocators_trace_project_folder
  (Some ([], x_descriptors)) tr =
Some (trX', initial_descriptors)
Htl: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) s []
Ht: input_valid_transition
  (preloaded_with_all_messages_vlsm FreeE) l
  (finite_trace_last is tr, iom) (
  s, oom)
proper_equivocator_descriptors IM final_descriptors
  final_state
  unfold composite_transition_item_projection_from_eq in Hproject_x; simpl in Hproject_x.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
s: state (preloaded_with_all_messages_vlsm FreeE)
iom, oom: option message
l: label (preloaded_with_all_messages_vlsm FreeE)
final_state:= finite_trace_last is
  (tr ++
   [{|
      l := l;
      input := iom;
      destination := s;
      output := oom
    |}]): state
  (composite_type (equivocator_IM IM))
initial_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  initial_descriptors is
x_descriptors: equivocator_descriptors IM
IHtr: proper_equivocator_descriptors IM x_descriptors
  (finite_trace_last is tr)
Htr: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) is tr
Hinit: initial_state_prop is
final_descriptors: equivocator_descriptors IM
x': list (composite_transition_item IM)
oitem': option (composite_transition_item IM)
ditem': equivocator_descriptors IM
Hproject_x: match
  equivocator_vlsm_transition_item_project
    (IM (projT1 l))
    {|
      l :=
        eq_rect_r
          (λ n : index,
             EquivocatorLabel (IM n))
          (projT2 l) eq_refl;
      input := iom;
      destination := s (projT1 l);
      output := oom
    |} (final_descriptors (projT1 l))
with
| Some (Some item', deqv') =>
    Some
      (Some
         {|
           l :=
             existT (projT1 l)
               (VLSM.l item');
           input := iom;
           destination :=
             equivocators_state_project
               IM final_descriptors s;
           output := oom
         |},
       equivocator_descriptors_update IM
         final_descriptors 
         (projT1 l) deqv')
| Some (None, deqv') =>
    Some
      (None,
       equivocator_descriptors_update IM
         final_descriptors 
         (projT1 l) deqv')
| None => None
end = Some (oitem', ditem')
Heqproject_x: Some (x', x_descriptors) =
match oitem' with
| Some item' => Some ([item'], ditem')
| None => Some ([], ditem')
end
trX': list (composite_transition_item IM)
Hproject_x': foldr equivocators_trace_project_folder
  (Some ([], x_descriptors)) tr =
Some (trX', initial_descriptors)
Htl: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) s []
Ht: input_valid_transition
  (preloaded_with_all_messages_vlsm FreeE) l
  (finite_trace_last is tr, iom) (
  s, oom)
proper_equivocator_descriptors IM final_descriptors
  final_state
  unfold eq_rect_r in Hproject_x; simpl in Hproject_x.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
s: state (preloaded_with_all_messages_vlsm FreeE)
iom, oom: option message
l: label (preloaded_with_all_messages_vlsm FreeE)
final_state:= finite_trace_last is
  (tr ++
   [{|
      l := l;
      input := iom;
      destination := s;
      output := oom
    |}]): state
  (composite_type (equivocator_IM IM))
initial_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  initial_descriptors is
x_descriptors: equivocator_descriptors IM
IHtr: proper_equivocator_descriptors IM x_descriptors
  (finite_trace_last is tr)
Htr: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) is tr
Hinit: initial_state_prop is
final_descriptors: equivocator_descriptors IM
x': list (composite_transition_item IM)
oitem': option (composite_transition_item IM)
ditem': equivocator_descriptors IM
Hproject_x: match
  equivocator_vlsm_transition_item_project
    (IM (projT1 l))
    {|
      l := projT2 l;
      input := iom;
      destination := s (projT1 l);
      output := oom
    |} (final_descriptors (projT1 l))
with
| Some (Some item', deqv') =>
    Some
      (Some
         {|
           l :=
             existT (projT1 l)
               (VLSM.l item');
           input := iom;
           destination :=
             equivocators_state_project
               IM final_descriptors s;
           output := oom
         |},
       equivocator_descriptors_update IM
         final_descriptors 
         (projT1 l) deqv')
| Some (None, deqv') =>
    Some
      (None,
       equivocator_descriptors_update IM
         final_descriptors 
         (projT1 l) deqv')
| None => None
end = Some (oitem', ditem')
Heqproject_x: Some (x', x_descriptors) =
match oitem' with
| Some item' => Some ([item'], ditem')
| None => Some ([], ditem')
end
trX': list (composite_transition_item IM)
Hproject_x': foldr equivocators_trace_project_folder
  (Some ([], x_descriptors)) tr =
Some (trX', initial_descriptors)
Htl: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) s []
Ht: input_valid_transition
  (preloaded_with_all_messages_vlsm FreeE) l
  (finite_trace_last is tr, iom) (
  s, oom)
proper_equivocator_descriptors IM final_descriptors
  final_state
  match type of Hproject_x with
  | context [equivocator_vlsm_transition_item_project ?X ?i ?c] =>
      destruct (equivocator_vlsm_transition_item_project X i c) as [(oitem'', ditem'') |]
        eqn: Heqprojecti; [| by congruence]
  end.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
s: state (preloaded_with_all_messages_vlsm FreeE)
iom, oom: option message
l: label (preloaded_with_all_messages_vlsm FreeE)
final_state:= finite_trace_last is
  (tr ++
   [{|
      l := l;
      input := iom;
      destination := s;
      output := oom
    |}]): state
  (composite_type (equivocator_IM IM))
initial_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  initial_descriptors is
x_descriptors: equivocator_descriptors IM
IHtr: proper_equivocator_descriptors IM x_descriptors
  (finite_trace_last is tr)
Htr: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) is tr
Hinit: initial_state_prop is
final_descriptors: equivocator_descriptors IM
x': list (composite_transition_item IM)
oitem': option (composite_transition_item IM)
ditem': equivocator_descriptors IM
oitem'': option transition_item
ditem'': MachineDescriptor (IM (projT1 l))
Heqprojecti: equivocator_vlsm_transition_item_project
  (IM (projT1 l))
  {|
    l := projT2 l;
    input := iom;
    destination := s (projT1 l);
    output := oom
  |} (final_descriptors (projT1 l)) =
Some (oitem'', ditem'')
Hproject_x: match oitem'' with
| Some item' =>
    Some
      (Some
         {|
           l :=
             existT (projT1 l)
               (VLSM.l item');
           input := iom;
           destination :=
             equivocators_state_project
               IM final_descriptors s;
           output := oom
         |},
       equivocator_descriptors_update IM
         final_descriptors 
         (projT1 l) ditem'')
| None =>
    Some
      (None,
       equivocator_descriptors_update IM
         final_descriptors 
         (projT1 l) ditem'')
end = Some (oitem', ditem')
Heqproject_x: Some (x', x_descriptors) =
match oitem' with
| Some item' => Some ([item'], ditem')
| None => Some ([], ditem')
end
trX': list (composite_transition_item IM)
Hproject_x': foldr equivocators_trace_project_folder
  (Some ([], x_descriptors)) tr =
Some (trX', initial_descriptors)
Htl: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) s []
Ht: input_valid_transition
  (preloaded_with_all_messages_vlsm FreeE) l
  (finite_trace_last is tr, iom) (
  s, oom)
proper_equivocator_descriptors IM final_descriptors
  final_state
  unfold equivocator_vlsm_transition_item_project in Heqprojecti.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
s: state (preloaded_with_all_messages_vlsm FreeE)
iom, oom: option message
l: label (preloaded_with_all_messages_vlsm FreeE)
final_state:= finite_trace_last is
  (tr ++
   [{|
      l := l;
      input := iom;
      destination := s;
      output := oom
    |}]): state
  (composite_type (equivocator_IM IM))
initial_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  initial_descriptors is
x_descriptors: equivocator_descriptors IM
IHtr: proper_equivocator_descriptors IM x_descriptors
  (finite_trace_last is tr)
Htr: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) is tr
Hinit: initial_state_prop is
final_descriptors: equivocator_descriptors IM
x': list (composite_transition_item IM)
oitem': option (composite_transition_item IM)
ditem': equivocator_descriptors IM
oitem'': option transition_item
ditem'': MachineDescriptor (IM (projT1 l))
Heqprojecti: match final_descriptors (projT1 l) with
| NewMachine _ =>
    Some
      (None,
       final_descriptors (projT1 l))
| Existing j =>
    match
      equivocator_state_project
        (s (projT1 l)) j
    with
    | Some sj =>
        match projT2 l with
        | Spawn sn =>
            if
             decide
               (j =
                equivocator_state_last
                  (s (projT1 l)))
            then
             Some (None, NewMachine sn)
            else Some (None, Existing j)
        | ContinueWith i lx =>
            if decide (i = j)
            then
             Some
               (Some
                  {|
                    l := lx;
                    input := iom;
                    destination := sj;
                    output := oom
                  |}, Existing i)
            else Some (None, Existing j)
        | ForkWith i lx =>
            if
             decide
               (j =
                equivocator_state_last
                  (s (projT1 l)))
            then
             Some
               (Some
                  {|
                    l := lx;
                    input := iom;
                    destination := sj;
                    output := oom
                  |}, Existing i)
            else Some (None, Existing j)
        end
    | None => None
    end
end = Some (oitem'', ditem'')
Hproject_x: match oitem'' with
| Some item' =>
    Some
      (Some
         {|
           l :=
             existT (projT1 l)
               (VLSM.l item');
           input := iom;
           destination :=
             equivocators_state_project
               IM final_descriptors s;
           output := oom
         |},
       equivocator_descriptors_update IM
         final_descriptors 
         (projT1 l) ditem'')
| None =>
    Some
      (None,
       equivocator_descriptors_update IM
         final_descriptors 
         (projT1 l) ditem'')
end = Some (oitem', ditem')
Heqproject_x: Some (x', x_descriptors) =
match oitem' with
| Some item' => Some ([item'], ditem')
| None => Some ([], ditem')
end
trX': list (composite_transition_item IM)
Hproject_x': foldr equivocators_trace_project_folder
  (Some ([], x_descriptors)) tr =
Some (trX', initial_descriptors)
Htl: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) s []
Ht: input_valid_transition
  (preloaded_with_all_messages_vlsm FreeE) l
  (finite_trace_last is tr, iom) (
  s, oom)
proper_equivocator_descriptors IM final_descriptors
  final_state
  unfold final_state in *.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
s: state (preloaded_with_all_messages_vlsm FreeE)
iom, oom: option message
l: label (preloaded_with_all_messages_vlsm FreeE)
final_state:= finite_trace_last is
  (tr ++
   [{|
      l := l;
      input := iom;
      destination := s;
      output := oom
    |}]): state
  (composite_type (equivocator_IM IM))
initial_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  initial_descriptors is
x_descriptors: equivocator_descriptors IM
IHtr: proper_equivocator_descriptors IM x_descriptors
  (finite_trace_last is tr)
Htr: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) is tr
Hinit: initial_state_prop is
final_descriptors: equivocator_descriptors IM
x': list (composite_transition_item IM)
oitem': option (composite_transition_item IM)
ditem': equivocator_descriptors IM
oitem'': option transition_item
ditem'': MachineDescriptor (IM (projT1 l))
Heqprojecti: match final_descriptors (projT1 l) with
| NewMachine _ =>
    Some
      (None,
       final_descriptors (projT1 l))
| Existing j =>
    match
      equivocator_state_project
        (s (projT1 l)) j
    with
    | Some sj =>
        match projT2 l with
        | Spawn sn =>
            if
             decide
               (j =
                equivocator_state_last
                  (s (projT1 l)))
            then
             Some (None, NewMachine sn)
            else Some (None, Existing j)
        | ContinueWith i lx =>
            if decide (i = j)
            then
             Some
               (Some
                  {|
                    l := lx;
                    input := iom;
                    destination := sj;
                    output := oom
                  |}, Existing i)
            else Some (None, Existing j)
        | ForkWith i lx =>
            if
             decide
               (j =
                equivocator_state_last
                  (s (projT1 l)))
            then
             Some
               (Some
                  {|
                    l := lx;
                    input := iom;
                    destination := sj;
                    output := oom
                  |}, Existing i)
            else Some (None, Existing j)
        end
    | None => None
    end
end = Some (oitem'', ditem'')
Hproject_x: match oitem'' with
| Some item' =>
    Some
      (Some
         {|
           l :=
             existT (projT1 l)
               (VLSM.l item');
           input := iom;
           destination :=
             equivocators_state_project
               IM final_descriptors s;
           output := oom
         |},
       equivocator_descriptors_update IM
         final_descriptors 
         (projT1 l) ditem'')
| None =>
    Some
      (None,
       equivocator_descriptors_update IM
         final_descriptors 
         (projT1 l) ditem'')
end = Some (oitem', ditem')
Heqproject_x: Some (x', x_descriptors) =
match oitem' with
| Some item' => Some ([item'], ditem')
| None => Some ([], ditem')
end
trX': list (composite_transition_item IM)
Hproject_x': foldr equivocators_trace_project_folder
  (Some ([], x_descriptors)) tr =
Some (trX', initial_descriptors)
Htl: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) s []
Ht: input_valid_transition
  (preloaded_with_all_messages_vlsm FreeE) l
  (finite_trace_last is tr, iom) (
  s, oom)
proper_equivocator_descriptors IM final_descriptors
  (finite_trace_last is
     (tr ++
      [{|
         l := l;
         input := iom;
         destination := s;
         output := oom
       |}])) clear final_state.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
s: state (preloaded_with_all_messages_vlsm FreeE)
iom, oom: option message
l: label (preloaded_with_all_messages_vlsm FreeE)
initial_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  initial_descriptors is
x_descriptors: equivocator_descriptors IM
IHtr: proper_equivocator_descriptors IM x_descriptors
  (finite_trace_last is tr)
Htr: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) is tr
Hinit: initial_state_prop is
final_descriptors: equivocator_descriptors IM
x': list (composite_transition_item IM)
oitem': option (composite_transition_item IM)
ditem': equivocator_descriptors IM
oitem'': option transition_item
ditem'': MachineDescriptor (IM (projT1 l))
Heqprojecti: match final_descriptors (projT1 l) with
| NewMachine _ =>
    Some
      (None,
       final_descriptors (projT1 l))
| Existing j =>
    match
      equivocator_state_project
        (s (projT1 l)) j
    with
    | Some sj =>
        match projT2 l with
        | Spawn sn =>
            if
             decide
               (j =
                equivocator_state_last
                  (s (projT1 l)))
            then
             Some (None, NewMachine sn)
            else Some (None, Existing j)
        | ContinueWith i lx =>
            if decide (i = j)
            then
             Some
               (Some
                  {|
                    l := lx;
                    input := iom;
                    destination := sj;
                    output := oom
                  |}, Existing i)
            else Some (None, Existing j)
        | ForkWith i lx =>
            if
             decide
               (j =
                equivocator_state_last
                  (s (projT1 l)))
            then
             Some
               (Some
                  {|
                    l := lx;
                    input := iom;
                    destination := sj;
                    output := oom
                  |}, Existing i)
            else Some (None, Existing j)
        end
    | None => None
    end
end = Some (oitem'', ditem'')
Hproject_x: match oitem'' with
| Some item' =>
    Some
      (Some
         {|
           l :=
             existT (projT1 l)
               (VLSM.l item');
           input := iom;
           destination :=
             equivocators_state_project
               IM final_descriptors s;
           output := oom
         |},
       equivocator_descriptors_update IM
         final_descriptors 
         (projT1 l) ditem'')
| None =>
    Some
      (None,
       equivocator_descriptors_update IM
         final_descriptors 
         (projT1 l) ditem'')
end = Some (oitem', ditem')
Heqproject_x: Some (x', x_descriptors) =
match oitem' with
| Some item' => Some ([item'], ditem')
| None => Some ([], ditem')
end
trX': list (composite_transition_item IM)
Hproject_x': foldr equivocators_trace_project_folder
  (Some ([], x_descriptors)) tr =
Some (trX', initial_descriptors)
Htl: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) s []
Ht: input_valid_transition
  (preloaded_with_all_messages_vlsm FreeE) l
  (finite_trace_last is tr, iom) (
  s, oom)
proper_equivocator_descriptors IM final_descriptors
  (finite_trace_last is
     (tr ++
      [{|
         l := l;
         input := iom;
         destination := s;
         output := oom
       |}]))
  rewrite finite_trace_last_is_last.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
s: state (preloaded_with_all_messages_vlsm FreeE)
iom, oom: option message
l: label (preloaded_with_all_messages_vlsm FreeE)
initial_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  initial_descriptors is
x_descriptors: equivocator_descriptors IM
IHtr: proper_equivocator_descriptors IM x_descriptors
  (finite_trace_last is tr)
Htr: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) is tr
Hinit: initial_state_prop is
final_descriptors: equivocator_descriptors IM
x': list (composite_transition_item IM)
oitem': option (composite_transition_item IM)
ditem': equivocator_descriptors IM
oitem'': option transition_item
ditem'': MachineDescriptor (IM (projT1 l))
Heqprojecti: match final_descriptors (projT1 l) with
| NewMachine _ =>
    Some
      (None,
       final_descriptors (projT1 l))
| Existing j =>
    match
      equivocator_state_project
        (s (projT1 l)) j
    with
    | Some sj =>
        match projT2 l with
        | Spawn sn =>
            if
             decide
               (j =
                equivocator_state_last
                  (s (projT1 l)))
            then
             Some (None, NewMachine sn)
            else Some (None, Existing j)
        | ContinueWith i lx =>
            if decide (i = j)
            then
             Some
               (Some
                  {|
                    l := lx;
                    input := iom;
                    destination := sj;
                    output := oom
                  |}, Existing i)
            else Some (None, Existing j)
        | ForkWith i lx =>
            if
             decide
               (j =
                equivocator_state_last
                  (s (projT1 l)))
            then
             Some
               (Some
                  {|
                    l := lx;
                    input := iom;
                    destination := sj;
                    output := oom
                  |}, Existing i)
            else Some (None, Existing j)
        end
    | None => None
    end
end = Some (oitem'', ditem'')
Hproject_x: match oitem'' with
| Some item' =>
    Some
      (Some
         {|
           l :=
             existT (projT1 l)
               (VLSM.l item');
           input := iom;
           destination :=
             equivocators_state_project
               IM final_descriptors s;
           output := oom
         |},
       equivocator_descriptors_update IM
         final_descriptors 
         (projT1 l) ditem'')
| None =>
    Some
      (None,
       equivocator_descriptors_update IM
         final_descriptors 
         (projT1 l) ditem'')
end = Some (oitem', ditem')
Heqproject_x: Some (x', x_descriptors) =
match oitem' with
| Some item' => Some ([item'], ditem')
| None => Some ([], ditem')
end
trX': list (composite_transition_item IM)
Hproject_x': foldr equivocators_trace_project_folder
  (Some ([], x_descriptors)) tr =
Some (trX', initial_descriptors)
Htl: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) s []
Ht: input_valid_transition
  (preloaded_with_all_messages_vlsm FreeE) l
  (finite_trace_last is tr, iom) (
  s, oom)
proper_equivocator_descriptors IM final_descriptors
  (destination
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |}) simpl.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
s: state (preloaded_with_all_messages_vlsm FreeE)
iom, oom: option message
l: label (preloaded_with_all_messages_vlsm FreeE)
initial_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  initial_descriptors is
x_descriptors: equivocator_descriptors IM
IHtr: proper_equivocator_descriptors IM x_descriptors
  (finite_trace_last is tr)
Htr: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) is tr
Hinit: initial_state_prop is
final_descriptors: equivocator_descriptors IM
x': list (composite_transition_item IM)
oitem': option (composite_transition_item IM)
ditem': equivocator_descriptors IM
oitem'': option transition_item
ditem'': MachineDescriptor (IM (projT1 l))
Heqprojecti: match final_descriptors (projT1 l) with
| NewMachine _ =>
    Some
      (None,
       final_descriptors (projT1 l))
| Existing j =>
    match
      equivocator_state_project
        (s (projT1 l)) j
    with
    | Some sj =>
        match projT2 l with
        | Spawn sn =>
            if
             decide
               (j =
                equivocator_state_last
                  (s (projT1 l)))
            then
             Some (None, NewMachine sn)
            else Some (None, Existing j)
        | ContinueWith i lx =>
            if decide (i = j)
            then
             Some
               (Some
                  {|
                    l := lx;
                    input := iom;
                    destination := sj;
                    output := oom
                  |}, Existing i)
            else Some (None, Existing j)
        | ForkWith i lx =>
            if
             decide
               (j =
                equivocator_state_last
                  (s (projT1 l)))
            then
             Some
               (Some
                  {|
                    l := lx;
                    input := iom;
                    destination := sj;
                    output := oom
                  |}, Existing i)
            else Some (None, Existing j)
        end
    | None => None
    end
end = Some (oitem'', ditem'')
Hproject_x: match oitem'' with
| Some item' =>
    Some
      (Some
         {|
           l :=
             existT (projT1 l)
               (VLSM.l item');
           input := iom;
           destination :=
             equivocators_state_project
               IM final_descriptors s;
           output := oom
         |},
       equivocator_descriptors_update IM
         final_descriptors 
         (projT1 l) ditem'')
| None =>
    Some
      (None,
       equivocator_descriptors_update IM
         final_descriptors 
         (projT1 l) ditem'')
end = Some (oitem', ditem')
Heqproject_x: Some (x', x_descriptors) =
match oitem' with
| Some item' => Some ([item'], ditem')
| None => Some ([], ditem')
end
trX': list (composite_transition_item IM)
Hproject_x': foldr equivocators_trace_project_folder
  (Some ([], x_descriptors)) tr =
Some (trX', initial_descriptors)
Htl: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) s []
Ht: input_valid_transition
  (preloaded_with_all_messages_vlsm FreeE) l
  (finite_trace_last is tr, iom) (
  s, oom)
proper_equivocator_descriptors IM final_descriptors s
  match type of Heqprojecti with
  | match ?fd with _ => _ end = _ => destruct fd as [sn | j] eqn: Hfinali
  end.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
s: state (preloaded_with_all_messages_vlsm FreeE)
iom, oom: option message
l: label (preloaded_with_all_messages_vlsm FreeE)
initial_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  initial_descriptors is
x_descriptors: equivocator_descriptors IM
IHtr: proper_equivocator_descriptors IM x_descriptors
  (finite_trace_last is tr)
Htr: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) is tr
Hinit: initial_state_prop is
final_descriptors: equivocator_descriptors IM
x': list (composite_transition_item IM)
oitem': option (composite_transition_item IM)
ditem': equivocator_descriptors IM
oitem'': option transition_item
ditem'': MachineDescriptor (IM (projT1 l))
sn: state (IM (projT1 l))
Hfinali: final_descriptors (projT1 l) = NewMachine sn
Heqprojecti: Some (None, NewMachine sn) =
Some (oitem'', ditem'')
Hproject_x: match oitem'' with
| Some item' =>
    Some
      (Some
         {|
           l :=
             existT (projT1 l)
               (VLSM.l item');
           input := iom;
           destination :=
             equivocators_state_project
               IM final_descriptors s;
           output := oom
         |},
       equivocator_descriptors_update IM
         final_descriptors 
         (projT1 l) ditem'')
| None =>
    Some
      (None,
       equivocator_descriptors_update IM
         final_descriptors 
         (projT1 l) ditem'')
end = Some (oitem', ditem')
Heqproject_x: Some (x', x_descriptors) =
match oitem' with
| Some item' => Some ([item'], ditem')
| None => Some ([], ditem')
end
trX': list (composite_transition_item IM)
Hproject_x': foldr equivocators_trace_project_folder
  (Some ([], x_descriptors)) tr =
Some (trX', initial_descriptors)
Htl: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) s []
Ht: input_valid_transition
  (preloaded_with_all_messages_vlsm FreeE) l
  (finite_trace_last is tr, iom) (
  s, oom)
proper_equivocator_descriptors IM final_descriptors s
message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
s: state (preloaded_with_all_messages_vlsm FreeE)
iom, oom: option message
l: label (preloaded_with_all_messages_vlsm FreeE)
initial_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  initial_descriptors is
x_descriptors: equivocator_descriptors IM
IHtr: proper_equivocator_descriptors IM x_descriptors
  (finite_trace_last is tr)
Htr: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) is tr
Hinit: initial_state_prop is
final_descriptors: equivocator_descriptors IM
x': list (composite_transition_item IM)
oitem': option (composite_transition_item IM)
ditem': equivocator_descriptors IM
oitem'': option transition_item
ditem'': MachineDescriptor (IM (projT1 l))
j: nat
Hfinali: final_descriptors (projT1 l) = Existing j
Heqprojecti: match
  equivocator_state_project
    (s (projT1 l)) j
with
| Some sj =>
    match projT2 l with
    | Spawn sn =>
        if
         decide
           (j =
            equivocator_state_last
              (s (projT1 l)))
        then Some (None, NewMachine sn)
        else Some (None, Existing j)
    | ContinueWith i lx =>
        if decide (i = j)
        then
         Some
           (Some
              {|
                l := lx;
                input := iom;
                destination := sj;
                output := oom
              |}, Existing i)
        else Some (None, Existing j)
    | ForkWith i lx =>
        if
         decide
           (j =
            equivocator_state_last
              (s (projT1 l)))
        then
         Some
           (Some
              {|
                l := lx;
                input := iom;
                destination := sj;
                output := oom
              |}, Existing i)
        else Some (None, Existing j)
    end
| None => None
end = Some (oitem'', ditem'')
Hproject_x: match oitem'' with
| Some item' =>
    Some
      (Some
         {|
           l :=
             existT (projT1 l)
               (VLSM.l item');
           input := iom;
           destination :=
             equivocators_state_project
               IM final_descriptors s;
           output := oom
         |},
       equivocator_descriptors_update IM
         final_descriptors 
         (projT1 l) ditem'')
| None =>
    Some
      (None,
       equivocator_descriptors_update IM
         final_descriptors 
         (projT1 l) ditem'')
end = Some (oitem', ditem')
Heqproject_x: Some (x', x_descriptors) =
match oitem' with
| Some item' => Some ([item'], ditem')
| None => Some ([], ditem')
end
trX': list (composite_transition_item IM)
Hproject_x': foldr equivocators_trace_project_folder
  (Some ([], x_descriptors)) tr =
Some (trX', initial_descriptors)
Htl: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) s []
Ht: input_valid_transition
  (preloaded_with_all_messages_vlsm FreeE) l
  (finite_trace_last is tr, iom) (
  s, oom)
proper_equivocator_descriptors IM final_descriptors s
  -message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
s: state (preloaded_with_all_messages_vlsm FreeE)
iom, oom: option message
l: label (preloaded_with_all_messages_vlsm FreeE)
initial_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  initial_descriptors is
x_descriptors: equivocator_descriptors IM
IHtr: proper_equivocator_descriptors IM x_descriptors
  (finite_trace_last is tr)
Htr: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) is tr
Hinit: initial_state_prop is
final_descriptors: equivocator_descriptors IM
x': list (composite_transition_item IM)
oitem': option (composite_transition_item IM)
ditem': equivocator_descriptors IM
oitem'': option transition_item
ditem'': MachineDescriptor (IM (projT1 l))
sn: state (IM (projT1 l))
Hfinali: final_descriptors (projT1 l) = NewMachine sn
Heqprojecti: Some (None, NewMachine sn) =
Some (oitem'', ditem'')
Hproject_x: match oitem'' with
| Some item' =>
    Some
      (Some
         {|
           l :=
             existT (projT1 l)
               (VLSM.l item');
           input := iom;
           destination :=
             equivocators_state_project
               IM final_descriptors s;
           output := oom
         |},
       equivocator_descriptors_update IM
         final_descriptors 
         (projT1 l) ditem'')
| None =>
    Some
      (None,
       equivocator_descriptors_update IM
         final_descriptors 
         (projT1 l) ditem'')
end = Some (oitem', ditem')
Heqproject_x: Some (x', x_descriptors) =
match oitem' with
| Some item' => Some ([item'], ditem')
| None => Some ([], ditem')
end
trX': list (composite_transition_item IM)
Hproject_x': foldr equivocators_trace_project_folder
  (Some ([], x_descriptors)) tr =
Some (trX', initial_descriptors)
Htl: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) s []
Ht: input_valid_transition
  (preloaded_with_all_messages_vlsm FreeE) l
  (finite_trace_last is tr, iom) (
  s, oom)
proper_equivocator_descriptors IM final_descriptors s inversion Heqprojecti.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
s: state (preloaded_with_all_messages_vlsm FreeE)
iom, oom: option message
l: label (preloaded_with_all_messages_vlsm FreeE)
initial_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  initial_descriptors is
x_descriptors: equivocator_descriptors IM
IHtr: proper_equivocator_descriptors IM x_descriptors
  (finite_trace_last is tr)
Htr: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) is tr
Hinit: initial_state_prop is
final_descriptors: equivocator_descriptors IM
x': list (composite_transition_item IM)
oitem': option (composite_transition_item IM)
ditem': equivocator_descriptors IM
oitem'': option transition_item
ditem'': MachineDescriptor (IM (projT1 l))
sn: state (IM (projT1 l))
Hfinali: final_descriptors (projT1 l) = NewMachine sn
Heqprojecti: Some (None, NewMachine sn) =
Some (oitem'', ditem'')
Hproject_x: match oitem'' with
| Some item' =>
    Some
      (Some
         {|
           l :=
             existT (projT1 l)
               (VLSM.l item');
           input := iom;
           destination :=
             equivocators_state_project
               IM final_descriptors s;
           output := oom
         |},
       equivocator_descriptors_update IM
         final_descriptors 
         (projT1 l) ditem'')
| None =>
    Some
      (None,
       equivocator_descriptors_update IM
         final_descriptors 
         (projT1 l) ditem'')
end = Some (oitem', ditem')
Heqproject_x: Some (x', x_descriptors) =
match oitem' with
| Some item' => Some ([item'], ditem')
| None => Some ([], ditem')
end
trX': list (composite_transition_item IM)
Hproject_x': foldr equivocators_trace_project_folder
  (Some ([], x_descriptors)) tr =
Some (trX', initial_descriptors)
Htl: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) s []
Ht: input_valid_transition
  (preloaded_with_all_messages_vlsm FreeE) l
  (finite_trace_last is tr, iom) (
  s, oom)
H2: None = oitem''
H3: NewMachine sn = ditem''
proper_equivocator_descriptors IM final_descriptors s subst.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
s: state (preloaded_with_all_messages_vlsm FreeE)
iom, oom: option message
l: label (preloaded_with_all_messages_vlsm FreeE)
initial_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  initial_descriptors is
x_descriptors: equivocator_descriptors IM
IHtr: proper_equivocator_descriptors IM x_descriptors
  (finite_trace_last is tr)
Htr: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) is tr
Hinit: initial_state_prop is
final_descriptors: equivocator_descriptors IM
x': list (composite_transition_item IM)
oitem': option (composite_transition_item IM)
ditem': equivocator_descriptors IM
sn: state (IM (projT1 l))
Hfinali: final_descriptors (projT1 l) = NewMachine sn
Heqprojecti: Some (None, NewMachine sn) =
Some (None, NewMachine sn)
Hproject_x: Some
  (None,
   equivocator_descriptors_update IM
     final_descriptors 
     (projT1 l) (NewMachine sn)) =
Some (oitem', ditem')
Heqproject_x: Some (x', x_descriptors) =
match oitem' with
| Some item' => Some ([item'], ditem')
| None => Some ([], ditem')
end
trX': list (composite_transition_item IM)
Hproject_x': foldr equivocators_trace_project_folder
  (Some ([], x_descriptors)) tr =
Some (trX', initial_descriptors)
Htl: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) s []
Ht: input_valid_transition
  (preloaded_with_all_messages_vlsm FreeE) l
  (finite_trace_last is tr, iom) (
  s, oom)
proper_equivocator_descriptors IM final_descriptors s clear Heqprojecti.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
s: state (preloaded_with_all_messages_vlsm FreeE)
iom, oom: option message
l: label (preloaded_with_all_messages_vlsm FreeE)
initial_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  initial_descriptors is
x_descriptors: equivocator_descriptors IM
IHtr: proper_equivocator_descriptors IM x_descriptors
  (finite_trace_last is tr)
Htr: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) is tr
Hinit: initial_state_prop is
final_descriptors: equivocator_descriptors IM
x': list (composite_transition_item IM)
oitem': option (composite_transition_item IM)
ditem': equivocator_descriptors IM
sn: state (IM (projT1 l))
Hfinali: final_descriptors (projT1 l) = NewMachine sn
Hproject_x: Some
  (None,
   equivocator_descriptors_update IM
     final_descriptors 
     (projT1 l) (NewMachine sn)) =
Some (oitem', ditem')
Heqproject_x: Some (x', x_descriptors) =
match oitem' with
| Some item' => Some ([item'], ditem')
| None => Some ([], ditem')
end
trX': list (composite_transition_item IM)
Hproject_x': foldr equivocators_trace_project_folder
  (Some ([], x_descriptors)) tr =
Some (trX', initial_descriptors)
Htl: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) s []
Ht: input_valid_transition
  (preloaded_with_all_messages_vlsm FreeE) l
  (finite_trace_last is tr, iom) (
  s, oom)
proper_equivocator_descriptors IM final_descriptors s
    inversion Hproject_x.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
s: state (preloaded_with_all_messages_vlsm FreeE)
iom, oom: option message
l: label (preloaded_with_all_messages_vlsm FreeE)
initial_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  initial_descriptors is
x_descriptors: equivocator_descriptors IM
IHtr: proper_equivocator_descriptors IM x_descriptors
  (finite_trace_last is tr)
Htr: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) is tr
Hinit: initial_state_prop is
final_descriptors: equivocator_descriptors IM
x': list (composite_transition_item IM)
oitem': option (composite_transition_item IM)
ditem': equivocator_descriptors IM
sn: state (IM (projT1 l))
Hfinali: final_descriptors (projT1 l) = NewMachine sn
Hproject_x: Some
  (None,
   equivocator_descriptors_update IM
     final_descriptors 
     (projT1 l) (NewMachine sn)) =
Some (oitem', ditem')
Heqproject_x: Some (x', x_descriptors) =
match oitem' with
| Some item' => Some ([item'], ditem')
| None => Some ([], ditem')
end
trX': list (composite_transition_item IM)
Hproject_x': foldr equivocators_trace_project_folder
  (Some ([], x_descriptors)) tr =
Some (trX', initial_descriptors)
Htl: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) s []
Ht: input_valid_transition
  (preloaded_with_all_messages_vlsm FreeE) l
  (finite_trace_last is tr, iom) (
  s, oom)
H2: None = oitem'
H3: equivocator_descriptors_update IM
  final_descriptors (projT1 l) 
  (NewMachine sn) = ditem'
proper_equivocator_descriptors IM final_descriptors s subst; clear Hproject_x.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
s: state (preloaded_with_all_messages_vlsm FreeE)
iom, oom: option message
l: label (preloaded_with_all_messages_vlsm FreeE)
initial_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  initial_descriptors is
x_descriptors: equivocator_descriptors IM
IHtr: proper_equivocator_descriptors IM x_descriptors
  (finite_trace_last is tr)
Htr: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) is tr
Hinit: initial_state_prop is
final_descriptors: equivocator_descriptors IM
x': list (composite_transition_item IM)
sn: state (IM (projT1 l))
Hfinali: final_descriptors (projT1 l) = NewMachine sn
Heqproject_x: Some (x', x_descriptors) =
Some
  ([],
   equivocator_descriptors_update IM
     final_descriptors 
     (projT1 l) (NewMachine sn))
trX': list (composite_transition_item IM)
Hproject_x': foldr equivocators_trace_project_folder
  (Some ([], x_descriptors)) tr =
Some (trX', initial_descriptors)
Htl: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) s []
Ht: input_valid_transition
  (preloaded_with_all_messages_vlsm FreeE) l
  (finite_trace_last is tr, iom) (
  s, oom)
proper_equivocator_descriptors IM final_descriptors s
    inversion Heqproject_x.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
s: state (preloaded_with_all_messages_vlsm FreeE)
iom, oom: option message
l: label (preloaded_with_all_messages_vlsm FreeE)
initial_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  initial_descriptors is
x_descriptors: equivocator_descriptors IM
IHtr: proper_equivocator_descriptors IM x_descriptors
  (finite_trace_last is tr)
Htr: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) is tr
Hinit: initial_state_prop is
final_descriptors: equivocator_descriptors IM
x': list (composite_transition_item IM)
sn: state (IM (projT1 l))
Hfinali: final_descriptors (projT1 l) = NewMachine sn
Heqproject_x: Some (x', x_descriptors) =
Some
  ([],
   equivocator_descriptors_update IM
     final_descriptors 
     (projT1 l) (NewMachine sn))
trX': list (composite_transition_item IM)
Hproject_x': foldr equivocators_trace_project_folder
  (Some ([], x_descriptors)) tr =
Some (trX', initial_descriptors)
Htl: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) s []
Ht: input_valid_transition
  (preloaded_with_all_messages_vlsm FreeE) l
  (finite_trace_last is tr, iom) (
  s, oom)
H2: x' = []
H3: x_descriptors =
equivocator_descriptors_update IM
  final_descriptors (projT1 l) 
  (NewMachine sn)
proper_equivocator_descriptors IM final_descriptors s subst.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
s: state (preloaded_with_all_messages_vlsm FreeE)
iom, oom: option message
l: label (preloaded_with_all_messages_vlsm FreeE)
initial_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  initial_descriptors is
final_descriptors: equivocator_descriptors IM
sn: state (IM (projT1 l))
IHtr: proper_equivocator_descriptors IM
  (equivocator_descriptors_update IM
     final_descriptors 
     (projT1 l) (NewMachine sn))
  (finite_trace_last is tr)
Htr: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) is tr
Hinit: initial_state_prop is
Hfinali: final_descriptors (projT1 l) = NewMachine sn
Heqproject_x: Some
  ([],
   equivocator_descriptors_update IM
     final_descriptors 
     (projT1 l) (NewMachine sn)) =
Some
  ([],
   equivocator_descriptors_update IM
     final_descriptors 
     (projT1 l) (NewMachine sn))
trX': list (composite_transition_item IM)
Hproject_x': foldr equivocators_trace_project_folder
  (Some
     ([],
      equivocator_descriptors_update IM
        final_descriptors 
        (projT1 l) (NewMachine sn))) tr =
Some (trX', initial_descriptors)
Htl: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) s []
Ht: input_valid_transition
  (preloaded_with_all_messages_vlsm FreeE) l
  (finite_trace_last is tr, iom) (
  s, oom)
proper_equivocator_descriptors IM final_descriptors s clear Heqproject_x.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
s: state (preloaded_with_all_messages_vlsm FreeE)
iom, oom: option message
l: label (preloaded_with_all_messages_vlsm FreeE)
initial_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  initial_descriptors is
final_descriptors: equivocator_descriptors IM
sn: state (IM (projT1 l))
IHtr: proper_equivocator_descriptors IM
  (equivocator_descriptors_update IM
     final_descriptors 
     (projT1 l) (NewMachine sn))
  (finite_trace_last is tr)
Htr: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) is tr
Hinit: initial_state_prop is
Hfinali: final_descriptors (projT1 l) = NewMachine sn
trX': list (composite_transition_item IM)
Hproject_x': foldr equivocators_trace_project_folder
  (Some
     ([],
      equivocator_descriptors_update IM
        final_descriptors 
        (projT1 l) (NewMachine sn))) tr =
Some (trX', initial_descriptors)
Htl: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) s []
Ht: input_valid_transition
  (preloaded_with_all_messages_vlsm FreeE) l
  (finite_trace_last is tr, iom) (
  s, oom)
proper_equivocator_descriptors IM final_descriptors s
    intro e.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
s: state (preloaded_with_all_messages_vlsm FreeE)
iom, oom: option message
l: label (preloaded_with_all_messages_vlsm FreeE)
initial_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  initial_descriptors is
final_descriptors: equivocator_descriptors IM
sn: state (IM (projT1 l))
IHtr: proper_equivocator_descriptors IM
  (equivocator_descriptors_update IM
     final_descriptors 
     (projT1 l) (NewMachine sn))
  (finite_trace_last is tr)
Htr: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) is tr
Hinit: initial_state_prop is
Hfinali: final_descriptors (projT1 l) = NewMachine sn
trX': list (composite_transition_item IM)
Hproject_x': foldr equivocators_trace_project_folder
  (Some
     ([],
      equivocator_descriptors_update IM
        final_descriptors 
        (projT1 l) (NewMachine sn))) tr =
Some (trX', initial_descriptors)
Htl: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) s []
Ht: input_valid_transition
  (preloaded_with_all_messages_vlsm FreeE) l
  (finite_trace_last is tr, iom) (
  s, oom)
e: index
proper_descriptor (IM e) (final_descriptors e) (s e) specialize (IHtr e).message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
s: state (preloaded_with_all_messages_vlsm FreeE)
iom, oom: option message
l: label (preloaded_with_all_messages_vlsm FreeE)
initial_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  initial_descriptors is
final_descriptors: equivocator_descriptors IM
sn: state (IM (projT1 l))
e: index
IHtr: proper_descriptor (IM e)
  (equivocator_descriptors_update IM
     final_descriptors 
     (projT1 l) (NewMachine sn) e)
  (finite_trace_last is tr e)
Htr: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) is tr
Hinit: initial_state_prop is
Hfinali: final_descriptors (projT1 l) = NewMachine sn
trX': list (composite_transition_item IM)
Hproject_x': foldr equivocators_trace_project_folder
  (Some
     ([],
      equivocator_descriptors_update IM
        final_descriptors 
        (projT1 l) (NewMachine sn))) tr =
Some (trX', initial_descriptors)
Htl: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) s []
Ht: input_valid_transition
  (preloaded_with_all_messages_vlsm FreeE) l
  (finite_trace_last is tr, iom) (
  s, oom)
proper_descriptor (IM e) (final_descriptors e) (s e)
    destruct (decide (e = projT1 l)).message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
s: state (preloaded_with_all_messages_vlsm FreeE)
iom, oom: option message
l: label (preloaded_with_all_messages_vlsm FreeE)
initial_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  initial_descriptors is
final_descriptors: equivocator_descriptors IM
sn: state (IM (projT1 l))
e: index
IHtr: proper_descriptor (IM e)
  (equivocator_descriptors_update IM
     final_descriptors 
     (projT1 l) (NewMachine sn) e)
  (finite_trace_last is tr e)
Htr: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) is tr
Hinit: initial_state_prop is
Hfinali: final_descriptors (projT1 l) = NewMachine sn
trX': list (composite_transition_item IM)
Hproject_x': foldr equivocators_trace_project_folder
  (Some
     ([],
      equivocator_descriptors_update IM
        final_descriptors 
        (projT1 l) (NewMachine sn))) tr =
Some (trX', initial_descriptors)
Htl: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) s []
Ht: input_valid_transition
  (preloaded_with_all_messages_vlsm FreeE) l
  (finite_trace_last is tr, iom) (
  s, oom)
e0: e = projT1 l
proper_descriptor (IM e) (final_descriptors e) (s e)
message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
s: state (preloaded_with_all_messages_vlsm FreeE)
iom, oom: option message
l: label (preloaded_with_all_messages_vlsm FreeE)
initial_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  initial_descriptors is
final_descriptors: equivocator_descriptors IM
sn: state (IM (projT1 l))
e: index
IHtr: proper_descriptor (IM e)
  (equivocator_descriptors_update IM
     final_descriptors 
     (projT1 l) (NewMachine sn) e)
  (finite_trace_last is tr e)
Htr: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) is tr
Hinit: initial_state_prop is
Hfinali: final_descriptors (projT1 l) = NewMachine sn
trX': list (composite_transition_item IM)
Hproject_x': foldr equivocators_trace_project_folder
  (Some
     ([],
      equivocator_descriptors_update IM
        final_descriptors 
        (projT1 l) (NewMachine sn))) tr =
Some (trX', initial_descriptors)
Htl: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) s []
Ht: input_valid_transition
  (preloaded_with_all_messages_vlsm FreeE) l
  (finite_trace_last is tr, iom) (
  s, oom)
n: e ≠ projT1 l
proper_descriptor (IM e) (final_descriptors e) (s e)
    +message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
s: state (preloaded_with_all_messages_vlsm FreeE)
iom, oom: option message
l: label (preloaded_with_all_messages_vlsm FreeE)
initial_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  initial_descriptors is
final_descriptors: equivocator_descriptors IM
sn: state (IM (projT1 l))
e: index
IHtr: proper_descriptor (IM e)
  (equivocator_descriptors_update IM
     final_descriptors 
     (projT1 l) (NewMachine sn) e)
  (finite_trace_last is tr e)
Htr: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) is tr
Hinit: initial_state_prop is
Hfinali: final_descriptors (projT1 l) = NewMachine sn
trX': list (composite_transition_item IM)
Hproject_x': foldr equivocators_trace_project_folder
  (Some
     ([],
      equivocator_descriptors_update IM
        final_descriptors 
        (projT1 l) (NewMachine sn))) tr =
Some (trX', initial_descriptors)
Htl: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) s []
Ht: input_valid_transition
  (preloaded_with_all_messages_vlsm FreeE) l
  (finite_trace_last is tr, iom) (
  s, oom)
e0: e = projT1 l
proper_descriptor (IM e) (final_descriptors e) (s e) subst.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
s: state (preloaded_with_all_messages_vlsm FreeE)
iom, oom: option message
l: label (preloaded_with_all_messages_vlsm FreeE)
initial_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  initial_descriptors is
final_descriptors: equivocator_descriptors IM
sn: state (IM (projT1 l))
IHtr: proper_descriptor (IM (projT1 l))
  (equivocator_descriptors_update IM
     final_descriptors 
     (projT1 l) (NewMachine sn) 
     (projT1 l))
  (finite_trace_last is tr (projT1 l))
Htr: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) is tr
Hinit: initial_state_prop is
Hfinali: final_descriptors (projT1 l) = NewMachine sn
trX': list (composite_transition_item IM)
Hproject_x': foldr equivocators_trace_project_folder
  (Some
     ([],
      equivocator_descriptors_update IM
        final_descriptors 
        (projT1 l) (NewMachine sn))) tr =
Some (trX', initial_descriptors)
Htl: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) s []
Ht: input_valid_transition
  (preloaded_with_all_messages_vlsm FreeE) l
  (finite_trace_last is tr, iom) (
  s, oom)
proper_descriptor (IM (projT1 l))
  (final_descriptors (projT1 l)) (s (projT1 l))
      rewrite equivocator_descriptors_update_eq in IHtr.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
s: state (preloaded_with_all_messages_vlsm FreeE)
iom, oom: option message
l: label (preloaded_with_all_messages_vlsm FreeE)
initial_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  initial_descriptors is
final_descriptors: equivocator_descriptors IM
sn: state (IM (projT1 l))
IHtr: proper_descriptor (IM (projT1 l))
  (NewMachine sn)
  (finite_trace_last is tr (projT1 l))
Htr: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) is tr
Hinit: initial_state_prop is
Hfinali: final_descriptors (projT1 l) = NewMachine sn
trX': list (composite_transition_item IM)
Hproject_x': foldr equivocators_trace_project_folder
  (Some
     ([],
      equivocator_descriptors_update IM
        final_descriptors 
        (projT1 l) (NewMachine sn))) tr =
Some (trX', initial_descriptors)
Htl: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) s []
Ht: input_valid_transition
  (preloaded_with_all_messages_vlsm FreeE) l
  (finite_trace_last is tr, iom) (
  s, oom)
proper_descriptor (IM (projT1 l))
  (final_descriptors (projT1 l)) (s (projT1 l))
      by cbn; rewrite Hfinali.
    +message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
s: state (preloaded_with_all_messages_vlsm FreeE)
iom, oom: option message
l: label (preloaded_with_all_messages_vlsm FreeE)
initial_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  initial_descriptors is
final_descriptors: equivocator_descriptors IM
sn: state (IM (projT1 l))
e: index
IHtr: proper_descriptor (IM e)
  (equivocator_descriptors_update IM
     final_descriptors 
     (projT1 l) (NewMachine sn) e)
  (finite_trace_last is tr e)
Htr: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) is tr
Hinit: initial_state_prop is
Hfinali: final_descriptors (projT1 l) = NewMachine sn
trX': list (composite_transition_item IM)
Hproject_x': foldr equivocators_trace_project_folder
  (Some
     ([],
      equivocator_descriptors_update IM
        final_descriptors 
        (projT1 l) (NewMachine sn))) tr =
Some (trX', initial_descriptors)
Htl: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) s []
Ht: input_valid_transition
  (preloaded_with_all_messages_vlsm FreeE) l
  (finite_trace_last is tr, iom) (
  s, oom)
n: e ≠ projT1 l
proper_descriptor (IM e) (final_descriptors e) (s e) state_update_simpl.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
s: state (preloaded_with_all_messages_vlsm FreeE)
iom, oom: option message
l: label (preloaded_with_all_messages_vlsm FreeE)
initial_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  initial_descriptors is
final_descriptors: equivocator_descriptors IM
sn: state (IM (projT1 l))
e: index
IHtr: proper_descriptor (IM e) 
  (final_descriptors e)
  (finite_trace_last is tr e)
Htr: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) is tr
Hinit: initial_state_prop is
Hfinali: final_descriptors (projT1 l) = NewMachine sn
trX': list (composite_transition_item IM)
Hproject_x': foldr equivocators_trace_project_folder
  (Some ([], final_descriptors)) tr =
Some (trX', initial_descriptors)
Htl: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) s []
Ht: input_valid_transition
  (preloaded_with_all_messages_vlsm FreeE) l
  (finite_trace_last is tr, iom) (
  s, oom)
n: e ≠ projT1 l
proper_descriptor (IM e) (final_descriptors e) (s e)
      destruct Ht as [Hv Ht]; cbn in Ht.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
s: state (preloaded_with_all_messages_vlsm FreeE)
iom, oom: option message
l: label (preloaded_with_all_messages_vlsm FreeE)
initial_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  initial_descriptors is
final_descriptors: equivocator_descriptors IM
sn: state (IM (projT1 l))
e: index
IHtr: proper_descriptor (IM e) 
  (final_descriptors e)
  (finite_trace_last is tr e)
Htr: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) is tr
Hinit: initial_state_prop is
Hfinali: final_descriptors (projT1 l) = NewMachine sn
trX': list (composite_transition_item IM)
Hproject_x': foldr equivocators_trace_project_folder
  (Some ([], final_descriptors)) tr =
Some (trX', initial_descriptors)
Htl: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) s []
Hv: input_valid
  (preloaded_with_all_messages_vlsm FreeE) l
  (finite_trace_last is tr, iom)
Ht: (let (i, li) := l in
 let (si', om') :=
   equivocator_transition 
     (IM i) li (finite_trace_last is tr i, iom) in
 (state_update (equivocator_IM IM)
    (finite_trace_last is tr) i si', om')) =
(s, oom)
n: e ≠ projT1 l
proper_descriptor (IM e) (final_descriptors e) (s e)
      destruct l as (i, li).message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
s: state (preloaded_with_all_messages_vlsm FreeE)
iom, oom: option message
i: index
li: label (equivocator_IM IM i)
initial_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  initial_descriptors is
final_descriptors: equivocator_descriptors IM
sn: state (IM (projT1 (existT i li)))
e: index
IHtr: proper_descriptor (IM e) 
  (final_descriptors e)
  (finite_trace_last is tr e)
Htr: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) is tr
Hinit: initial_state_prop is
Hfinali: final_descriptors (projT1 (existT i li)) =
NewMachine sn
trX': list (composite_transition_item IM)
Hproject_x': foldr equivocators_trace_project_folder
  (Some ([], final_descriptors)) tr =
Some (trX', initial_descriptors)
Htl: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) s []
Hv: input_valid
  (preloaded_with_all_messages_vlsm FreeE)
  (existT i li) (finite_trace_last is tr, iom)
Ht: (let (si', om') :=
   equivocator_transition 
     (IM i) li (finite_trace_last is tr i, iom) in
 (state_update (equivocator_IM IM)
    (finite_trace_last is tr) i si', om')) =
(s, oom)
n: e ≠ projT1 (existT i li)
proper_descriptor (IM e) (final_descriptors e) (s e)
      case_match.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
s: state (preloaded_with_all_messages_vlsm FreeE)
iom, oom: option message
i: index
li: label (equivocator_IM IM i)
initial_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  initial_descriptors is
final_descriptors: equivocator_descriptors IM
sn: state (IM (projT1 (existT i li)))
e: index
IHtr: proper_descriptor (IM e) 
  (final_descriptors e)
  (finite_trace_last is tr e)
Htr: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) is tr
Hinit: initial_state_prop is
Hfinali: final_descriptors (projT1 (existT i li)) =
NewMachine sn
trX': list (composite_transition_item IM)
Hproject_x': foldr equivocators_trace_project_folder
  (Some ([], final_descriptors)) tr =
Some (trX', initial_descriptors)
Htl: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) s []
Hv: input_valid
  (preloaded_with_all_messages_vlsm FreeE)
  (existT i li) (finite_trace_last is tr, iom)
e0: equivocator_state (IM i)
o: option message
H1: equivocator_transition 
  (IM i) li (finite_trace_last is tr i, iom) =
(e0, o)
Ht: (state_update (equivocator_IM IM)
   (finite_trace_last is tr) i e0, o) = (
s, oom)
n: e ≠ projT1 (existT i li)
proper_descriptor (IM e) (final_descriptors e) (s e)
      inversion Ht.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
s: state (preloaded_with_all_messages_vlsm FreeE)
iom, oom: option message
i: index
li: label (equivocator_IM IM i)
initial_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  initial_descriptors is
final_descriptors: equivocator_descriptors IM
sn: state (IM (projT1 (existT i li)))
e: index
IHtr: proper_descriptor (IM e) 
  (final_descriptors e)
  (finite_trace_last is tr e)
Htr: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) is tr
Hinit: initial_state_prop is
Hfinali: final_descriptors (projT1 (existT i li)) =
NewMachine sn
trX': list (composite_transition_item IM)
Hproject_x': foldr equivocators_trace_project_folder
  (Some ([], final_descriptors)) tr =
Some (trX', initial_descriptors)
Htl: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) s []
Hv: input_valid
  (preloaded_with_all_messages_vlsm FreeE)
  (existT i li) (finite_trace_last is tr, iom)
e0: equivocator_state (IM i)
o: option message
H1: equivocator_transition 
  (IM i) li (finite_trace_last is tr i, iom) =
(e0, o)
Ht: (state_update (equivocator_IM IM)
   (finite_trace_last is tr) i e0, o) = (
s, oom)
n: e ≠ projT1 (existT i li)
H3: state_update (equivocator_IM IM)
  (finite_trace_last is tr) i e0 = s
H4: o = oom
proper_descriptor (IM e) (final_descriptors e)
  (state_update (equivocator_IM IM)
     (finite_trace_last is tr) i e0 e) subst.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
iom, oom: option message
i: index
li: label (equivocator_IM IM i)
initial_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  initial_descriptors is
final_descriptors: equivocator_descriptors IM
sn: state (IM (projT1 (existT i li)))
e: index
IHtr: proper_descriptor (IM e) 
  (final_descriptors e)
  (finite_trace_last is tr e)
Htr: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) is tr
Hinit: initial_state_prop is
Hfinali: final_descriptors (projT1 (existT i li)) =
NewMachine sn
trX': list (composite_transition_item IM)
Hproject_x': foldr equivocators_trace_project_folder
  (Some ([], final_descriptors)) tr =
Some (trX', initial_descriptors)
e0: equivocator_state (IM i)
Htl: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE)
  (state_update (equivocator_IM IM)
     (finite_trace_last is tr) i e0) []
Hv: input_valid
  (preloaded_with_all_messages_vlsm FreeE)
  (existT i li) (finite_trace_last is tr, iom)
Ht: (state_update (equivocator_IM IM)
   (finite_trace_last is tr) i e0, oom) =
(state_update (equivocator_IM IM)
   (finite_trace_last is tr) i e0, oom)
H1: equivocator_transition 
  (IM i) li (finite_trace_last is tr i, iom) =
(e0, oom)
n: e ≠ projT1 (existT i li)
proper_descriptor (IM e) (final_descriptors e)
  (state_update (equivocator_IM IM)
     (finite_trace_last is tr) i e0 e) simpl in n.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
iom, oom: option message
i: index
li: label (equivocator_IM IM i)
initial_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  initial_descriptors is
final_descriptors: equivocator_descriptors IM
sn: state (IM (projT1 (existT i li)))
e: index
IHtr: proper_descriptor (IM e) 
  (final_descriptors e)
  (finite_trace_last is tr e)
Htr: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) is tr
Hinit: initial_state_prop is
Hfinali: final_descriptors (projT1 (existT i li)) =
NewMachine sn
trX': list (composite_transition_item IM)
Hproject_x': foldr equivocators_trace_project_folder
  (Some ([], final_descriptors)) tr =
Some (trX', initial_descriptors)
e0: equivocator_state (IM i)
Htl: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE)
  (state_update (equivocator_IM IM)
     (finite_trace_last is tr) i e0) []
Hv: input_valid
  (preloaded_with_all_messages_vlsm FreeE)
  (existT i li) (finite_trace_last is tr, iom)
Ht: (state_update (equivocator_IM IM)
   (finite_trace_last is tr) i e0, oom) =
(state_update (equivocator_IM IM)
   (finite_trace_last is tr) i e0, oom)
H1: equivocator_transition 
  (IM i) li (finite_trace_last is tr i, iom) =
(e0, oom)
n: e ≠ i
proper_descriptor (IM e) (final_descriptors e)
  (state_update (equivocator_IM IM)
     (finite_trace_last is tr) i e0 e)
      by state_update_simpl.
  -message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
s: state (preloaded_with_all_messages_vlsm FreeE)
iom, oom: option message
l: label (preloaded_with_all_messages_vlsm FreeE)
initial_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  initial_descriptors is
x_descriptors: equivocator_descriptors IM
IHtr: proper_equivocator_descriptors IM x_descriptors
  (finite_trace_last is tr)
Htr: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) is tr
Hinit: initial_state_prop is
final_descriptors: equivocator_descriptors IM
x': list (composite_transition_item IM)
oitem': option (composite_transition_item IM)
ditem': equivocator_descriptors IM
oitem'': option transition_item
ditem'': MachineDescriptor (IM (projT1 l))
j: nat
Hfinali: final_descriptors (projT1 l) = Existing j
Heqprojecti: match
  equivocator_state_project
    (s (projT1 l)) j
with
| Some sj =>
    match projT2 l with
    | Spawn sn =>
        if
         decide
           (j =
            equivocator_state_last
              (s (projT1 l)))
        then Some (None, NewMachine sn)
        else Some (None, Existing j)
    | ContinueWith i lx =>
        if decide (i = j)
        then
         Some
           (Some
              {|
                l := lx;
                input := iom;
                destination := sj;
                output := oom
              |}, Existing i)
        else Some (None, Existing j)
    | ForkWith i lx =>
        if
         decide
           (j =
            equivocator_state_last
              (s (projT1 l)))
        then
         Some
           (Some
              {|
                l := lx;
                input := iom;
                destination := sj;
                output := oom
              |}, Existing i)
        else Some (None, Existing j)
    end
| None => None
end = Some (oitem'', ditem'')
Hproject_x: match oitem'' with
| Some item' =>
    Some
      (Some
         {|
           l :=
             existT (projT1 l)
               (VLSM.l item');
           input := iom;
           destination :=
             equivocators_state_project
               IM final_descriptors s;
           output := oom
         |},
       equivocator_descriptors_update IM
         final_descriptors 
         (projT1 l) ditem'')
| None =>
    Some
      (None,
       equivocator_descriptors_update IM
         final_descriptors 
         (projT1 l) ditem'')
end = Some (oitem', ditem')
Heqproject_x: Some (x', x_descriptors) =
match oitem' with
| Some item' => Some ([item'], ditem')
| None => Some ([], ditem')
end
trX': list (composite_transition_item IM)
Hproject_x': foldr equivocators_trace_project_folder
  (Some ([], x_descriptors)) tr =
Some (trX', initial_descriptors)
Htl: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) s []
Ht: input_valid_transition
  (preloaded_with_all_messages_vlsm FreeE) l
  (finite_trace_last is tr, iom) (
  s, oom)
proper_equivocator_descriptors IM final_descriptors s destruct l as (i, li).message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
s: state (preloaded_with_all_messages_vlsm FreeE)
iom, oom: option message
i: index
li: label (equivocator_IM IM i)
initial_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  initial_descriptors is
x_descriptors: equivocator_descriptors IM
IHtr: proper_equivocator_descriptors IM x_descriptors
  (finite_trace_last is tr)
Htr: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) is tr
Hinit: initial_state_prop is
final_descriptors: equivocator_descriptors IM
x': list (composite_transition_item IM)
oitem': option (composite_transition_item IM)
ditem': equivocator_descriptors IM
oitem'': option transition_item
ditem'': MachineDescriptor
  (IM (projT1 (existT i li)))
j: nat
Hfinali: final_descriptors (projT1 (existT i li)) =
Existing j
Heqprojecti: match
  equivocator_state_project
    (s (projT1 (existT i li))) j
with
| Some sj =>
    match projT2 (existT i li) with
    | Spawn sn =>
        if
         decide
           (j =
            equivocator_state_last
              (s (projT1 (existT i li))))
        then Some (None, NewMachine sn)
        else Some (None, Existing j)
    | ContinueWith i0 lx =>
        if decide (i0 = j)
        then
         Some
           (Some
              {|
                l := lx;
                input := iom;
                destination := sj;
                output := oom
              |}, Existing i0)
        else Some (None, Existing j)
    | ForkWith i0 lx =>
        if
         decide
           (j =
            equivocator_state_last
              (s (projT1 (existT i li))))
        then
         Some
           (Some
              {|
                l := lx;
                input := iom;
                destination := sj;
                output := oom
              |}, Existing i0)
        else Some (None, Existing j)
    end
| None => None
end = Some (oitem'', ditem'')
Hproject_x: match oitem'' with
| Some item' =>
    Some
      (Some
         {|
           l :=
             existT
               (projT1 (existT i li))
               (l item');
           input := iom;
           destination :=
             equivocators_state_project
               IM final_descriptors s;
           output := oom
         |},
       equivocator_descriptors_update IM
         final_descriptors
         (projT1 (existT i li)) ditem'')
| None =>
    Some
      (None,
       equivocator_descriptors_update IM
         final_descriptors
         (projT1 (existT i li)) ditem'')
end = Some (oitem', ditem')
Heqproject_x: Some (x', x_descriptors) =
match oitem' with
| Some item' => Some ([item'], ditem')
| None => Some ([], ditem')
end
trX': list (composite_transition_item IM)
Hproject_x': foldr equivocators_trace_project_folder
  (Some ([], x_descriptors)) tr =
Some (trX', initial_descriptors)
Htl: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) s []
Ht: input_valid_transition
  (preloaded_with_all_messages_vlsm FreeE)
  (existT i li) (finite_trace_last is tr, iom)
  (s, oom)
proper_equivocator_descriptors IM final_descriptors s
    unfold projT1, projT2 in Heqprojecti.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
s: state (preloaded_with_all_messages_vlsm FreeE)
iom, oom: option message
i: index
li: label (equivocator_IM IM i)
initial_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  initial_descriptors is
x_descriptors: equivocator_descriptors IM
IHtr: proper_equivocator_descriptors IM x_descriptors
  (finite_trace_last is tr)
Htr: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) is tr
Hinit: initial_state_prop is
final_descriptors: equivocator_descriptors IM
x': list (composite_transition_item IM)
oitem': option (composite_transition_item IM)
ditem': equivocator_descriptors IM
oitem'': option transition_item
ditem'': MachineDescriptor
  (IM (projT1 (existT i li)))
j: nat
Hfinali: final_descriptors (projT1 (existT i li)) =
Existing j
Heqprojecti: match
  equivocator_state_project (s i) j
with
| Some sj =>
    match li with
    | Spawn sn =>
        if
         decide
           (j =
            equivocator_state_last (s i))
        then Some (None, NewMachine sn)
        else Some (None, Existing j)
    | ContinueWith i0 lx =>
        if decide (i0 = j)
        then
         Some
           (Some
              {|
                l := lx;
                input := iom;
                destination := sj;
                output := oom
              |}, Existing i0)
        else Some (None, Existing j)
    | ForkWith i0 lx =>
        if
         decide
           (j =
            equivocator_state_last (s i))
        then
         Some
           (Some
              {|
                l := lx;
                input := iom;
                destination := sj;
                output := oom
              |}, Existing i0)
        else Some (None, Existing j)
    end
| None => None
end = Some (oitem'', ditem'')
Hproject_x: match oitem'' with
| Some item' =>
    Some
      (Some
         {|
           l :=
             existT
               (projT1 (existT i li))
               (l item');
           input := iom;
           destination :=
             equivocators_state_project
               IM final_descriptors s;
           output := oom
         |},
       equivocator_descriptors_update IM
         final_descriptors
         (projT1 (existT i li)) ditem'')
| None =>
    Some
      (None,
       equivocator_descriptors_update IM
         final_descriptors
         (projT1 (existT i li)) ditem'')
end = Some (oitem', ditem')
Heqproject_x: Some (x', x_descriptors) =
match oitem' with
| Some item' => Some ([item'], ditem')
| None => Some ([], ditem')
end
trX': list (composite_transition_item IM)
Hproject_x': foldr equivocators_trace_project_folder
  (Some ([], x_descriptors)) tr =
Some (trX', initial_descriptors)
Htl: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) s []
Ht: input_valid_transition
  (preloaded_with_all_messages_vlsm FreeE)
  (existT i li) (finite_trace_last is tr, iom)
  (s, oom)
proper_equivocator_descriptors IM final_descriptors s
    destruct Ht as [Hv Ht].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
s: state (preloaded_with_all_messages_vlsm FreeE)
iom, oom: option message
i: index
li: label (equivocator_IM IM i)
initial_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  initial_descriptors is
x_descriptors: equivocator_descriptors IM
IHtr: proper_equivocator_descriptors IM x_descriptors
  (finite_trace_last is tr)
Htr: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) is tr
Hinit: initial_state_prop is
final_descriptors: equivocator_descriptors IM
x': list (composite_transition_item IM)
oitem': option (composite_transition_item IM)
ditem': equivocator_descriptors IM
oitem'': option transition_item
ditem'': MachineDescriptor
  (IM (projT1 (existT i li)))
j: nat
Hfinali: final_descriptors (projT1 (existT i li)) =
Existing j
Heqprojecti: match
  equivocator_state_project (s i) j
with
| Some sj =>
    match li with
    | Spawn sn =>
        if
         decide
           (j =
            equivocator_state_last (s i))
        then Some (None, NewMachine sn)
        else Some (None, Existing j)
    | ContinueWith i0 lx =>
        if decide (i0 = j)
        then
         Some
           (Some
              {|
                l := lx;
                input := iom;
                destination := sj;
                output := oom
              |}, Existing i0)
        else Some (None, Existing j)
    | ForkWith i0 lx =>
        if
         decide
           (j =
            equivocator_state_last (s i))
        then
         Some
           (Some
              {|
                l := lx;
                input := iom;
                destination := sj;
                output := oom
              |}, Existing i0)
        else Some (None, Existing j)
    end
| None => None
end = Some (oitem'', ditem'')
Hproject_x: match oitem'' with
| Some item' =>
    Some
      (Some
         {|
           l :=
             existT
               (projT1 (existT i li))
               (l item');
           input := iom;
           destination :=
             equivocators_state_project
               IM final_descriptors s;
           output := oom
         |},
       equivocator_descriptors_update IM
         final_descriptors
         (projT1 (existT i li)) ditem'')
| None =>
    Some
      (None,
       equivocator_descriptors_update IM
         final_descriptors
         (projT1 (existT i li)) ditem'')
end = Some (oitem', ditem')
Heqproject_x: Some (x', x_descriptors) =
match oitem' with
| Some item' => Some ([item'], ditem')
| None => Some ([], ditem')
end
trX': list (composite_transition_item IM)
Hproject_x': foldr equivocators_trace_project_folder
  (Some ([], x_descriptors)) tr =
Some (trX', initial_descriptors)
Htl: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) s []
Hv: input_valid
  (preloaded_with_all_messages_vlsm FreeE)
  (existT i li) (finite_trace_last is tr, iom)
Ht: transition (existT i li)
  (finite_trace_last is tr, iom) = (
s, oom)
proper_equivocator_descriptors IM final_descriptors s
    cbn in Ht.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
s: state (preloaded_with_all_messages_vlsm FreeE)
iom, oom: option message
i: index
li: label (equivocator_IM IM i)
initial_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  initial_descriptors is
x_descriptors: equivocator_descriptors IM
IHtr: proper_equivocator_descriptors IM x_descriptors
  (finite_trace_last is tr)
Htr: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) is tr
Hinit: initial_state_prop is
final_descriptors: equivocator_descriptors IM
x': list (composite_transition_item IM)
oitem': option (composite_transition_item IM)
ditem': equivocator_descriptors IM
oitem'': option transition_item
ditem'': MachineDescriptor
  (IM (projT1 (existT i li)))
j: nat
Hfinali: final_descriptors (projT1 (existT i li)) =
Existing j
Heqprojecti: match
  equivocator_state_project (s i) j
with
| Some sj =>
    match li with
    | Spawn sn =>
        if
         decide
           (j =
            equivocator_state_last (s i))
        then Some (None, NewMachine sn)
        else Some (None, Existing j)
    | ContinueWith i0 lx =>
        if decide (i0 = j)
        then
         Some
           (Some
              {|
                l := lx;
                input := iom;
                destination := sj;
                output := oom
              |}, Existing i0)
        else Some (None, Existing j)
    | ForkWith i0 lx =>
        if
         decide
           (j =
            equivocator_state_last (s i))
        then
         Some
           (Some
              {|
                l := lx;
                input := iom;
                destination := sj;
                output := oom
              |}, Existing i0)
        else Some (None, Existing j)
    end
| None => None
end = Some (oitem'', ditem'')
Hproject_x: match oitem'' with
| Some item' =>
    Some
      (Some
         {|
           l :=
             existT
               (projT1 (existT i li))
               (l item');
           input := iom;
           destination :=
             equivocators_state_project
               IM final_descriptors s;
           output := oom
         |},
       equivocator_descriptors_update IM
         final_descriptors
         (projT1 (existT i li)) ditem'')
| None =>
    Some
      (None,
       equivocator_descriptors_update IM
         final_descriptors
         (projT1 (existT i li)) ditem'')
end = Some (oitem', ditem')
Heqproject_x: Some (x', x_descriptors) =
match oitem' with
| Some item' => Some ([item'], ditem')
| None => Some ([], ditem')
end
trX': list (composite_transition_item IM)
Hproject_x': foldr equivocators_trace_project_folder
  (Some ([], x_descriptors)) tr =
Some (trX', initial_descriptors)
Htl: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) s []
Hv: input_valid
  (preloaded_with_all_messages_vlsm FreeE)
  (existT i li) (finite_trace_last is tr, iom)
Ht: (let (si', om') :=
   equivocator_transition 
     (IM i) li (finite_trace_last is tr i, iom) in
 (state_update (equivocator_IM IM)
    (finite_trace_last is tr) i si', om')) =
(s, oom)
proper_equivocator_descriptors IM final_descriptors s
    destruct (equivocator_transition _ _ _) as (si', om') eqn: Ht'.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
s: state (preloaded_with_all_messages_vlsm FreeE)
iom, oom: option message
i: index
li: label (equivocator_IM IM i)
initial_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  initial_descriptors is
x_descriptors: equivocator_descriptors IM
IHtr: proper_equivocator_descriptors IM x_descriptors
  (finite_trace_last is tr)
Htr: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) is tr
Hinit: initial_state_prop is
final_descriptors: equivocator_descriptors IM
x': list (composite_transition_item IM)
oitem': option (composite_transition_item IM)
ditem': equivocator_descriptors IM
oitem'': option transition_item
ditem'': MachineDescriptor
  (IM (projT1 (existT i li)))
j: nat
Hfinali: final_descriptors (projT1 (existT i li)) =
Existing j
Heqprojecti: match
  equivocator_state_project (s i) j
with
| Some sj =>
    match li with
    | Spawn sn =>
        if
         decide
           (j =
            equivocator_state_last (s i))
        then Some (None, NewMachine sn)
        else Some (None, Existing j)
    | ContinueWith i0 lx =>
        if decide (i0 = j)
        then
         Some
           (Some
              {|
                l := lx;
                input := iom;
                destination := sj;
                output := oom
              |}, Existing i0)
        else Some (None, Existing j)
    | ForkWith i0 lx =>
        if
         decide
           (j =
            equivocator_state_last (s i))
        then
         Some
           (Some
              {|
                l := lx;
                input := iom;
                destination := sj;
                output := oom
              |}, Existing i0)
        else Some (None, Existing j)
    end
| None => None
end = Some (oitem'', ditem'')
Hproject_x: match oitem'' with
| Some item' =>
    Some
      (Some
         {|
           l :=
             existT
               (projT1 (existT i li))
               (l item');
           input := iom;
           destination :=
             equivocators_state_project
               IM final_descriptors s;
           output := oom
         |},
       equivocator_descriptors_update IM
         final_descriptors
         (projT1 (existT i li)) ditem'')
| None =>
    Some
      (None,
       equivocator_descriptors_update IM
         final_descriptors
         (projT1 (existT i li)) ditem'')
end = Some (oitem', ditem')
Heqproject_x: Some (x', x_descriptors) =
match oitem' with
| Some item' => Some ([item'], ditem')
| None => Some ([], ditem')
end
trX': list (composite_transition_item IM)
Hproject_x': foldr equivocators_trace_project_folder
  (Some ([], x_descriptors)) tr =
Some (trX', initial_descriptors)
Htl: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) s []
Hv: input_valid
  (preloaded_with_all_messages_vlsm FreeE)
  (existT i li) (finite_trace_last is tr, iom)
si': equivocator_state (IM i)
om': option message
Ht': equivocator_transition 
  (IM i) li (finite_trace_last is tr i, iom) =
(si', om')
Ht: (state_update (equivocator_IM IM)
   (finite_trace_last is tr) i si', om') =
(s, oom)
proper_equivocator_descriptors IM final_descriptors s
    inversion Ht.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
s: state (preloaded_with_all_messages_vlsm FreeE)
iom, oom: option message
i: index
li: label (equivocator_IM IM i)
initial_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  initial_descriptors is
x_descriptors: equivocator_descriptors IM
IHtr: proper_equivocator_descriptors IM x_descriptors
  (finite_trace_last is tr)
Htr: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) is tr
Hinit: initial_state_prop is
final_descriptors: equivocator_descriptors IM
x': list (composite_transition_item IM)
oitem': option (composite_transition_item IM)
ditem': equivocator_descriptors IM
oitem'': option transition_item
ditem'': MachineDescriptor
  (IM (projT1 (existT i li)))
j: nat
Hfinali: final_descriptors (projT1 (existT i li)) =
Existing j
Heqprojecti: match
  equivocator_state_project (s i) j
with
| Some sj =>
    match li with
    | Spawn sn =>
        if
         decide
           (j =
            equivocator_state_last (s i))
        then Some (None, NewMachine sn)
        else Some (None, Existing j)
    | ContinueWith i0 lx =>
        if decide (i0 = j)
        then
         Some
           (Some
              {|
                l := lx;
                input := iom;
                destination := sj;
                output := oom
              |}, Existing i0)
        else Some (None, Existing j)
    | ForkWith i0 lx =>
        if
         decide
           (j =
            equivocator_state_last (s i))
        then
         Some
           (Some
              {|
                l := lx;
                input := iom;
                destination := sj;
                output := oom
              |}, Existing i0)
        else Some (None, Existing j)
    end
| None => None
end = Some (oitem'', ditem'')
Hproject_x: match oitem'' with
| Some item' =>
    Some
      (Some
         {|
           l :=
             existT
               (projT1 (existT i li))
               (l item');
           input := iom;
           destination :=
             equivocators_state_project
               IM final_descriptors s;
           output := oom
         |},
       equivocator_descriptors_update IM
         final_descriptors
         (projT1 (existT i li)) ditem'')
| None =>
    Some
      (None,
       equivocator_descriptors_update IM
         final_descriptors
         (projT1 (existT i li)) ditem'')
end = Some (oitem', ditem')
Heqproject_x: Some (x', x_descriptors) =
match oitem' with
| Some item' => Some ([item'], ditem')
| None => Some ([], ditem')
end
trX': list (composite_transition_item IM)
Hproject_x': foldr equivocators_trace_project_folder
  (Some ([], x_descriptors)) tr =
Some (trX', initial_descriptors)
Htl: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) s []
Hv: input_valid
  (preloaded_with_all_messages_vlsm FreeE)
  (existT i li) (finite_trace_last is tr, iom)
si': equivocator_state (IM i)
om': option message
Ht': equivocator_transition 
  (IM i) li (finite_trace_last is tr i, iom) =
(si', om')
Ht: (state_update (equivocator_IM IM)
   (finite_trace_last is tr) i si', om') =
(s, oom)
H2: state_update (equivocator_IM IM)
  (finite_trace_last is tr) i si' = s
H3: om' = oom
proper_equivocator_descriptors IM final_descriptors
  (state_update (equivocator_IM IM)
     (finite_trace_last is tr) i si') subst om'.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
s: state (preloaded_with_all_messages_vlsm FreeE)
iom, oom: option message
i: index
li: label (equivocator_IM IM i)
initial_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  initial_descriptors is
x_descriptors: equivocator_descriptors IM
IHtr: proper_equivocator_descriptors IM x_descriptors
  (finite_trace_last is tr)
Htr: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) is tr
Hinit: initial_state_prop is
final_descriptors: equivocator_descriptors IM
x': list (composite_transition_item IM)
oitem': option (composite_transition_item IM)
ditem': equivocator_descriptors IM
oitem'': option transition_item
ditem'': MachineDescriptor
  (IM (projT1 (existT i li)))
j: nat
Hfinali: final_descriptors (projT1 (existT i li)) =
Existing j
Heqprojecti: match
  equivocator_state_project (s i) j
with
| Some sj =>
    match li with
    | Spawn sn =>
        if
         decide
           (j =
            equivocator_state_last (s i))
        then Some (None, NewMachine sn)
        else Some (None, Existing j)
    | ContinueWith i0 lx =>
        if decide (i0 = j)
        then
         Some
           (Some
              {|
                l := lx;
                input := iom;
                destination := sj;
                output := oom
              |}, Existing i0)
        else Some (None, Existing j)
    | ForkWith i0 lx =>
        if
         decide
           (j =
            equivocator_state_last (s i))
        then
         Some
           (Some
              {|
                l := lx;
                input := iom;
                destination := sj;
                output := oom
              |}, Existing i0)
        else Some (None, Existing j)
    end
| None => None
end = Some (oitem'', ditem'')
Hproject_x: match oitem'' with
| Some item' =>
    Some
      (Some
         {|
           l :=
             existT
               (projT1 (existT i li))
               (l item');
           input := iom;
           destination :=
             equivocators_state_project
               IM final_descriptors s;
           output := oom
         |},
       equivocator_descriptors_update IM
         final_descriptors
         (projT1 (existT i li)) ditem'')
| None =>
    Some
      (None,
       equivocator_descriptors_update IM
         final_descriptors
         (projT1 (existT i li)) ditem'')
end = Some (oitem', ditem')
Heqproject_x: Some (x', x_descriptors) =
match oitem' with
| Some item' => Some ([item'], ditem')
| None => Some ([], ditem')
end
trX': list (composite_transition_item IM)
Hproject_x': foldr equivocators_trace_project_folder
  (Some ([], x_descriptors)) tr =
Some (trX', initial_descriptors)
Htl: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) s []
Hv: input_valid
  (preloaded_with_all_messages_vlsm FreeE)
  (existT i li) (finite_trace_last is tr, iom)
si': equivocator_state (IM i)
Ht: (state_update (equivocator_IM IM)
   (finite_trace_last is tr) i si', oom) =
(s, oom)
Ht': equivocator_transition 
  (IM i) li (finite_trace_last is tr i, iom) =
(si', oom)
H2: state_update (equivocator_IM IM)
  (finite_trace_last is tr) i si' = s
proper_equivocator_descriptors IM final_descriptors
  (state_update (equivocator_IM IM)
     (finite_trace_last is tr) i si') clear Ht.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
s: state (preloaded_with_all_messages_vlsm FreeE)
iom, oom: option message
i: index
li: label (equivocator_IM IM i)
initial_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  initial_descriptors is
x_descriptors: equivocator_descriptors IM
IHtr: proper_equivocator_descriptors IM x_descriptors
  (finite_trace_last is tr)
Htr: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) is tr
Hinit: initial_state_prop is
final_descriptors: equivocator_descriptors IM
x': list (composite_transition_item IM)
oitem': option (composite_transition_item IM)
ditem': equivocator_descriptors IM
oitem'': option transition_item
ditem'': MachineDescriptor
  (IM (projT1 (existT i li)))
j: nat
Hfinali: final_descriptors (projT1 (existT i li)) =
Existing j
Heqprojecti: match
  equivocator_state_project (s i) j
with
| Some sj =>
    match li with
    | Spawn sn =>
        if
         decide
           (j =
            equivocator_state_last (s i))
        then Some (None, NewMachine sn)
        else Some (None, Existing j)
    | ContinueWith i0 lx =>
        if decide (i0 = j)
        then
         Some
           (Some
              {|
                l := lx;
                input := iom;
                destination := sj;
                output := oom
              |}, Existing i0)
        else Some (None, Existing j)
    | ForkWith i0 lx =>
        if
         decide
           (j =
            equivocator_state_last (s i))
        then
         Some
           (Some
              {|
                l := lx;
                input := iom;
                destination := sj;
                output := oom
              |}, Existing i0)
        else Some (None, Existing j)
    end
| None => None
end = Some (oitem'', ditem'')
Hproject_x: match oitem'' with
| Some item' =>
    Some
      (Some
         {|
           l :=
             existT
               (projT1 (existT i li))
               (l item');
           input := iom;
           destination :=
             equivocators_state_project
               IM final_descriptors s;
           output := oom
         |},
       equivocator_descriptors_update IM
         final_descriptors
         (projT1 (existT i li)) ditem'')
| None =>
    Some
      (None,
       equivocator_descriptors_update IM
         final_descriptors
         (projT1 (existT i li)) ditem'')
end = Some (oitem', ditem')
Heqproject_x: Some (x', x_descriptors) =
match oitem' with
| Some item' => Some ([item'], ditem')
| None => Some ([], ditem')
end
trX': list (composite_transition_item IM)
Hproject_x': foldr equivocators_trace_project_folder
  (Some ([], x_descriptors)) tr =
Some (trX', initial_descriptors)
Htl: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) s []
Hv: input_valid
  (preloaded_with_all_messages_vlsm FreeE)
  (existT i li) (finite_trace_last is tr, iom)
si': equivocator_state (IM i)
Ht': equivocator_transition 
  (IM i) li (finite_trace_last is tr i, iom) =
(si', oom)
H2: state_update (equivocator_IM IM)
  (finite_trace_last is tr) i si' = s
proper_equivocator_descriptors IM final_descriptors
  (state_update (equivocator_IM IM)
     (finite_trace_last is tr) i si')
    replace (s i) with si' in * by (subst; state_update_simpl; done).message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
s: state (preloaded_with_all_messages_vlsm FreeE)
iom, oom: option message
i: index
li: label (equivocator_IM IM i)
initial_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  initial_descriptors is
x_descriptors: equivocator_descriptors IM
IHtr: proper_equivocator_descriptors IM x_descriptors
  (finite_trace_last is tr)
Htr: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) is tr
Hinit: initial_state_prop is
final_descriptors: equivocator_descriptors IM
x': list (composite_transition_item IM)
oitem': option (composite_transition_item IM)
ditem': equivocator_descriptors IM
oitem'': option transition_item
ditem'': MachineDescriptor
  (IM (projT1 (existT i li)))
j: nat
Hfinali: final_descriptors (projT1 (existT i li)) =
Existing j
si': equivocator_state (IM i)
Heqprojecti: match
  equivocator_state_project si' j
with
| Some sj =>
    match li with
    | Spawn sn =>
        if
         decide
           (j =
            equivocator_state_last si')
        then Some (None, NewMachine sn)
        else Some (None, Existing j)
    | ContinueWith i0 lx =>
        if decide (i0 = j)
        then
         Some
           (Some
              {|
                l := lx;
                input := iom;
                destination := sj;
                output := oom
              |}, Existing i0)
        else Some (None, Existing j)
    | ForkWith i0 lx =>
        if
         decide
           (j =
            equivocator_state_last si')
        then
         Some
           (Some
              {|
                l := lx;
                input := iom;
                destination := sj;
                output := oom
              |}, Existing i0)
        else Some (None, Existing j)
    end
| None => None
end = Some (oitem'', ditem'')
Hproject_x: match oitem'' with
| Some item' =>
    Some
      (Some
         {|
           l :=
             existT
               (projT1 (existT i li))
               (l item');
           input := iom;
           destination :=
             equivocators_state_project
               IM final_descriptors s;
           output := oom
         |},
       equivocator_descriptors_update IM
         final_descriptors
         (projT1 (existT i li)) ditem'')
| None =>
    Some
      (None,
       equivocator_descriptors_update IM
         final_descriptors
         (projT1 (existT i li)) ditem'')
end = Some (oitem', ditem')
Heqproject_x: Some (x', x_descriptors) =
match oitem' with
| Some item' => Some ([item'], ditem')
| None => Some ([], ditem')
end
trX': list (composite_transition_item IM)
Hproject_x': foldr equivocators_trace_project_folder
  (Some ([], x_descriptors)) tr =
Some (trX', initial_descriptors)
Htl: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) s []
Hv: input_valid
  (preloaded_with_all_messages_vlsm FreeE)
  (existT i li) (finite_trace_last is tr, iom)
Ht': equivocator_transition 
  (IM i) li (finite_trace_last is tr i, iom) =
(si', oom)
H2: state_update (equivocator_IM IM)
  (finite_trace_last is tr) i si' = s
proper_equivocator_descriptors IM final_descriptors
  (state_update (equivocator_IM IM)
     (finite_trace_last is tr) i si')
    destruct (equivocator_state_project si' j) as [si'j |] eqn: Hj; [| done].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
s: state (preloaded_with_all_messages_vlsm FreeE)
iom, oom: option message
i: index
li: label (equivocator_IM IM i)
initial_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  initial_descriptors is
x_descriptors: equivocator_descriptors IM
IHtr: proper_equivocator_descriptors IM x_descriptors
  (finite_trace_last is tr)
Htr: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) is tr
Hinit: initial_state_prop is
final_descriptors: equivocator_descriptors IM
x': list (composite_transition_item IM)
oitem': option (composite_transition_item IM)
ditem': equivocator_descriptors IM
oitem'': option transition_item
ditem'': MachineDescriptor
  (IM (projT1 (existT i li)))
j: nat
Hfinali: final_descriptors (projT1 (existT i li)) =
Existing j
si': equivocator_state (IM i)
si'j: state (IM i)
Hj: equivocator_state_project si' j = Some si'j
Heqprojecti: match li with
| Spawn sn =>
    if
     decide
       (j = equivocator_state_last si')
    then Some (None, NewMachine sn)
    else Some (None, Existing j)
| ContinueWith i0 lx =>
    if decide (i0 = j)
    then
     Some
       (Some
          {|
            l := lx;
            input := iom;
            destination := si'j;
            output := oom
          |}, Existing i0)
    else Some (None, Existing j)
| ForkWith i0 lx =>
    if
     decide
       (j = equivocator_state_last si')
    then
     Some
       (Some
          {|
            l := lx;
            input := iom;
            destination := si'j;
            output := oom
          |}, Existing i0)
    else Some (None, Existing j)
end = Some (oitem'', ditem'')
Hproject_x: match oitem'' with
| Some item' =>
    Some
      (Some
         {|
           l :=
             existT
               (projT1 (existT i li))
               (l item');
           input := iom;
           destination :=
             equivocators_state_project
               IM final_descriptors s;
           output := oom
         |},
       equivocator_descriptors_update IM
         final_descriptors
         (projT1 (existT i li)) ditem'')
| None =>
    Some
      (None,
       equivocator_descriptors_update IM
         final_descriptors
         (projT1 (existT i li)) ditem'')
end = Some (oitem', ditem')
Heqproject_x: Some (x', x_descriptors) =
match oitem' with
| Some item' => Some ([item'], ditem')
| None => Some ([], ditem')
end
trX': list (composite_transition_item IM)
Hproject_x': foldr equivocators_trace_project_folder
  (Some ([], x_descriptors)) tr =
Some (trX', initial_descriptors)
Htl: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) s []
Hv: input_valid
  (preloaded_with_all_messages_vlsm FreeE)
  (existT i li) (finite_trace_last is tr, iom)
Ht': equivocator_transition 
  (IM i) li (finite_trace_last is tr i, iom) =
(si', oom)
H2: state_update (equivocator_IM IM)
  (finite_trace_last is tr) i si' = s
proper_equivocator_descriptors IM final_descriptors
  (state_update (equivocator_IM IM)
     (finite_trace_last is tr) i si')
    by destruct li as [ndi | idi li | idi li]
    ; destruct (decide _)
    ; inversion Heqprojecti; subst; clear Heqprojecti
    ; inversion Hproject_x; subst; clear Hproject_x
    ; inversion Heqproject_x; subst; clear Heqproject_x
    ; intro eqv; specialize (IHtr eqv)
    ; (destruct (decide (i = eqv)); subst; state_update_simpl
       ; cbn in *; [rewrite ?Hfinali; eexists |]; done).
Qed.

A corollary of preloaded_equivocators_valid_trace_from_project selecting only the proper_equivocator_descriptors property.

Lemma preloaded_equivocators_valid_trace_project_proper_initial
  (is : composite_state (equivocator_IM IM))
  (tr : list (composite_transition_item (equivocator_IM IM)))
  (final_state := finite_trace_last is tr)
  (Htr : finite_constrained_trace_from FreeE is tr)
  (final_descriptors : equivocator_descriptors IM)
  (trX : list (composite_transition_item IM))
  (initial_descriptors : equivocator_descriptors IM)
  (Hproject : equivocators_trace_project final_descriptors tr = Some (trX, initial_descriptors))
  (Hproper : proper_equivocator_descriptors IM final_descriptors final_state)
  : proper_equivocator_descriptors IM initial_descriptors is.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
final_state:= finite_trace_last is tr: state
  (composite_type (equivocator_IM IM))
Htr: finite_constrained_trace_from FreeE is tr
final_descriptors: equivocator_descriptors IM
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hproject: equivocators_trace_project
  final_descriptors tr =
Some (trX, initial_descriptors)
Hproper: proper_equivocator_descriptors IM
  final_descriptors final_state
proper_equivocator_descriptors IM initial_descriptors
  is
Proof.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
final_state:= finite_trace_last is tr: state
  (composite_type (equivocator_IM IM))
Htr: finite_constrained_trace_from FreeE is tr
final_descriptors: equivocator_descriptors IM
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hproject: equivocators_trace_project
  final_descriptors tr =
Some (trX, initial_descriptors)
Hproper: proper_equivocator_descriptors IM
  final_descriptors final_state
proper_equivocator_descriptors IM initial_descriptors
  is
  destruct
    (preloaded_equivocators_valid_trace_from_project
      final_descriptors is tr Hproper Htr)
    as [_trX [_initial_descriptors [_Hproject [Hiproper _]]]].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
final_state:= finite_trace_last is tr: state
  (composite_type (equivocator_IM IM))
Htr: finite_constrained_trace_from FreeE is tr
final_descriptors: equivocator_descriptors IM
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hproject: equivocators_trace_project
  final_descriptors tr =
Some (trX, initial_descriptors)
Hproper: proper_equivocator_descriptors IM
  final_descriptors final_state
_trX: list (composite_transition_item IM)
_initial_descriptors: equivocator_descriptors IM
_Hproject: equivocators_trace_project
  final_descriptors tr =
Some (_trX, _initial_descriptors)
Hiproper: proper_equivocator_descriptors IM
  _initial_descriptors is
proper_equivocator_descriptors IM initial_descriptors
  is
  rewrite Hproject in _Hproject.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
final_state:= finite_trace_last is tr: state
  (composite_type (equivocator_IM IM))
Htr: finite_constrained_trace_from FreeE is tr
final_descriptors: equivocator_descriptors IM
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hproject: equivocators_trace_project
  final_descriptors tr =
Some (trX, initial_descriptors)
Hproper: proper_equivocator_descriptors IM
  final_descriptors final_state
_trX: list (composite_transition_item IM)
_initial_descriptors: equivocator_descriptors IM
_Hproject: Some (trX, initial_descriptors) =
Some (_trX, _initial_descriptors)
Hiproper: proper_equivocator_descriptors IM
  _initial_descriptors is
proper_equivocator_descriptors IM initial_descriptors
  is
  by inversion _Hproject; subst.
Qed.

Lemma equivocators_trace_project_output_reflecting_inv
  (is : composite_state (equivocator_IM IM))
  (tr : list (composite_transition_item (equivocator_IM IM)))
  (Htr : finite_constrained_trace_from (free_composite_vlsm (equivocator_IM IM)) is tr)
  (m : message)
  (Hbbs : Exists (field_selector output m) tr)
  : exists
    (final_descriptors initial_descriptors : equivocator_descriptors IM)
    (trX : list (composite_transition_item IM)),
    not_equivocating_equivocator_descriptors IM final_descriptors (finite_trace_last is tr) /\
    equivocators_trace_project final_descriptors tr = Some (trX, initial_descriptors) /\
    Exists (field_selector output m) trX.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_constrained_trace_from
  (free_composite_vlsm (equivocator_IM IM)) is
  tr
m: message
Hbbs: Exists (field_selector output m) tr
∃ (final_descriptors
   initial_descriptors : equivocator_descriptors IM) 
  (trX : list (composite_transition_item IM)),
  not_equivocating_equivocator_descriptors IM
    final_descriptors (finite_trace_last is tr)
  ∧ equivocators_trace_project final_descriptors tr =
    Some (trX, initial_descriptors)
    ∧ Exists (field_selector output m) trX
Proof.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_constrained_trace_from
  (free_composite_vlsm (equivocator_IM IM)) is
  tr
m: message
Hbbs: Exists (field_selector output m) tr
∃ (final_descriptors
   initial_descriptors : equivocator_descriptors IM) 
  (trX : list (composite_transition_item IM)),
  not_equivocating_equivocator_descriptors IM
    final_descriptors (finite_trace_last is tr)
  ∧ equivocators_trace_project final_descriptors tr =
    Some (trX, initial_descriptors)
    ∧ Exists (field_selector output m) trX
  apply Exists_exists in Hbbs.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_constrained_trace_from
  (free_composite_vlsm (equivocator_IM IM)) is
  tr
m: message
Hbbs: ∃ x : transition_item,
  x ∈ tr ∧ field_selector output m x
∃ (final_descriptors
   initial_descriptors : equivocator_descriptors IM) 
  (trX : list (composite_transition_item IM)),
  not_equivocating_equivocator_descriptors IM
    final_descriptors (finite_trace_last is tr)
  ∧ equivocators_trace_project final_descriptors tr =
    Some (trX, initial_descriptors)
    ∧ Exists (field_selector output m) trX
  destruct Hbbs as [item [Hitem Hm]].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_constrained_trace_from
  (free_composite_vlsm (equivocator_IM IM)) is
  tr
m: message
item: transition_item
Hitem: item ∈ tr
Hm: field_selector output m item
∃ (final_descriptors
   initial_descriptors : equivocator_descriptors IM) 
  (trX : list (composite_transition_item IM)),
  not_equivocating_equivocator_descriptors IM
    final_descriptors (finite_trace_last is tr)
  ∧ equivocators_trace_project final_descriptors tr =
    Some (trX, initial_descriptors)
    ∧ Exists (field_selector output m) trX simpl in Hm.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_constrained_trace_from
  (free_composite_vlsm (equivocator_IM IM)) is
  tr
m: message
item: transition_item
Hitem: item ∈ tr
Hm: output item = Some m
∃ (final_descriptors
   initial_descriptors : equivocator_descriptors IM) 
  (trX : list (composite_transition_item IM)),
  not_equivocating_equivocator_descriptors IM
    final_descriptors (finite_trace_last is tr)
  ∧ equivocators_trace_project final_descriptors tr =
    Some (trX, initial_descriptors)
    ∧ Exists (field_selector output m) trX
  apply (finite_trace_projection_list_in (equivocator_IM IM)) in Hitem.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_constrained_trace_from
  (free_composite_vlsm (equivocator_IM IM)) is
  tr
m: message
item: transition_item
Hitem: {|
  l := projT2 (l item);
  input := input item;
  destination :=
    destination item (projT1 (l item));
  output := output item
|}
∈ VLSM_projection_finite_trace_project
    (preloaded_component_projection
       (equivocator_IM IM) 
       (projT1 (l item))) tr
Hm: output item = Some m
∃ (final_descriptors
   initial_descriptors : equivocator_descriptors IM) 
  (trX : list (composite_transition_item IM)),
  not_equivocating_equivocator_descriptors IM
    final_descriptors (finite_trace_last is tr)
  ∧ equivocators_trace_project final_descriptors tr =
    Some (trX, initial_descriptors)
    ∧ Exists (field_selector output m) trX
  destruct item.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_constrained_trace_from
  (free_composite_vlsm (equivocator_IM IM)) is
  tr
m: message
l: label (composite_type (equivocator_IM IM))
input: option message
destination: state
  (composite_type (equivocator_IM IM))
output: option message
Hitem: {|
  l :=
    projT2
      (VLSM.l
         {|
           l := l;
           input := input;
           destination := destination;
           output := output
         |});
  input :=
    VLSM.input
      {|
        l := l;
        input := input;
        destination := destination;
        output := output
      |};
  destination :=
    VLSM.destination
      {|
        l := l;
        input := input;
        destination := destination;
        output := output
      |}
      (projT1
         (VLSM.l
            {|
              l := l;
              input := input;
              destination := destination;
              output := output
            |}));
  output :=
    VLSM.output
      {|
        l := l;
        input := input;
        destination := destination;
        output := output
      |}
|}
∈ VLSM_projection_finite_trace_project
    (preloaded_component_projection
       (equivocator_IM IM)
       (projT1
          (VLSM.l
             {|
               l := l;
               input := input;
               destination := destination;
               output := output
             |}))) tr
Hm: VLSM.output
  {|
    l := l;
    input := input;
    destination := destination;
    output := output
  |} = Some m
∃ (final_descriptors
   initial_descriptors : equivocator_descriptors IM) 
  (trX : list (composite_transition_item IM)),
  not_equivocating_equivocator_descriptors IM
    final_descriptors (finite_trace_last is tr)
  ∧ equivocators_trace_project final_descriptors tr =
    Some (trX, initial_descriptors)
    ∧ Exists (field_selector VLSM.output m) trX simpl in *.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_constrained_trace_from
  (free_composite_vlsm (equivocator_IM IM)) is
  tr
m: message
l: composite_label (equivocator_IM IM)
input: option message
destination: composite_state (equivocator_IM IM)
output: option message
Hitem: {|
  l := projT2 l;
  input := input;
  destination := destination (projT1 l);
  output := output
|}
∈ VLSM_projection_finite_trace_project
    (preloaded_component_projection
       (equivocator_IM IM) 
       (projT1 l)) tr
Hm: output = Some m
∃ (final_descriptors
   initial_descriptors : equivocator_descriptors IM) 
  (trX : list (composite_transition_item IM)),
  not_equivocating_equivocator_descriptors IM
    final_descriptors (finite_trace_last is tr)
  ∧ equivocators_trace_project final_descriptors tr =
    Some (trX, initial_descriptors)
    ∧ Exists (field_selector VLSM.output m) trX destruct l as (i, li).message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_constrained_trace_from
  (free_composite_vlsm (equivocator_IM IM)) is
  tr
m: message
i: index
li: label (equivocator_IM IM i)
input: option message
destination: composite_state (equivocator_IM IM)
output: option message
Hitem: {|
  l := projT2 (existT i li);
  input := input;
  destination :=
    destination (projT1 (existT i li));
  output := output
|}
∈ VLSM_projection_finite_trace_project
    (preloaded_component_projection
       (equivocator_IM IM)
       (projT1 (existT i li))) tr
Hm: output = Some m
∃ (final_descriptors
   initial_descriptors : equivocator_descriptors IM) 
  (trX : list (composite_transition_item IM)),
  not_equivocating_equivocator_descriptors IM
    final_descriptors (finite_trace_last is tr)
  ∧ equivocators_trace_project final_descriptors tr =
    Some (trX, initial_descriptors)
    ∧ Exists (field_selector VLSM.output m) trX simpl in *.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_constrained_trace_from
  (free_composite_vlsm (equivocator_IM IM)) is
  tr
m: message
i: index
li: EquivocatorLabel (IM i)
input: option message
destination: composite_state (equivocator_IM IM)
output: option message
Hitem: {|
  l := li;
  input := input;
  destination := destination i;
  output := output
|}
∈ VLSM_projection_finite_trace_project
    (preloaded_component_projection
       (equivocator_IM IM) i) tr
Hm: output = Some m
∃ (final_descriptors
   initial_descriptors : equivocator_descriptors IM) 
  (trX : list (composite_transition_item IM)),
  not_equivocating_equivocator_descriptors IM
    final_descriptors (finite_trace_last is tr)
  ∧ equivocators_trace_project final_descriptors tr =
    Some (trX, initial_descriptors)
    ∧ Exists (field_selector VLSM.output m) trX
  specialize
    (preloaded_finite_valid_trace_projection (equivocator_IM IM) i _ _ Htr)
    as Htri.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_constrained_trace_from
  (free_composite_vlsm (equivocator_IM IM)) is
  tr
m: message
i: index
li: EquivocatorLabel (IM i)
input: option message
destination: composite_state (equivocator_IM IM)
output: option message
Hitem: {|
  l := li;
  input := input;
  destination := destination i;
  output := output
|}
∈ VLSM_projection_finite_trace_project
    (preloaded_component_projection
       (equivocator_IM IM) i) tr
Hm: output = Some m
Htri: finite_constrained_trace_from
  (equivocator_IM IM i) 
  (is i)
  (VLSM_projection_finite_trace_project
     (preloaded_component_projection
        (equivocator_IM IM) i) tr)
∃ (final_descriptors
   initial_descriptors : equivocator_descriptors IM) 
  (trX : list (composite_transition_item IM)),
  not_equivocating_equivocator_descriptors IM
    final_descriptors (finite_trace_last is tr)
  ∧ equivocators_trace_project final_descriptors tr =
    Some (trX, initial_descriptors)
    ∧ Exists (field_selector VLSM.output m) trX
  specialize
    (equivocator_vlsm_trace_project_output_reflecting_inv (IM i) _ _ Htri m) as Hex.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_constrained_trace_from
  (free_composite_vlsm (equivocator_IM IM)) is
  tr
m: message
i: index
li: EquivocatorLabel (IM i)
input: option message
destination: composite_state (equivocator_IM IM)
output: option message
Hitem: {|
  l := li;
  input := input;
  destination := destination i;
  output := output
|}
∈ VLSM_projection_finite_trace_project
    (preloaded_component_projection
       (equivocator_IM IM) i) tr
Hm: output = Some m
Htri: finite_constrained_trace_from
  (equivocator_IM IM i) 
  (is i)
  (VLSM_projection_finite_trace_project
     (preloaded_component_projection
        (equivocator_IM IM) i) tr)
Hex: Exists (field_selector VLSM.output m)
  (VLSM_projection_finite_trace_project
     (preloaded_component_projection
        (equivocator_IM IM) i) tr)
→ ∃ (j i0 : MachineDescriptor (IM i)) 
    (_ : proper_descriptor (IM i) i0 (is i)) 
    (_ : existing_descriptor 
           (IM i) j
           (finite_trace_last 
              (is i)
              (VLSM_projection_finite_trace_project
                 (preloaded_component_projection
                    (equivocator_IM IM) i) tr))) 
    (trX : list transition_item) 
    (_ : equivocator_vlsm_trace_project 
           (IM i)
           (VLSM_projection_finite_trace_project
              (preloaded_component_projection
                 (equivocator_IM IM) i) tr) j =
         Some (trX, i0)),
    Exists (field_selector VLSM.output m) trX
∃ (final_descriptors
   initial_descriptors : equivocator_descriptors IM) 
  (trX : list (composite_transition_item IM)),
  not_equivocating_equivocator_descriptors IM
    final_descriptors (finite_trace_last is tr)
  ∧ equivocators_trace_project final_descriptors tr =
    Some (trX, initial_descriptors)
    ∧ Exists (field_selector VLSM.output m) trX
  spec Hex; [by apply Exists_exists; eexists _ |].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_constrained_trace_from
  (free_composite_vlsm (equivocator_IM IM)) is
  tr
m: message
i: index
li: EquivocatorLabel (IM i)
input: option message
destination: composite_state (equivocator_IM IM)
output: option message
Hitem: {|
  l := li;
  input := input;
  destination := destination i;
  output := output
|}
∈ VLSM_projection_finite_trace_project
    (preloaded_component_projection
       (equivocator_IM IM) i) tr
Hm: output = Some m
Htri: finite_constrained_trace_from
  (equivocator_IM IM i) 
  (is i)
  (VLSM_projection_finite_trace_project
     (preloaded_component_projection
        (equivocator_IM IM) i) tr)
Hex: ∃ (j i0 : MachineDescriptor (IM i)) 
  (_ : proper_descriptor (IM i) i0 (is i)) 
  (_ : existing_descriptor 
         (IM i) j
         (finite_trace_last 
            (is i)
            (VLSM_projection_finite_trace_project
               (preloaded_component_projection
                  (equivocator_IM IM) i) tr))) 
  (trX : list transition_item) 
  (_ : equivocator_vlsm_trace_project 
         (IM i)
         (VLSM_projection_finite_trace_project
            (preloaded_component_projection
               (equivocator_IM IM) i) tr) j =
       Some (trX, i0)),
  Exists (field_selector VLSM.output m) trX
∃ (final_descriptors
   initial_descriptors : equivocator_descriptors IM) 
  (trX : list (composite_transition_item IM)),
  not_equivocating_equivocator_descriptors IM
    final_descriptors (finite_trace_last is tr)
  ∧ equivocators_trace_project final_descriptors tr =
    Some (trX, initial_descriptors)
    ∧ Exists (field_selector VLSM.output m) trX
  destruct Hex as [eqv_final [eqv_init [Heqv_init [Heqv_final [trXi [Hprojecti Hex]]]]]].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_constrained_trace_from
  (free_composite_vlsm (equivocator_IM IM)) is
  tr
m: message
i: index
li: EquivocatorLabel (IM i)
input: option message
destination: composite_state (equivocator_IM IM)
output: option message
Hitem: {|
  l := li;
  input := input;
  destination := destination i;
  output := output
|}
∈ VLSM_projection_finite_trace_project
    (preloaded_component_projection
       (equivocator_IM IM) i) tr
Hm: output = Some m
Htri: finite_constrained_trace_from
  (equivocator_IM IM i) 
  (is i)
  (VLSM_projection_finite_trace_project
     (preloaded_component_projection
        (equivocator_IM IM) i) tr)
eqv_final, eqv_init: MachineDescriptor (IM i)
Heqv_init: proper_descriptor (IM i) eqv_init (is i)
Heqv_final: existing_descriptor (IM i) eqv_final
  (finite_trace_last (is i)
     (VLSM_projection_finite_trace_project
        (preloaded_component_projection
           (equivocator_IM IM) i) tr))
trXi: list transition_item
Hprojecti: equivocator_vlsm_trace_project 
  (IM i)
  (VLSM_projection_finite_trace_project
     (preloaded_component_projection
        (equivocator_IM IM) i) tr)
  eqv_final = Some (trXi, eqv_init)
Hex: Exists (field_selector VLSM.output m) trXi
∃ (final_descriptors
   initial_descriptors : equivocator_descriptors IM) 
  (trX : list (composite_transition_item IM)),
  not_equivocating_equivocator_descriptors IM
    final_descriptors (finite_trace_last is tr)
  ∧ equivocators_trace_project final_descriptors tr =
    Some (trX, initial_descriptors)
    ∧ Exists (field_selector VLSM.output m) trX
  specialize (VLSM_projection_finite_trace_last
    (preloaded_component_projection (equivocator_IM IM) i) _ _ Htr) as Hlst.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_constrained_trace_from
  (free_composite_vlsm (equivocator_IM IM)) is
  tr
m: message
i: index
li: EquivocatorLabel (IM i)
input: option message
destination: composite_state (equivocator_IM IM)
output: option message
Hitem: {|
  l := li;
  input := input;
  destination := destination i;
  output := output
|}
∈ VLSM_projection_finite_trace_project
    (preloaded_component_projection
       (equivocator_IM IM) i) tr
Hm: output = Some m
Htri: finite_constrained_trace_from
  (equivocator_IM IM i) 
  (is i)
  (VLSM_projection_finite_trace_project
     (preloaded_component_projection
        (equivocator_IM IM) i) tr)
eqv_final, eqv_init: MachineDescriptor (IM i)
Heqv_init: proper_descriptor (IM i) eqv_init (is i)
Heqv_final: existing_descriptor (IM i) eqv_final
  (finite_trace_last (is i)
     (VLSM_projection_finite_trace_project
        (preloaded_component_projection
           (equivocator_IM IM) i) tr))
trXi: list transition_item
Hprojecti: equivocator_vlsm_trace_project 
  (IM i)
  (VLSM_projection_finite_trace_project
     (preloaded_component_projection
        (equivocator_IM IM) i) tr)
  eqv_final = Some (trXi, eqv_init)
Hex: Exists (field_selector VLSM.output m) trXi
Hlst: (λ s : state
         (preloaded_with_all_messages_vlsm
            (free_composite_vlsm
               (equivocator_IM IM))), 
   s i) (finite_trace_last is tr) =
finite_trace_last
  ((λ s : state
            (preloaded_with_all_messages_vlsm
               (free_composite_vlsm
                  (equivocator_IM IM))), 
      s i) is)
  (VLSM_projection_finite_trace_project
     (preloaded_component_projection
        (equivocator_IM IM) i) tr)
∃ (final_descriptors
   initial_descriptors : equivocator_descriptors IM) 
  (trX : list (composite_transition_item IM)),
  not_equivocating_equivocator_descriptors IM
    final_descriptors (finite_trace_last is tr)
  ∧ equivocators_trace_project final_descriptors tr =
    Some (trX, initial_descriptors)
    ∧ Exists (field_selector VLSM.output m) trX
  simpl in Hlst, Heqv_final.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_constrained_trace_from
  (free_composite_vlsm (equivocator_IM IM)) is
  tr
m: message
i: index
li: EquivocatorLabel (IM i)
input: option message
destination: composite_state (equivocator_IM IM)
output: option message
Hitem: {|
  l := li;
  input := input;
  destination := destination i;
  output := output
|}
∈ VLSM_projection_finite_trace_project
    (preloaded_component_projection
       (equivocator_IM IM) i) tr
Hm: output = Some m
Htri: finite_constrained_trace_from
  (equivocator_IM IM i) 
  (is i)
  (VLSM_projection_finite_trace_project
     (preloaded_component_projection
        (equivocator_IM IM) i) tr)
eqv_final, eqv_init: MachineDescriptor (IM i)
Heqv_init: proper_descriptor (IM i) eqv_init (is i)
Heqv_final: existing_descriptor (IM i) eqv_final
  (finite_trace_last (is i)
     (VLSM_projection_finite_trace_project
        (preloaded_component_projection
           (equivocator_IM IM) i) tr))
trXi: list transition_item
Hprojecti: equivocator_vlsm_trace_project 
  (IM i)
  (VLSM_projection_finite_trace_project
     (preloaded_component_projection
        (equivocator_IM IM) i) tr)
  eqv_final = Some (trXi, eqv_init)
Hex: Exists (field_selector VLSM.output m) trXi
Hlst: finite_trace_last is tr i =
finite_trace_last (is i)
  (VLSM_projection_finite_trace_project
     (preloaded_component_projection
        (equivocator_IM IM) i) tr)
∃ (final_descriptors
   initial_descriptors : equivocator_descriptors IM) 
  (trX : list (composite_transition_item IM)),
  not_equivocating_equivocator_descriptors IM
    final_descriptors (finite_trace_last is tr)
  ∧ equivocators_trace_project final_descriptors tr =
    Some (trX, initial_descriptors)
    ∧ Exists (field_selector VLSM.output m) trX rewrite <- Hlst in Heqv_final.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_constrained_trace_from
  (free_composite_vlsm (equivocator_IM IM)) is
  tr
m: message
i: index
li: EquivocatorLabel (IM i)
input: option message
destination: composite_state (equivocator_IM IM)
output: option message
Hitem: {|
  l := li;
  input := input;
  destination := destination i;
  output := output
|}
∈ VLSM_projection_finite_trace_project
    (preloaded_component_projection
       (equivocator_IM IM) i) tr
Hm: output = Some m
Htri: finite_constrained_trace_from
  (equivocator_IM IM i) 
  (is i)
  (VLSM_projection_finite_trace_project
     (preloaded_component_projection
        (equivocator_IM IM) i) tr)
eqv_final, eqv_init: MachineDescriptor (IM i)
Heqv_init: proper_descriptor (IM i) eqv_init (is i)
Heqv_final: existing_descriptor (IM i) eqv_final
  (finite_trace_last is tr i)
trXi: list transition_item
Hprojecti: equivocator_vlsm_trace_project 
  (IM i)
  (VLSM_projection_finite_trace_project
     (preloaded_component_projection
        (equivocator_IM IM) i) tr)
  eqv_final = Some (trXi, eqv_init)
Hex: Exists (field_selector VLSM.output m) trXi
Hlst: finite_trace_last is tr i =
finite_trace_last (is i)
  (VLSM_projection_finite_trace_project
     (preloaded_component_projection
        (equivocator_IM IM) i) tr)
∃ (final_descriptors
   initial_descriptors : equivocator_descriptors IM) 
  (trX : list (composite_transition_item IM)),
  not_equivocating_equivocator_descriptors IM
    final_descriptors (finite_trace_last is tr)
  ∧ equivocators_trace_project final_descriptors tr =
    Some (trX, initial_descriptors)
    ∧ Exists (field_selector VLSM.output m) trX clear Hlst.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_constrained_trace_from
  (free_composite_vlsm (equivocator_IM IM)) is
  tr
m: message
i: index
li: EquivocatorLabel (IM i)
input: option message
destination: composite_state (equivocator_IM IM)
output: option message
Hitem: {|
  l := li;
  input := input;
  destination := destination i;
  output := output
|}
∈ VLSM_projection_finite_trace_project
    (preloaded_component_projection
       (equivocator_IM IM) i) tr
Hm: output = Some m
Htri: finite_constrained_trace_from
  (equivocator_IM IM i) 
  (is i)
  (VLSM_projection_finite_trace_project
     (preloaded_component_projection
        (equivocator_IM IM) i) tr)
eqv_final, eqv_init: MachineDescriptor (IM i)
Heqv_init: proper_descriptor (IM i) eqv_init (is i)
Heqv_final: existing_descriptor (IM i) eqv_final
  (finite_trace_last is tr i)
trXi: list transition_item
Hprojecti: equivocator_vlsm_trace_project 
  (IM i)
  (VLSM_projection_finite_trace_project
     (preloaded_component_projection
        (equivocator_IM IM) i) tr)
  eqv_final = Some (trXi, eqv_init)
Hex: Exists (field_selector VLSM.output m) trXi
∃ (final_descriptors
   initial_descriptors : equivocator_descriptors IM) 
  (trX : list (composite_transition_item IM)),
  not_equivocating_equivocator_descriptors IM
    final_descriptors (finite_trace_last is tr)
  ∧ equivocators_trace_project final_descriptors tr =
    Some (trX, initial_descriptors)
    ∧ Exists (field_selector VLSM.output m) trX
  remember (finite_trace_last is tr) as final.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_constrained_trace_from
  (free_composite_vlsm (equivocator_IM IM)) is
  tr
m: message
i: index
li: EquivocatorLabel (IM i)
input: option message
destination: composite_state (equivocator_IM IM)
output: option message
Hitem: {|
  l := li;
  input := input;
  destination := destination i;
  output := output
|}
∈ VLSM_projection_finite_trace_project
    (preloaded_component_projection
       (equivocator_IM IM) i) tr
Hm: output = Some m
Htri: finite_constrained_trace_from
  (equivocator_IM IM i) 
  (is i)
  (VLSM_projection_finite_trace_project
     (preloaded_component_projection
        (equivocator_IM IM) i) tr)
eqv_final, eqv_init: MachineDescriptor (IM i)
Heqv_init: proper_descriptor (IM i) eqv_init (is i)
final: state (composite_type (equivocator_IM IM))
Heqfinal: final = finite_trace_last is tr
Heqv_final: existing_descriptor (IM i) eqv_final
  (final i)
trXi: list transition_item
Hprojecti: equivocator_vlsm_trace_project 
  (IM i)
  (VLSM_projection_finite_trace_project
     (preloaded_component_projection
        (equivocator_IM IM) i) tr)
  eqv_final = Some (trXi, eqv_init)
Hex: Exists (field_selector VLSM.output m) trXi
∃ (final_descriptors
   initial_descriptors : equivocator_descriptors IM) 
  (trX : list (composite_transition_item IM)),
  not_equivocating_equivocator_descriptors IM
    final_descriptors final
  ∧ equivocators_trace_project final_descriptors tr =
    Some (trX, initial_descriptors)
    ∧ Exists (field_selector VLSM.output m) trX
  remember (equivocator_descriptors_update _ (zero_descriptor IM) i eqv_final) as final_descriptors.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_constrained_trace_from
  (free_composite_vlsm (equivocator_IM IM)) is
  tr
m: message
i: index
li: EquivocatorLabel (IM i)
input: option message
destination: composite_state (equivocator_IM IM)
output: option message
Hitem: {|
  l := li;
  input := input;
  destination := destination i;
  output := output
|}
∈ VLSM_projection_finite_trace_project
    (preloaded_component_projection
       (equivocator_IM IM) i) tr
Hm: output = Some m
Htri: finite_constrained_trace_from
  (equivocator_IM IM i) 
  (is i)
  (VLSM_projection_finite_trace_project
     (preloaded_component_projection
        (equivocator_IM IM) i) tr)
eqv_final, eqv_init: MachineDescriptor (IM i)
Heqv_init: proper_descriptor (IM i) eqv_init (is i)
final: state (composite_type (equivocator_IM IM))
Heqfinal: final = finite_trace_last is tr
Heqv_final: existing_descriptor (IM i) eqv_final
  (final i)
trXi: list transition_item
Hprojecti: equivocator_vlsm_trace_project 
  (IM i)
  (VLSM_projection_finite_trace_project
     (preloaded_component_projection
        (equivocator_IM IM) i) tr)
  eqv_final = Some (trXi, eqv_init)
Hex: Exists (field_selector VLSM.output m) trXi
final_descriptors: ∀ j : index,
  MachineDescriptor (IM j)
Heqfinal_descriptors: final_descriptors =
equivocator_descriptors_update
  IM (zero_descriptor IM) i
  eqv_final
∃ (final_descriptors
   initial_descriptors : equivocator_descriptors IM) 
  (trX : list (composite_transition_item IM)),
  not_equivocating_equivocator_descriptors IM
    final_descriptors final
  ∧ equivocators_trace_project final_descriptors tr =
    Some (trX, initial_descriptors)
    ∧ Exists (field_selector VLSM.output m) trX
  assert (Hfinal_descriptors : not_equivocating_equivocator_descriptors IM final_descriptors final).message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_constrained_trace_from
  (free_composite_vlsm (equivocator_IM IM)) is
  tr
m: message
i: index
li: EquivocatorLabel (IM i)
input: option message
destination: composite_state (equivocator_IM IM)
output: option message
Hitem: {|
  l := li;
  input := input;
  destination := destination i;
  output := output
|}
∈ VLSM_projection_finite_trace_project
    (preloaded_component_projection
       (equivocator_IM IM) i) tr
Hm: output = Some m
Htri: finite_constrained_trace_from
  (equivocator_IM IM i) 
  (is i)
  (VLSM_projection_finite_trace_project
     (preloaded_component_projection
        (equivocator_IM IM) i) tr)
eqv_final, eqv_init: MachineDescriptor (IM i)
Heqv_init: proper_descriptor (IM i) eqv_init (is i)
final: state (composite_type (equivocator_IM IM))
Heqfinal: final = finite_trace_last is tr
Heqv_final: existing_descriptor (IM i) eqv_final
  (final i)
trXi: list transition_item
Hprojecti: equivocator_vlsm_trace_project 
  (IM i)
  (VLSM_projection_finite_trace_project
     (preloaded_component_projection
        (equivocator_IM IM) i) tr)
  eqv_final = Some (trXi, eqv_init)
Hex: Exists (field_selector VLSM.output m) trXi
final_descriptors: ∀ j : index,
  MachineDescriptor (IM j)
Heqfinal_descriptors: final_descriptors =
equivocator_descriptors_update
  IM (zero_descriptor IM) i
  eqv_final
not_equivocating_equivocator_descriptors IM
  final_descriptors final
message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_constrained_trace_from
  (free_composite_vlsm (equivocator_IM IM)) is
  tr
m: message
i: index
li: EquivocatorLabel (IM i)
input: option message
destination: composite_state (equivocator_IM IM)
output: option message
Hitem: {|
  l := li;
  input := input;
  destination := destination i;
  output := output
|}
∈ VLSM_projection_finite_trace_project
    (preloaded_component_projection
       (equivocator_IM IM) i) tr
Hm: output = Some m
Htri: finite_constrained_trace_from
  (equivocator_IM IM i) 
  (is i)
  (VLSM_projection_finite_trace_project
     (preloaded_component_projection
        (equivocator_IM IM) i) tr)
eqv_final, eqv_init: MachineDescriptor (IM i)
Heqv_init: proper_descriptor (IM i) eqv_init (is i)
final: state (composite_type (equivocator_IM IM))
Heqfinal: final = finite_trace_last is tr
Heqv_final: existing_descriptor (IM i) eqv_final
  (final i)
trXi: list transition_item
Hprojecti: equivocator_vlsm_trace_project 
  (IM i)
  (VLSM_projection_finite_trace_project
     (preloaded_component_projection
        (equivocator_IM IM) i) tr)
  eqv_final = Some (trXi, eqv_init)
Hex: Exists (field_selector VLSM.output m) trXi
final_descriptors: ∀ j : index,
  MachineDescriptor (IM j)
Heqfinal_descriptors: final_descriptors =
equivocator_descriptors_update
  IM (zero_descriptor IM) i
  eqv_final
Hfinal_descriptors: not_equivocating_equivocator_descriptors
  IM final_descriptors final
∃ (final_descriptors
   initial_descriptors : 
   equivocator_descriptors IM) 
  (trX : list (composite_transition_item IM)),
  not_equivocating_equivocator_descriptors IM
    final_descriptors final
  ∧ equivocators_trace_project final_descriptors tr =
    Some (trX, initial_descriptors)
    ∧ Exists (field_selector VLSM.output m) trX
  {message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_constrained_trace_from
  (free_composite_vlsm (equivocator_IM IM)) is
  tr
m: message
i: index
li: EquivocatorLabel (IM i)
input: option message
destination: composite_state (equivocator_IM IM)
output: option message
Hitem: {|
  l := li;
  input := input;
  destination := destination i;
  output := output
|}
∈ VLSM_projection_finite_trace_project
    (preloaded_component_projection
       (equivocator_IM IM) i) tr
Hm: output = Some m
Htri: finite_constrained_trace_from
  (equivocator_IM IM i) 
  (is i)
  (VLSM_projection_finite_trace_project
     (preloaded_component_projection
        (equivocator_IM IM) i) tr)
eqv_final, eqv_init: MachineDescriptor (IM i)
Heqv_init: proper_descriptor (IM i) eqv_init (is i)
final: state (composite_type (equivocator_IM IM))
Heqfinal: final = finite_trace_last is tr
Heqv_final: existing_descriptor (IM i) eqv_final
  (final i)
trXi: list transition_item
Hprojecti: equivocator_vlsm_trace_project 
  (IM i)
  (VLSM_projection_finite_trace_project
     (preloaded_component_projection
        (equivocator_IM IM) i) tr)
  eqv_final = Some (trXi, eqv_init)
Hex: Exists (field_selector VLSM.output m) trXi
final_descriptors: ∀ j : index,
  MachineDescriptor (IM j)
Heqfinal_descriptors: final_descriptors =
equivocator_descriptors_update
  IM (zero_descriptor IM) i
  eqv_final
not_equivocating_equivocator_descriptors IM
  final_descriptors final
    intro eqv.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_constrained_trace_from
  (free_composite_vlsm (equivocator_IM IM)) is
  tr
m: message
i: index
li: EquivocatorLabel (IM i)
input: option message
destination: composite_state (equivocator_IM IM)
output: option message
Hitem: {|
  l := li;
  input := input;
  destination := destination i;
  output := output
|}
∈ VLSM_projection_finite_trace_project
    (preloaded_component_projection
       (equivocator_IM IM) i) tr
Hm: output = Some m
Htri: finite_constrained_trace_from
  (equivocator_IM IM i) 
  (is i)
  (VLSM_projection_finite_trace_project
     (preloaded_component_projection
        (equivocator_IM IM) i) tr)
eqv_final, eqv_init: MachineDescriptor (IM i)
Heqv_init: proper_descriptor (IM i) eqv_init (is i)
final: state (composite_type (equivocator_IM IM))
Heqfinal: final = finite_trace_last is tr
Heqv_final: existing_descriptor (IM i) eqv_final
  (final i)
trXi: list transition_item
Hprojecti: equivocator_vlsm_trace_project 
  (IM i)
  (VLSM_projection_finite_trace_project
     (preloaded_component_projection
        (equivocator_IM IM) i) tr)
  eqv_final = Some (trXi, eqv_init)
Hex: Exists (field_selector VLSM.output m) trXi
final_descriptors: ∀ j : index,
  MachineDescriptor (IM j)
Heqfinal_descriptors: final_descriptors =
equivocator_descriptors_update
  IM (zero_descriptor IM) i
  eqv_final
eqv: index
existing_descriptor (IM eqv) (final_descriptors eqv)
  (final eqv) subst final_descriptors.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_constrained_trace_from
  (free_composite_vlsm (equivocator_IM IM)) is
  tr
m: message
i: index
li: EquivocatorLabel (IM i)
input: option message
destination: composite_state (equivocator_IM IM)
output: option message
Hitem: {|
  l := li;
  input := input;
  destination := destination i;
  output := output
|}
∈ VLSM_projection_finite_trace_project
    (preloaded_component_projection
       (equivocator_IM IM) i) tr
Hm: output = Some m
Htri: finite_constrained_trace_from
  (equivocator_IM IM i) 
  (is i)
  (VLSM_projection_finite_trace_project
     (preloaded_component_projection
        (equivocator_IM IM) i) tr)
eqv_final, eqv_init: MachineDescriptor (IM i)
Heqv_init: proper_descriptor (IM i) eqv_init (is i)
final: state (composite_type (equivocator_IM IM))
Heqfinal: final = finite_trace_last is tr
Heqv_final: existing_descriptor (IM i) eqv_final
  (final i)
trXi: list transition_item
Hprojecti: equivocator_vlsm_trace_project 
  (IM i)
  (VLSM_projection_finite_trace_project
     (preloaded_component_projection
        (equivocator_IM IM) i) tr)
  eqv_final = Some (trXi, eqv_init)
Hex: Exists (field_selector VLSM.output m) trXi
eqv: index
existing_descriptor (IM eqv)
  (equivocator_descriptors_update IM
     (zero_descriptor IM) i eqv_final eqv) (final eqv)
    destruct (decide (i = eqv)); subst; state_update_simpl; [done |].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_constrained_trace_from
  (free_composite_vlsm (equivocator_IM IM)) is
  tr
m: message
i: index
li: EquivocatorLabel (IM i)
input: option message
destination: composite_state (equivocator_IM IM)
Hitem: {|
  l := li;
  input := input;
  destination := destination i;
  output := Some m
|}
∈ VLSM_projection_finite_trace_project
    (preloaded_component_projection
       (equivocator_IM IM) i) tr
Htri: finite_constrained_trace_from
  (equivocator_IM IM i) 
  (is i)
  (VLSM_projection_finite_trace_project
     (preloaded_component_projection
        (equivocator_IM IM) i) tr)
eqv_final, eqv_init: MachineDescriptor (IM i)
Heqv_init: proper_descriptor (IM i) eqv_init (is i)
Heqv_final: existing_descriptor (IM i) eqv_final
  (finite_trace_last is tr i)
trXi: list transition_item
Hprojecti: equivocator_vlsm_trace_project 
  (IM i)
  (VLSM_projection_finite_trace_project
     (preloaded_component_projection
        (equivocator_IM IM) i) tr)
  eqv_final = Some (trXi, eqv_init)
Hex: Exists (field_selector output m) trXi
eqv: index
n: i ≠ eqv
existing_descriptor (IM eqv) (zero_descriptor IM eqv)
  (finite_trace_last is tr eqv)
    by apply zero_descriptor_proper.
  }message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_constrained_trace_from
  (free_composite_vlsm (equivocator_IM IM)) is
  tr
m: message
i: index
li: EquivocatorLabel (IM i)
input: option message
destination: composite_state (equivocator_IM IM)
output: option message
Hitem: {|
  l := li;
  input := input;
  destination := destination i;
  output := output
|}
∈ VLSM_projection_finite_trace_project
    (preloaded_component_projection
       (equivocator_IM IM) i) tr
Hm: output = Some m
Htri: finite_constrained_trace_from
  (equivocator_IM IM i) 
  (is i)
  (VLSM_projection_finite_trace_project
     (preloaded_component_projection
        (equivocator_IM IM) i) tr)
eqv_final, eqv_init: MachineDescriptor (IM i)
Heqv_init: proper_descriptor (IM i) eqv_init (is i)
final: state (composite_type (equivocator_IM IM))
Heqfinal: final = finite_trace_last is tr
Heqv_final: existing_descriptor (IM i) eqv_final
  (final i)
trXi: list transition_item
Hprojecti: equivocator_vlsm_trace_project 
  (IM i)
  (VLSM_projection_finite_trace_project
     (preloaded_component_projection
        (equivocator_IM IM) i) tr)
  eqv_final = Some (trXi, eqv_init)
Hex: Exists (field_selector VLSM.output m) trXi
final_descriptors: ∀ j : index,
  MachineDescriptor (IM j)
Heqfinal_descriptors: final_descriptors =
equivocator_descriptors_update
  IM (zero_descriptor IM) i
  eqv_final
Hfinal_descriptors: not_equivocating_equivocator_descriptors
  IM final_descriptors final
∃ (final_descriptors
   initial_descriptors : equivocator_descriptors IM) 
  (trX : list (composite_transition_item IM)),
  not_equivocating_equivocator_descriptors IM
    final_descriptors final
  ∧ equivocators_trace_project final_descriptors tr =
    Some (trX, initial_descriptors)
    ∧ Exists (field_selector VLSM.output m) trX
  exists final_descriptors.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_constrained_trace_from
  (free_composite_vlsm (equivocator_IM IM)) is
  tr
m: message
i: index
li: EquivocatorLabel (IM i)
input: option message
destination: composite_state (equivocator_IM IM)
output: option message
Hitem: {|
  l := li;
  input := input;
  destination := destination i;
  output := output
|}
∈ VLSM_projection_finite_trace_project
    (preloaded_component_projection
       (equivocator_IM IM) i) tr
Hm: output = Some m
Htri: finite_constrained_trace_from
  (equivocator_IM IM i) 
  (is i)
  (VLSM_projection_finite_trace_project
     (preloaded_component_projection
        (equivocator_IM IM) i) tr)
eqv_final, eqv_init: MachineDescriptor (IM i)
Heqv_init: proper_descriptor (IM i) eqv_init (is i)
final: state (composite_type (equivocator_IM IM))
Heqfinal: final = finite_trace_last is tr
Heqv_final: existing_descriptor (IM i) eqv_final
  (final i)
trXi: list transition_item
Hprojecti: equivocator_vlsm_trace_project 
  (IM i)
  (VLSM_projection_finite_trace_project
     (preloaded_component_projection
        (equivocator_IM IM) i) tr)
  eqv_final = Some (trXi, eqv_init)
Hex: Exists (field_selector VLSM.output m) trXi
final_descriptors: ∀ j : index,
  MachineDescriptor (IM j)
Heqfinal_descriptors: final_descriptors =
equivocator_descriptors_update
  IM (zero_descriptor IM) i
  eqv_final
Hfinal_descriptors: not_equivocating_equivocator_descriptors
  IM final_descriptors final
∃ (initial_descriptors : equivocator_descriptors IM) 
  (trX : list (composite_transition_item IM)),
  not_equivocating_equivocator_descriptors IM
    final_descriptors final
  ∧ equivocators_trace_project final_descriptors tr =
    Some (trX, initial_descriptors)
    ∧ Exists (field_selector VLSM.output m) trX
  subst final.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_constrained_trace_from
  (free_composite_vlsm (equivocator_IM IM)) is
  tr
m: message
i: index
li: EquivocatorLabel (IM i)
input: option message
destination: composite_state (equivocator_IM IM)
output: option message
Hitem: {|
  l := li;
  input := input;
  destination := destination i;
  output := output
|}
∈ VLSM_projection_finite_trace_project
    (preloaded_component_projection
       (equivocator_IM IM) i) tr
Hm: output = Some m
Htri: finite_constrained_trace_from
  (equivocator_IM IM i) 
  (is i)
  (VLSM_projection_finite_trace_project
     (preloaded_component_projection
        (equivocator_IM IM) i) tr)
eqv_final, eqv_init: MachineDescriptor (IM i)
Heqv_init: proper_descriptor (IM i) eqv_init (is i)
Heqv_final: existing_descriptor (IM i) eqv_final
  (finite_trace_last is tr i)
trXi: list transition_item
Hprojecti: equivocator_vlsm_trace_project 
  (IM i)
  (VLSM_projection_finite_trace_project
     (preloaded_component_projection
        (equivocator_IM IM) i) tr)
  eqv_final = Some (trXi, eqv_init)
Hex: Exists (field_selector VLSM.output m) trXi
final_descriptors: ∀ j : index,
  MachineDescriptor (IM j)
Heqfinal_descriptors: final_descriptors =
equivocator_descriptors_update
  IM (zero_descriptor IM) i
  eqv_final
Hfinal_descriptors: not_equivocating_equivocator_descriptors
  IM final_descriptors
  (finite_trace_last is tr)
∃ (initial_descriptors : equivocator_descriptors IM) 
  (trX : list (composite_transition_item IM)),
  not_equivocating_equivocator_descriptors IM
    final_descriptors (finite_trace_last is tr)
  ∧ equivocators_trace_project final_descriptors tr =
    Some (trX, initial_descriptors)
    ∧ Exists (field_selector VLSM.output m) trX
  assert (Hfinal_descriptors_proper :
    proper_equivocator_descriptors IM final_descriptors (finite_trace_last is tr)).message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_constrained_trace_from
  (free_composite_vlsm (equivocator_IM IM)) is
  tr
m: message
i: index
li: EquivocatorLabel (IM i)
input: option message
destination: composite_state (equivocator_IM IM)
output: option message
Hitem: {|
  l := li;
  input := input;
  destination := destination i;
  output := output
|}
∈ VLSM_projection_finite_trace_project
    (preloaded_component_projection
       (equivocator_IM IM) i) tr
Hm: output = Some m
Htri: finite_constrained_trace_from
  (equivocator_IM IM i) 
  (is i)
  (VLSM_projection_finite_trace_project
     (preloaded_component_projection
        (equivocator_IM IM) i) tr)
eqv_final, eqv_init: MachineDescriptor (IM i)
Heqv_init: proper_descriptor (IM i) eqv_init (is i)
Heqv_final: existing_descriptor (IM i) eqv_final
  (finite_trace_last is tr i)
trXi: list transition_item
Hprojecti: equivocator_vlsm_trace_project 
  (IM i)
  (VLSM_projection_finite_trace_project
     (preloaded_component_projection
        (equivocator_IM IM) i) tr)
  eqv_final = Some (trXi, eqv_init)
Hex: Exists (field_selector VLSM.output m) trXi
final_descriptors: ∀ j : index,
  MachineDescriptor (IM j)
Heqfinal_descriptors: final_descriptors =
equivocator_descriptors_update
  IM (zero_descriptor IM) i
  eqv_final
Hfinal_descriptors: not_equivocating_equivocator_descriptors
  IM final_descriptors
  (finite_trace_last is tr)
proper_equivocator_descriptors IM final_descriptors
  (finite_trace_last is tr)
message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_constrained_trace_from
  (free_composite_vlsm (equivocator_IM IM)) is
  tr
m: message
i: index
li: EquivocatorLabel (IM i)
input: option message
destination: composite_state (equivocator_IM IM)
output: option message
Hitem: {|
  l := li;
  input := input;
  destination := destination i;
  output := output
|}
∈ VLSM_projection_finite_trace_project
    (preloaded_component_projection
       (equivocator_IM IM) i) tr
Hm: output = Some m
Htri: finite_constrained_trace_from
  (equivocator_IM IM i) 
  (is i)
  (VLSM_projection_finite_trace_project
     (preloaded_component_projection
        (equivocator_IM IM) i) tr)
eqv_final, eqv_init: MachineDescriptor (IM i)
Heqv_init: proper_descriptor (IM i) eqv_init (is i)
Heqv_final: existing_descriptor (IM i) eqv_final
  (finite_trace_last is tr i)
trXi: list transition_item
Hprojecti: equivocator_vlsm_trace_project 
  (IM i)
  (VLSM_projection_finite_trace_project
     (preloaded_component_projection
        (equivocator_IM IM) i) tr)
  eqv_final = Some (trXi, eqv_init)
Hex: Exists (field_selector VLSM.output m) trXi
final_descriptors: ∀ j : index,
  MachineDescriptor (IM j)
Heqfinal_descriptors: final_descriptors =
equivocator_descriptors_update
  IM (zero_descriptor IM) i
  eqv_final
Hfinal_descriptors: not_equivocating_equivocator_descriptors
  IM final_descriptors
  (finite_trace_last is tr)
Hfinal_descriptors_proper: proper_equivocator_descriptors
  IM final_descriptors
  (finite_trace_last is tr)
∃ (initial_descriptors : equivocator_descriptors IM) 
  (trX : list (composite_transition_item IM)),
  not_equivocating_equivocator_descriptors IM
    final_descriptors (finite_trace_last is tr)
  ∧ equivocators_trace_project final_descriptors tr =
    Some (trX, initial_descriptors)
    ∧ Exists (field_selector VLSM.output m) trX
  {message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_constrained_trace_from
  (free_composite_vlsm (equivocator_IM IM)) is
  tr
m: message
i: index
li: EquivocatorLabel (IM i)
input: option message
destination: composite_state (equivocator_IM IM)
output: option message
Hitem: {|
  l := li;
  input := input;
  destination := destination i;
  output := output
|}
∈ VLSM_projection_finite_trace_project
    (preloaded_component_projection
       (equivocator_IM IM) i) tr
Hm: output = Some m
Htri: finite_constrained_trace_from
  (equivocator_IM IM i) 
  (is i)
  (VLSM_projection_finite_trace_project
     (preloaded_component_projection
        (equivocator_IM IM) i) tr)
eqv_final, eqv_init: MachineDescriptor (IM i)
Heqv_init: proper_descriptor (IM i) eqv_init (is i)
Heqv_final: existing_descriptor (IM i) eqv_final
  (finite_trace_last is tr i)
trXi: list transition_item
Hprojecti: equivocator_vlsm_trace_project 
  (IM i)
  (VLSM_projection_finite_trace_project
     (preloaded_component_projection
        (equivocator_IM IM) i) tr)
  eqv_final = Some (trXi, eqv_init)
Hex: Exists (field_selector VLSM.output m) trXi
final_descriptors: ∀ j : index,
  MachineDescriptor (IM j)
Heqfinal_descriptors: final_descriptors =
equivocator_descriptors_update
  IM (zero_descriptor IM) i
  eqv_final
Hfinal_descriptors: not_equivocating_equivocator_descriptors
  IM final_descriptors
  (finite_trace_last is tr)
proper_equivocator_descriptors IM final_descriptors
  (finite_trace_last is tr) by apply not_equivocating_equivocator_descriptors_proper. }message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_constrained_trace_from
  (free_composite_vlsm (equivocator_IM IM)) is
  tr
m: message
i: index
li: EquivocatorLabel (IM i)
input: option message
destination: composite_state (equivocator_IM IM)
output: option message
Hitem: {|
  l := li;
  input := input;
  destination := destination i;
  output := output
|}
∈ VLSM_projection_finite_trace_project
    (preloaded_component_projection
       (equivocator_IM IM) i) tr
Hm: output = Some m
Htri: finite_constrained_trace_from
  (equivocator_IM IM i) 
  (is i)
  (VLSM_projection_finite_trace_project
     (preloaded_component_projection
        (equivocator_IM IM) i) tr)
eqv_final, eqv_init: MachineDescriptor (IM i)
Heqv_init: proper_descriptor (IM i) eqv_init (is i)
Heqv_final: existing_descriptor (IM i) eqv_final
  (finite_trace_last is tr i)
trXi: list transition_item
Hprojecti: equivocator_vlsm_trace_project 
  (IM i)
  (VLSM_projection_finite_trace_project
     (preloaded_component_projection
        (equivocator_IM IM) i) tr)
  eqv_final = Some (trXi, eqv_init)
Hex: Exists (field_selector VLSM.output m) trXi
final_descriptors: ∀ j : index,
  MachineDescriptor (IM j)
Heqfinal_descriptors: final_descriptors =
equivocator_descriptors_update
  IM (zero_descriptor IM) i
  eqv_final
Hfinal_descriptors: not_equivocating_equivocator_descriptors
  IM final_descriptors
  (finite_trace_last is tr)
Hfinal_descriptors_proper: proper_equivocator_descriptors
  IM final_descriptors
  (finite_trace_last is tr)
∃ (initial_descriptors : equivocator_descriptors IM) 
  (trX : list (composite_transition_item IM)),
  not_equivocating_equivocator_descriptors IM
    final_descriptors (finite_trace_last is tr)
  ∧ equivocators_trace_project final_descriptors tr =
    Some (trX, initial_descriptors)
    ∧ Exists (field_selector VLSM.output m) trX
  destruct (preloaded_equivocators_valid_trace_from_project  _ _ _ Hfinal_descriptors_proper Htr)
    as [trX [initial_descriptors [Hproject_tr _]]].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_constrained_trace_from
  (free_composite_vlsm (equivocator_IM IM)) is
  tr
m: message
i: index
li: EquivocatorLabel (IM i)
input: option message
destination: composite_state (equivocator_IM IM)
output: option message
Hitem: {|
  l := li;
  input := input;
  destination := destination i;
  output := output
|}
∈ VLSM_projection_finite_trace_project
    (preloaded_component_projection
       (equivocator_IM IM) i) tr
Hm: output = Some m
Htri: finite_constrained_trace_from
  (equivocator_IM IM i) 
  (is i)
  (VLSM_projection_finite_trace_project
     (preloaded_component_projection
        (equivocator_IM IM) i) tr)
eqv_final, eqv_init: MachineDescriptor (IM i)
Heqv_init: proper_descriptor (IM i) eqv_init (is i)
Heqv_final: existing_descriptor (IM i) eqv_final
  (finite_trace_last is tr i)
trXi: list transition_item
Hprojecti: equivocator_vlsm_trace_project 
  (IM i)
  (VLSM_projection_finite_trace_project
     (preloaded_component_projection
        (equivocator_IM IM) i) tr)
  eqv_final = Some (trXi, eqv_init)
Hex: Exists (field_selector VLSM.output m) trXi
final_descriptors: ∀ j : index,
  MachineDescriptor (IM j)
Heqfinal_descriptors: final_descriptors =
equivocator_descriptors_update
  IM (zero_descriptor IM) i
  eqv_final
Hfinal_descriptors: not_equivocating_equivocator_descriptors
  IM final_descriptors
  (finite_trace_last is tr)
Hfinal_descriptors_proper: proper_equivocator_descriptors
  IM final_descriptors
  (finite_trace_last is tr)
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hproject_tr: equivocators_trace_project
  final_descriptors tr =
Some (trX, initial_descriptors)
∃ (initial_descriptors : equivocator_descriptors IM) 
  (trX : list (composite_transition_item IM)),
  not_equivocating_equivocator_descriptors IM
    final_descriptors (finite_trace_last is tr)
  ∧ equivocators_trace_project final_descriptors tr =
    Some (trX, initial_descriptors)
    ∧ Exists (field_selector VLSM.output m) trX
  exists initial_descriptors, trX.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_constrained_trace_from
  (free_composite_vlsm (equivocator_IM IM)) is
  tr
m: message
i: index
li: EquivocatorLabel (IM i)
input: option message
destination: composite_state (equivocator_IM IM)
output: option message
Hitem: {|
  l := li;
  input := input;
  destination := destination i;
  output := output
|}
∈ VLSM_projection_finite_trace_project
    (preloaded_component_projection
       (equivocator_IM IM) i) tr
Hm: output = Some m
Htri: finite_constrained_trace_from
  (equivocator_IM IM i) 
  (is i)
  (VLSM_projection_finite_trace_project
     (preloaded_component_projection
        (equivocator_IM IM) i) tr)
eqv_final, eqv_init: MachineDescriptor (IM i)
Heqv_init: proper_descriptor (IM i) eqv_init (is i)
Heqv_final: existing_descriptor (IM i) eqv_final
  (finite_trace_last is tr i)
trXi: list transition_item
Hprojecti: equivocator_vlsm_trace_project 
  (IM i)
  (VLSM_projection_finite_trace_project
     (preloaded_component_projection
        (equivocator_IM IM) i) tr)
  eqv_final = Some (trXi, eqv_init)
Hex: Exists (field_selector VLSM.output m) trXi
final_descriptors: ∀ j : index,
  MachineDescriptor (IM j)
Heqfinal_descriptors: final_descriptors =
equivocator_descriptors_update
  IM (zero_descriptor IM) i
  eqv_final
Hfinal_descriptors: not_equivocating_equivocator_descriptors
  IM final_descriptors
  (finite_trace_last is tr)
Hfinal_descriptors_proper: proper_equivocator_descriptors
  IM final_descriptors
  (finite_trace_last is tr)
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hproject_tr: equivocators_trace_project
  final_descriptors tr =
Some (trX, initial_descriptors)
not_equivocating_equivocator_descriptors IM
  final_descriptors (finite_trace_last is tr)
∧ equivocators_trace_project final_descriptors tr =
  Some (trX, initial_descriptors)
  ∧ Exists (field_selector VLSM.output m) trX split; [done |].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_constrained_trace_from
  (free_composite_vlsm (equivocator_IM IM)) is
  tr
m: message
i: index
li: EquivocatorLabel (IM i)
input: option message
destination: composite_state (equivocator_IM IM)
output: option message
Hitem: {|
  l := li;
  input := input;
  destination := destination i;
  output := output
|}
∈ VLSM_projection_finite_trace_project
    (preloaded_component_projection
       (equivocator_IM IM) i) tr
Hm: output = Some m
Htri: finite_constrained_trace_from
  (equivocator_IM IM i) 
  (is i)
  (VLSM_projection_finite_trace_project
     (preloaded_component_projection
        (equivocator_IM IM) i) tr)
eqv_final, eqv_init: MachineDescriptor (IM i)
Heqv_init: proper_descriptor (IM i) eqv_init (is i)
Heqv_final: existing_descriptor (IM i) eqv_final
  (finite_trace_last is tr i)
trXi: list transition_item
Hprojecti: equivocator_vlsm_trace_project 
  (IM i)
  (VLSM_projection_finite_trace_project
     (preloaded_component_projection
        (equivocator_IM IM) i) tr)
  eqv_final = Some (trXi, eqv_init)
Hex: Exists (field_selector VLSM.output m) trXi
final_descriptors: ∀ j : index,
  MachineDescriptor (IM j)
Heqfinal_descriptors: final_descriptors =
equivocator_descriptors_update
  IM (zero_descriptor IM) i
  eqv_final
Hfinal_descriptors: not_equivocating_equivocator_descriptors
  IM final_descriptors
  (finite_trace_last is tr)
Hfinal_descriptors_proper: proper_equivocator_descriptors
  IM final_descriptors
  (finite_trace_last is tr)
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hproject_tr: equivocators_trace_project
  final_descriptors tr =
Some (trX, initial_descriptors)
equivocators_trace_project final_descriptors tr =
Some (trX, initial_descriptors)
∧ Exists (field_selector VLSM.output m) trX split; [done |].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_constrained_trace_from
  (free_composite_vlsm (equivocator_IM IM)) is
  tr
m: message
i: index
li: EquivocatorLabel (IM i)
input: option message
destination: composite_state (equivocator_IM IM)
output: option message
Hitem: {|
  l := li;
  input := input;
  destination := destination i;
  output := output
|}
∈ VLSM_projection_finite_trace_project
    (preloaded_component_projection
       (equivocator_IM IM) i) tr
Hm: output = Some m
Htri: finite_constrained_trace_from
  (equivocator_IM IM i) 
  (is i)
  (VLSM_projection_finite_trace_project
     (preloaded_component_projection
        (equivocator_IM IM) i) tr)
eqv_final, eqv_init: MachineDescriptor (IM i)
Heqv_init: proper_descriptor (IM i) eqv_init (is i)
Heqv_final: existing_descriptor (IM i) eqv_final
  (finite_trace_last is tr i)
trXi: list transition_item
Hprojecti: equivocator_vlsm_trace_project 
  (IM i)
  (VLSM_projection_finite_trace_project
     (preloaded_component_projection
        (equivocator_IM IM) i) tr)
  eqv_final = Some (trXi, eqv_init)
Hex: Exists (field_selector VLSM.output m) trXi
final_descriptors: ∀ j : index,
  MachineDescriptor (IM j)
Heqfinal_descriptors: final_descriptors =
equivocator_descriptors_update
  IM (zero_descriptor IM) i
  eqv_final
Hfinal_descriptors: not_equivocating_equivocator_descriptors
  IM final_descriptors
  (finite_trace_last is tr)
Hfinal_descriptors_proper: proper_equivocator_descriptors
  IM final_descriptors
  (finite_trace_last is tr)
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hproject_tr: equivocators_trace_project
  final_descriptors tr =
Some (trX, initial_descriptors)
Exists (field_selector VLSM.output m) trX
  specialize (equivocators_trace_project_finite_trace_projection_list_commute
    i final_descriptors initial_descriptors
      eqv_init tr trX trXi Hproject_tr)
    as Hcommute.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_constrained_trace_from
  (free_composite_vlsm (equivocator_IM IM)) is
  tr
m: message
i: index
li: EquivocatorLabel (IM i)
input: option message
destination: composite_state (equivocator_IM IM)
output: option message
Hitem: {|
  l := li;
  input := input;
  destination := destination i;
  output := output
|}
∈ VLSM_projection_finite_trace_project
    (preloaded_component_projection
       (equivocator_IM IM) i) tr
Hm: output = Some m
Htri: finite_constrained_trace_from
  (equivocator_IM IM i) 
  (is i)
  (VLSM_projection_finite_trace_project
     (preloaded_component_projection
        (equivocator_IM IM) i) tr)
eqv_final, eqv_init: MachineDescriptor (IM i)
Heqv_init: proper_descriptor (IM i) eqv_init (is i)
Heqv_final: existing_descriptor (IM i) eqv_final
  (finite_trace_last is tr i)
trXi: list transition_item
Hprojecti: equivocator_vlsm_trace_project 
  (IM i)
  (VLSM_projection_finite_trace_project
     (preloaded_component_projection
        (equivocator_IM IM) i) tr)
  eqv_final = Some (trXi, eqv_init)
Hex: Exists (field_selector VLSM.output m) trXi
final_descriptors: ∀ j : index,
  MachineDescriptor (IM j)
Heqfinal_descriptors: final_descriptors =
equivocator_descriptors_update
  IM (zero_descriptor IM) i
  eqv_final
Hfinal_descriptors: not_equivocating_equivocator_descriptors
  IM final_descriptors
  (finite_trace_last is tr)
Hfinal_descriptors_proper: proper_equivocator_descriptors
  IM final_descriptors
  (finite_trace_last is tr)
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hproject_tr: equivocators_trace_project
  final_descriptors tr =
Some (trX, initial_descriptors)
Hcommute: equivocator_vlsm_trace_project 
  (IM i)
  (finite_trace_projection_list
     (equivocator_IM IM) i tr)
  (final_descriptors i) =
Some (trXi, eqv_init)
→ initial_descriptors i = eqv_init
  ∧ finite_trace_projection_list IM i trX =
    trXi
Exists (field_selector VLSM.output m) trX
  assert (Hfinali : final_descriptors i = eqv_final) by (subst; state_update_simpl; done).message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_constrained_trace_from
  (free_composite_vlsm (equivocator_IM IM)) is
  tr
m: message
i: index
li: EquivocatorLabel (IM i)
input: option message
destination: composite_state (equivocator_IM IM)
output: option message
Hitem: {|
  l := li;
  input := input;
  destination := destination i;
  output := output
|}
∈ VLSM_projection_finite_trace_project
    (preloaded_component_projection
       (equivocator_IM IM) i) tr
Hm: output = Some m
Htri: finite_constrained_trace_from
  (equivocator_IM IM i) 
  (is i)
  (VLSM_projection_finite_trace_project
     (preloaded_component_projection
        (equivocator_IM IM) i) tr)
eqv_final, eqv_init: MachineDescriptor (IM i)
Heqv_init: proper_descriptor (IM i) eqv_init (is i)
Heqv_final: existing_descriptor (IM i) eqv_final
  (finite_trace_last is tr i)
trXi: list transition_item
Hprojecti: equivocator_vlsm_trace_project 
  (IM i)
  (VLSM_projection_finite_trace_project
     (preloaded_component_projection
        (equivocator_IM IM) i) tr)
  eqv_final = Some (trXi, eqv_init)
Hex: Exists (field_selector VLSM.output m) trXi
final_descriptors: ∀ j : index,
  MachineDescriptor (IM j)
Heqfinal_descriptors: final_descriptors =
equivocator_descriptors_update
  IM (zero_descriptor IM) i
  eqv_final
Hfinal_descriptors: not_equivocating_equivocator_descriptors
  IM final_descriptors
  (finite_trace_last is tr)
Hfinal_descriptors_proper: proper_equivocator_descriptors
  IM final_descriptors
  (finite_trace_last is tr)
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hproject_tr: equivocators_trace_project
  final_descriptors tr =
Some (trX, initial_descriptors)
Hcommute: equivocator_vlsm_trace_project 
  (IM i)
  (finite_trace_projection_list
     (equivocator_IM IM) i tr)
  (final_descriptors i) =
Some (trXi, eqv_init)
→ initial_descriptors i = eqv_init
  ∧ finite_trace_projection_list IM i trX =
    trXi
Hfinali: final_descriptors i = eqv_final
Exists (field_selector VLSM.output m) trX
  rewrite Hfinali in Hcommute.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_constrained_trace_from
  (free_composite_vlsm (equivocator_IM IM)) is
  tr
m: message
i: index
li: EquivocatorLabel (IM i)
input: option message
destination: composite_state (equivocator_IM IM)
output: option message
Hitem: {|
  l := li;
  input := input;
  destination := destination i;
  output := output
|}
∈ VLSM_projection_finite_trace_project
    (preloaded_component_projection
       (equivocator_IM IM) i) tr
Hm: output = Some m
Htri: finite_constrained_trace_from
  (equivocator_IM IM i) 
  (is i)
  (VLSM_projection_finite_trace_project
     (preloaded_component_projection
        (equivocator_IM IM) i) tr)
eqv_final, eqv_init: MachineDescriptor (IM i)
Heqv_init: proper_descriptor (IM i) eqv_init (is i)
Heqv_final: existing_descriptor (IM i) eqv_final
  (finite_trace_last is tr i)
trXi: list transition_item
Hprojecti: equivocator_vlsm_trace_project 
  (IM i)
  (VLSM_projection_finite_trace_project
     (preloaded_component_projection
        (equivocator_IM IM) i) tr)
  eqv_final = Some (trXi, eqv_init)
Hex: Exists (field_selector VLSM.output m) trXi
final_descriptors: ∀ j : index,
  MachineDescriptor (IM j)
Heqfinal_descriptors: final_descriptors =
equivocator_descriptors_update
  IM (zero_descriptor IM) i
  eqv_final
Hfinal_descriptors: not_equivocating_equivocator_descriptors
  IM final_descriptors
  (finite_trace_last is tr)
Hfinal_descriptors_proper: proper_equivocator_descriptors
  IM final_descriptors
  (finite_trace_last is tr)
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hproject_tr: equivocators_trace_project
  final_descriptors tr =
Some (trX, initial_descriptors)
Hcommute: equivocator_vlsm_trace_project 
  (IM i)
  (finite_trace_projection_list
     (equivocator_IM IM) i tr) eqv_final =
Some (trXi, eqv_init)
→ initial_descriptors i = eqv_init
  ∧ finite_trace_projection_list IM i trX =
    trXi
Hfinali: final_descriptors i = eqv_final
Exists (field_selector VLSM.output m) trX
  specialize (Hcommute Hprojecti).message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_constrained_trace_from
  (free_composite_vlsm (equivocator_IM IM)) is
  tr
m: message
i: index
li: EquivocatorLabel (IM i)
input: option message
destination: composite_state (equivocator_IM IM)
output: option message
Hitem: {|
  l := li;
  input := input;
  destination := destination i;
  output := output
|}
∈ VLSM_projection_finite_trace_project
    (preloaded_component_projection
       (equivocator_IM IM) i) tr
Hm: output = Some m
Htri: finite_constrained_trace_from
  (equivocator_IM IM i) 
  (is i)
  (VLSM_projection_finite_trace_project
     (preloaded_component_projection
        (equivocator_IM IM) i) tr)
eqv_final, eqv_init: MachineDescriptor (IM i)
Heqv_init: proper_descriptor (IM i) eqv_init (is i)
Heqv_final: existing_descriptor (IM i) eqv_final
  (finite_trace_last is tr i)
trXi: list transition_item
Hprojecti: equivocator_vlsm_trace_project 
  (IM i)
  (VLSM_projection_finite_trace_project
     (preloaded_component_projection
        (equivocator_IM IM) i) tr)
  eqv_final = Some (trXi, eqv_init)
Hex: Exists (field_selector VLSM.output m) trXi
final_descriptors: ∀ j : index,
  MachineDescriptor (IM j)
Heqfinal_descriptors: final_descriptors =
equivocator_descriptors_update
  IM (zero_descriptor IM) i
  eqv_final
Hfinal_descriptors: not_equivocating_equivocator_descriptors
  IM final_descriptors
  (finite_trace_last is tr)
Hfinal_descriptors_proper: proper_equivocator_descriptors
  IM final_descriptors
  (finite_trace_last is tr)
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hproject_tr: equivocators_trace_project
  final_descriptors tr =
Some (trX, initial_descriptors)
Hcommute: initial_descriptors i = eqv_init
∧ finite_trace_projection_list IM i trX =
  trXi
Hfinali: final_descriptors i = eqv_final
Exists (field_selector VLSM.output m) trX
  destruct Hcommute as [Hiniti Hcommute].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_constrained_trace_from
  (free_composite_vlsm (equivocator_IM IM)) is
  tr
m: message
i: index
li: EquivocatorLabel (IM i)
input: option message
destination: composite_state (equivocator_IM IM)
output: option message
Hitem: {|
  l := li;
  input := input;
  destination := destination i;
  output := output
|}
∈ VLSM_projection_finite_trace_project
    (preloaded_component_projection
       (equivocator_IM IM) i) tr
Hm: output = Some m
Htri: finite_constrained_trace_from
  (equivocator_IM IM i) 
  (is i)
  (VLSM_projection_finite_trace_project
     (preloaded_component_projection
        (equivocator_IM IM) i) tr)
eqv_final, eqv_init: MachineDescriptor (IM i)
Heqv_init: proper_descriptor (IM i) eqv_init (is i)
Heqv_final: existing_descriptor (IM i) eqv_final
  (finite_trace_last is tr i)
trXi: list transition_item
Hprojecti: equivocator_vlsm_trace_project 
  (IM i)
  (VLSM_projection_finite_trace_project
     (preloaded_component_projection
        (equivocator_IM IM) i) tr)
  eqv_final = Some (trXi, eqv_init)
Hex: Exists (field_selector VLSM.output m) trXi
final_descriptors: ∀ j : index,
  MachineDescriptor (IM j)
Heqfinal_descriptors: final_descriptors =
equivocator_descriptors_update
  IM (zero_descriptor IM) i
  eqv_final
Hfinal_descriptors: not_equivocating_equivocator_descriptors
  IM final_descriptors
  (finite_trace_last is tr)
Hfinal_descriptors_proper: proper_equivocator_descriptors
  IM final_descriptors
  (finite_trace_last is tr)
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hproject_tr: equivocators_trace_project
  final_descriptors tr =
Some (trX, initial_descriptors)
Hiniti: initial_descriptors i = eqv_init
Hcommute: finite_trace_projection_list IM i trX =
trXi
Hfinali: final_descriptors i = eqv_final
Exists (field_selector VLSM.output m) trX
  clear -Hex Hcommute.message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
m: message
i: index
trXi: list transition_item
Hex: Exists (field_selector output m) trXi
trX: list (composite_transition_item IM)
Hcommute: finite_trace_projection_list IM i trX =
trXi
Exists (field_selector output m) trX subst.message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
m: message
i: index
trX: list (composite_transition_item IM)
Hex: Exists (field_selector output m)
  (finite_trace_projection_list IM i trX)
Exists (field_selector output m) trX
  apply Exists_exists in Hex.message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
m: message
i: index
trX: list (composite_transition_item IM)
Hex: ∃ x : transition_item,
  x ∈ finite_trace_projection_list IM i trX
  ∧ field_selector output m x
Exists (field_selector output m) trX destruct Hex as [x [Hx Hm]].message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
m: message
i: index
trX: list (composite_transition_item IM)
x: transition_item
Hx: x ∈ finite_trace_projection_list IM i trX
Hm: field_selector output m x
Exists (field_selector output m) trX
  apply (finite_trace_projection_list_in_rev IM) in Hx.message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
m: message
i: index
trX: list (composite_transition_item IM)
x: transition_item
Hx: ∃ itemX : composite_transition_item IM,
  itemX ∈ trX
  ∧ output itemX = output x
    ∧ input itemX = input x
      ∧ destination itemX i = destination x
        ∧ (∃ Hl1 : i = projT1 (l itemX),
             eq_rect_r
               (λ n : index, label (IM n))
               (projT2 (l itemX)) Hl1 = 
             l x)
Hm: field_selector output m x
Exists (field_selector output m) trX
  destruct Hx as [itemX [HitemX [Houtput _]]].message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
m: message
i: index
trX: list (composite_transition_item IM)
x: transition_item
itemX: composite_transition_item IM
HitemX: itemX ∈ trX
Houtput: output itemX = output x
Hm: field_selector output m x
Exists (field_selector output m) trX
  apply Exists_exists.message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
m: message
i: index
trX: list (composite_transition_item IM)
x: transition_item
itemX: composite_transition_item IM
HitemX: itemX ∈ trX
Houtput: output itemX = output x
Hm: field_selector output m x
∃ x : transition_item,
  x ∈ trX ∧ field_selector output m x exists itemX.message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
m: message
i: index
trX: list (composite_transition_item IM)
x: transition_item
itemX: composite_transition_item IM
HitemX: itemX ∈ trX
Houtput: output itemX = output x
Hm: field_selector output m x
itemX ∈ trX ∧ field_selector output m itemX split; [done |].message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
m: message
i: index
trX: list (composite_transition_item IM)
x: transition_item
itemX: composite_transition_item IM
HitemX: itemX ∈ trX
Houtput: output itemX = output x
Hm: field_selector output m x
field_selector output m itemX
  by simpl; rewrite Houtput.
Qed.

Lemma equivocators_trace_project_output_reflecting_iff
  (is : composite_state (equivocator_IM IM))
  (tr : list (composite_transition_item (equivocator_IM IM)))
  (Htr : finite_constrained_trace_from (free_composite_vlsm (equivocator_IM IM)) is tr)
  (m : message)
  : Exists (field_selector output m) tr
  <-> exists
    (final_descriptors initial_descriptors : equivocator_descriptors IM)
    (trX : list (composite_transition_item IM)),
    not_equivocating_equivocator_descriptors IM final_descriptors (finite_trace_last is tr) /\
    equivocators_trace_project final_descriptors tr = Some (trX, initial_descriptors) /\
    Exists (field_selector output m) trX.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_constrained_trace_from
  (free_composite_vlsm (equivocator_IM IM)) is
  tr
m: message
Exists (field_selector output m) tr
↔ (∃ (final_descriptors
      initial_descriptors : equivocator_descriptors IM) 
     (trX : list (composite_transition_item IM)),
     not_equivocating_equivocator_descriptors IM
       final_descriptors (finite_trace_last is tr)
     ∧ equivocators_trace_project final_descriptors tr =
       Some (trX, initial_descriptors)
       ∧ Exists (field_selector output m) trX)
Proof.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_constrained_trace_from
  (free_composite_vlsm (equivocator_IM IM)) is
  tr
m: message
Exists (field_selector output m) tr
↔ (∃ (final_descriptors
      initial_descriptors : equivocator_descriptors IM) 
     (trX : list (composite_transition_item IM)),
     not_equivocating_equivocator_descriptors IM
       final_descriptors (finite_trace_last is tr)
     ∧ equivocators_trace_project final_descriptors tr =
       Some (trX, initial_descriptors)
       ∧ Exists (field_selector output m) trX)
  split; [by apply equivocators_trace_project_output_reflecting_inv |].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_constrained_trace_from
  (free_composite_vlsm (equivocator_IM IM)) is
  tr
m: message
(∃ (final_descriptors
    initial_descriptors : equivocator_descriptors IM) 
   (trX : list (composite_transition_item IM)),
   not_equivocating_equivocator_descriptors IM
     final_descriptors (finite_trace_last is tr)
   ∧ equivocators_trace_project final_descriptors tr =
     Some (trX, initial_descriptors)
     ∧ Exists (field_selector output m) trX)
→ Exists (field_selector output m) tr
  intros [final_descriptors [initial_descriptors [trX [Hfinal_descriptors [Hpr_tr Hex]]]]].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_constrained_trace_from
  (free_composite_vlsm (equivocator_IM IM)) is
  tr
m: message
final_descriptors, initial_descriptors: equivocator_descriptors IM
trX: list (composite_transition_item IM)
Hfinal_descriptors: not_equivocating_equivocator_descriptors
  IM final_descriptors
  (finite_trace_last is tr)
Hpr_tr: equivocators_trace_project final_descriptors
  tr = Some (trX, initial_descriptors)
Hex: Exists (field_selector output m) trX
Exists (field_selector output m) tr
  apply Exists_exists in Hex.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_constrained_trace_from
  (free_composite_vlsm (equivocator_IM IM)) is
  tr
m: message
final_descriptors, initial_descriptors: equivocator_descriptors IM
trX: list (composite_transition_item IM)
Hfinal_descriptors: not_equivocating_equivocator_descriptors
  IM final_descriptors
  (finite_trace_last is tr)
Hpr_tr: equivocators_trace_project final_descriptors
  tr = Some (trX, initial_descriptors)
Hex: ∃ x : transition_item,
  x ∈ trX ∧ field_selector output m x
Exists (field_selector output m) tr
  destruct Hex as [itemX [HitemX Hm]].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_constrained_trace_from
  (free_composite_vlsm (equivocator_IM IM)) is
  tr
m: message
final_descriptors, initial_descriptors: equivocator_descriptors IM
trX: list (composite_transition_item IM)
Hfinal_descriptors: not_equivocating_equivocator_descriptors
  IM final_descriptors
  (finite_trace_last is tr)
Hpr_tr: equivocators_trace_project final_descriptors
  tr = Some (trX, initial_descriptors)
itemX: transition_item
HitemX: itemX ∈ trX
Hm: field_selector output m itemX
Exists (field_selector output m) tr
  apply elem_of_list_split in HitemX.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_constrained_trace_from
  (free_composite_vlsm (equivocator_IM IM)) is
  tr
m: message
final_descriptors, initial_descriptors: equivocator_descriptors IM
trX: list (composite_transition_item IM)
Hfinal_descriptors: not_equivocating_equivocator_descriptors
  IM final_descriptors
  (finite_trace_last is tr)
Hpr_tr: equivocators_trace_project final_descriptors
  tr = Some (trX, initial_descriptors)
itemX: transition_item
HitemX: ∃ l1 l2 : list transition_item,
  trX = l1 ++ itemX :: l2
Hm: field_selector output m itemX
Exists (field_selector output m) tr
  destruct HitemX as [preX [sufX Heq_trX]].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_constrained_trace_from
  (free_composite_vlsm (equivocator_IM IM)) is
  tr
m: message
final_descriptors, initial_descriptors: equivocator_descriptors IM
trX: list (composite_transition_item IM)
Hfinal_descriptors: not_equivocating_equivocator_descriptors
  IM final_descriptors
  (finite_trace_last is tr)
Hpr_tr: equivocators_trace_project final_descriptors
  tr = Some (trX, initial_descriptors)
itemX: transition_item
preX, sufX: list transition_item
Heq_trX: trX = preX ++ itemX :: sufX
Hm: field_selector output m itemX
Exists (field_selector output m) tr
  subst.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_constrained_trace_from
  (free_composite_vlsm (equivocator_IM IM)) is
  tr
m: message
final_descriptors, initial_descriptors: equivocator_descriptors IM
Hfinal_descriptors: not_equivocating_equivocator_descriptors
  IM final_descriptors
  (finite_trace_last is tr)
itemX: transition_item
preX, sufX: list transition_item
Hpr_tr: equivocators_trace_project final_descriptors
  tr =
Some
  (preX ++ itemX :: sufX, initial_descriptors)
Hm: field_selector output m itemX
Exists (field_selector output m) tr
  apply equivocators_trace_project_app_inv_item in Hpr_tr.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_constrained_trace_from
  (free_composite_vlsm (equivocator_IM IM)) is
  tr
m: message
final_descriptors, initial_descriptors: equivocator_descriptors IM
Hfinal_descriptors: not_equivocating_equivocator_descriptors
  IM final_descriptors
  (finite_trace_last is tr)
itemX: transition_item
preX, sufX: list transition_item
Hpr_tr: ∃ (pre
   suf : list
           (composite_transition_item
              (equivocator_IM IM))) 
  (item : composite_transition_item
            (equivocator_IM IM)) 
  (item_descriptors
   pre_descriptors : equivocator_descriptors
                      IM),
  equivocators_trace_project
    final_descriptors suf =
  Some (sufX, item_descriptors)
  ∧ equivocators_transition_item_project
      item_descriptors item =
    Some (Some itemX, pre_descriptors)
    ∧ equivocators_trace_project
        pre_descriptors pre =
      Some (preX, initial_descriptors)
      ∧ tr = pre ++ [item] ++ suf
Hm: field_selector output m itemX
Exists (field_selector output m) tr
  destruct Hpr_tr as [pre [suf [item [item_descriptors [pre_descriptors [_ [Hpr_item [_ Heqtr]]]]]]]].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_constrained_trace_from
  (free_composite_vlsm (equivocator_IM IM)) is
  tr
m: message
final_descriptors, initial_descriptors: equivocator_descriptors IM
Hfinal_descriptors: not_equivocating_equivocator_descriptors
  IM final_descriptors
  (finite_trace_last is tr)
itemX: transition_item
preX, sufX: list transition_item
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
item: composite_transition_item (equivocator_IM IM)
item_descriptors, pre_descriptors: equivocator_descriptors IM
Hpr_item: equivocators_transition_item_project
  item_descriptors item =
Some (Some itemX, pre_descriptors)
Heqtr: tr = pre ++ [item] ++ suf
Hm: field_selector output m itemX
Exists (field_selector output m) tr
  subst.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
item: composite_transition_item (equivocator_IM IM)
Htr: finite_constrained_trace_from
  (free_composite_vlsm (equivocator_IM IM)) is
  (pre ++ [item] ++ suf)
m: message
final_descriptors, initial_descriptors: equivocator_descriptors IM
Hfinal_descriptors: not_equivocating_equivocator_descriptors
  IM final_descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf))
itemX: transition_item
preX, sufX: list transition_item
item_descriptors, pre_descriptors: equivocator_descriptors IM
Hpr_item: equivocators_transition_item_project
  item_descriptors item =
Some (Some itemX, pre_descriptors)
Hm: field_selector output m itemX
Exists (field_selector output m)
  (pre ++ [item] ++ suf)
  rewrite !Exists_app.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
item: composite_transition_item (equivocator_IM IM)
Htr: finite_constrained_trace_from
  (free_composite_vlsm (equivocator_IM IM)) is
  (pre ++ [item] ++ suf)
m: message
final_descriptors, initial_descriptors: equivocator_descriptors IM
Hfinal_descriptors: not_equivocating_equivocator_descriptors
  IM final_descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf))
itemX: transition_item
preX, sufX: list transition_item
item_descriptors, pre_descriptors: equivocator_descriptors IM
Hpr_item: equivocators_transition_item_project
  item_descriptors item =
Some (Some itemX, pre_descriptors)
Hm: field_selector output m itemX
Exists (field_selector output m) pre
∨ Exists (field_selector output m) [item]
  ∨ Exists (field_selector output m) suf right.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
item: composite_transition_item (equivocator_IM IM)
Htr: finite_constrained_trace_from
  (free_composite_vlsm (equivocator_IM IM)) is
  (pre ++ [item] ++ suf)
m: message
final_descriptors, initial_descriptors: equivocator_descriptors IM
Hfinal_descriptors: not_equivocating_equivocator_descriptors
  IM final_descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf))
itemX: transition_item
preX, sufX: list transition_item
item_descriptors, pre_descriptors: equivocator_descriptors IM
Hpr_item: equivocators_transition_item_project
  item_descriptors item =
Some (Some itemX, pre_descriptors)
Hm: field_selector output m itemX
Exists (field_selector output m) [item]
∨ Exists (field_selector output m) suf left.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
item: composite_transition_item (equivocator_IM IM)
Htr: finite_constrained_trace_from
  (free_composite_vlsm (equivocator_IM IM)) is
  (pre ++ [item] ++ suf)
m: message
final_descriptors, initial_descriptors: equivocator_descriptors IM
Hfinal_descriptors: not_equivocating_equivocator_descriptors
  IM final_descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf))
itemX: transition_item
preX, sufX: list transition_item
item_descriptors, pre_descriptors: equivocator_descriptors IM
Hpr_item: equivocators_transition_item_project
  item_descriptors item =
Some (Some itemX, pre_descriptors)
Hm: field_selector output m itemX
Exists (field_selector output m) [item] constructor.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
item: composite_transition_item (equivocator_IM IM)
Htr: finite_constrained_trace_from
  (free_composite_vlsm (equivocator_IM IM)) is
  (pre ++ [item] ++ suf)
m: message
final_descriptors, initial_descriptors: equivocator_descriptors IM
Hfinal_descriptors: not_equivocating_equivocator_descriptors
  IM final_descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf))
itemX: transition_item
preX, sufX: list transition_item
item_descriptors, pre_descriptors: equivocator_descriptors IM
Hpr_item: equivocators_transition_item_project
  item_descriptors item =
Some (Some itemX, pre_descriptors)
Hm: field_selector output m itemX
field_selector output m item
  apply equivocators_transition_item_project_inv_characterization in Hpr_item.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
item: composite_transition_item (equivocator_IM IM)
Htr: finite_constrained_trace_from
  (free_composite_vlsm (equivocator_IM IM)) is
  (pre ++ [item] ++ suf)
m: message
final_descriptors, initial_descriptors: equivocator_descriptors IM
Hfinal_descriptors: not_equivocating_equivocator_descriptors
  IM final_descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf))
itemX: transition_item
preX, sufX: list transition_item
item_descriptors, pre_descriptors: equivocator_descriptors IM
Hpr_item: (∃ Hex : existing_equivocator_label
           (IM (projT1 (l item)))
           (projT2 (l item)),
   existT (projT1 (l item))
     (existing_equivocator_label_extract
        (IM (projT1 (l item)))
        (projT2 (l item)) Hex) = 
   l itemX)
∧ input item = input itemX
  ∧ output item = output itemX
    ∧ equivocators_state_project IM
        item_descriptors 
        (destination item) =
      destination itemX
Hm: field_selector output m itemX
field_selector output m item
  destruct Hpr_item as [_ [_ [Heqoutput _]]].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
item: composite_transition_item (equivocator_IM IM)
Htr: finite_constrained_trace_from
  (free_composite_vlsm (equivocator_IM IM)) is
  (pre ++ [item] ++ suf)
m: message
final_descriptors, initial_descriptors: equivocator_descriptors IM
Hfinal_descriptors: not_equivocating_equivocator_descriptors
  IM final_descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf))
itemX: transition_item
preX, sufX: list transition_item
item_descriptors, pre_descriptors: equivocator_descriptors IM
Heqoutput: output item = output itemX
Hm: field_selector output m itemX
field_selector output m item
  by simpl in *; congruence.
Qed.

Projecting a preloaded valid trace of the composition of equivocators using proper_equivocator_descriptors one obtains a preloaded valid trace of a free composition.

Lemma pre_equivocators_valid_trace_project
  (is final_state : state (equivocators_no_equivocations_vlsm IM))
  (tr : list (composite_transition_item (equivocator_IM IM)))
  (Htr : finite_constrained_trace_init_to FreeE is final_state tr)
  (final_descriptors : equivocator_descriptors IM)
  (Hproper : proper_equivocator_descriptors IM final_descriptors final_state)
  : exists
    (initial_descriptors : equivocator_descriptors IM),
    proper_equivocator_descriptors IM initial_descriptors is /\
    exists
    (isX := equivocators_state_project IM initial_descriptors is)
    (final_stateX := equivocators_state_project IM final_descriptors final_state)
    (trX : list (composite_transition_item IM)),
    equivocators_trace_project final_descriptors tr = Some (trX, initial_descriptors) /\
    finite_constrained_trace_init_to Free isX final_stateX trX.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is, final_state: state
  (equivocators_no_equivocations_vlsm IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_constrained_trace_init_to FreeE is
  final_state tr
final_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  final_descriptors final_state
∃ initial_descriptors : equivocator_descriptors IM,
  proper_equivocator_descriptors IM
    initial_descriptors is
  ∧ (let isX :=
       equivocators_state_project IM
         initial_descriptors is in
     let final_stateX :=
       equivocators_state_project IM final_descriptors
         final_state in
     ∃ trX : list (composite_transition_item IM),
       equivocators_trace_project final_descriptors tr =
       Some (trX, initial_descriptors)
       ∧ finite_constrained_trace_init_to Free isX
           final_stateX trX)
Proof.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is, final_state: state
  (equivocators_no_equivocations_vlsm IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_constrained_trace_init_to FreeE is
  final_state tr
final_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  final_descriptors final_state
∃ initial_descriptors : equivocator_descriptors IM,
  proper_equivocator_descriptors IM
    initial_descriptors is
  ∧ (let isX :=
       equivocators_state_project IM
         initial_descriptors is in
     let final_stateX :=
       equivocators_state_project IM final_descriptors
         final_state in
     ∃ trX : list (composite_transition_item IM),
       equivocators_trace_project final_descriptors tr =
       Some (trX, initial_descriptors)
       ∧ finite_constrained_trace_init_to Free isX
           final_stateX trX)
  generalize dependent final_descriptors.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is, final_state: state
  (equivocators_no_equivocations_vlsm IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_constrained_trace_init_to FreeE is
  final_state tr
∀ final_descriptors : equivocator_descriptors IM,
  proper_equivocator_descriptors IM final_descriptors
    final_state
  → ∃ initial_descriptors : equivocator_descriptors IM,
      proper_equivocator_descriptors IM
        initial_descriptors is
      ∧ (let isX :=
           equivocators_state_project IM
             initial_descriptors is in
         let final_stateX :=
           equivocators_state_project IM
             final_descriptors final_state in
         ∃ trX : list (composite_transition_item IM),
           equivocators_trace_project
             final_descriptors tr =
           Some (trX, initial_descriptors)
           ∧ finite_constrained_trace_init_to Free isX
               final_stateX trX)
  generalize dependent final_state.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: state (equivocators_no_equivocations_vlsm IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
∀ final_state : state
                  (equivocators_no_equivocations_vlsm
                     IM),
  finite_constrained_trace_init_to FreeE is
    final_state tr
  → ∀ final_descriptors : equivocator_descriptors IM,
      proper_equivocator_descriptors IM
        final_descriptors final_state
      → ∃ initial_descriptors : equivocator_descriptors
                                  IM,
          proper_equivocator_descriptors IM
            initial_descriptors is
          ∧ (let isX :=
               equivocators_state_project IM
                 initial_descriptors is in
             let final_stateX :=
               equivocators_state_project IM
                 final_descriptors final_state in
             ∃ trX : list
                       (composite_transition_item IM),
               equivocators_trace_project
                 final_descriptors tr =
               Some (trX, initial_descriptors)
               ∧ finite_constrained_trace_init_to Free
                   isX final_stateX trX)
  induction tr using rev_ind; intros.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is, final_state: state
  (equivocators_no_equivocations_vlsm IM)
Htr: finite_constrained_trace_init_to FreeE is
  final_state []
final_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  final_descriptors final_state
∃ initial_descriptors : equivocator_descriptors IM,
  proper_equivocator_descriptors IM
    initial_descriptors is
  ∧ (let isX :=
       equivocators_state_project IM
         initial_descriptors is in
     let final_stateX :=
       equivocators_state_project IM final_descriptors
         final_state in
     ∃ trX : list (composite_transition_item IM),
       equivocators_trace_project final_descriptors [] =
       Some (trX, initial_descriptors)
       ∧ finite_constrained_trace_init_to Free isX
           final_stateX trX)
message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: state (equivocators_no_equivocations_vlsm IM)
x: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
IHtr: ∀ final_state : state
                  (equivocators_no_equivocations_vlsm
                     IM),
  finite_constrained_trace_init_to FreeE is
    final_state tr
  → ∀ final_descriptors : 
       equivocator_descriptors IM,
      proper_equivocator_descriptors IM
        final_descriptors final_state
      → ∃ initial_descriptors : 
           equivocator_descriptors IM,
          proper_equivocator_descriptors IM
            initial_descriptors is
          ∧ (let isX :=
               equivocators_state_project IM
                 initial_descriptors is in
             let final_stateX :=
               equivocators_state_project IM
                 final_descriptors final_state
               in
             ∃ trX : list
                      (composite_transition_item
                      IM),
               equivocators_trace_project
                 final_descriptors tr =
               Some (trX, initial_descriptors)
               ∧ finite_constrained_trace_init_to
                   Free isX final_stateX trX)
final_state: state
  (equivocators_no_equivocations_vlsm IM)
Htr: finite_constrained_trace_init_to FreeE is
  final_state (tr ++ [x])
final_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  final_descriptors final_state
∃ initial_descriptors : equivocator_descriptors IM,
  proper_equivocator_descriptors IM
    initial_descriptors is
  ∧ (let isX :=
       equivocators_state_project IM
         initial_descriptors is in
     let final_stateX :=
       equivocators_state_project IM final_descriptors
         final_state in
     ∃ trX : list (composite_transition_item IM),
       equivocators_trace_project final_descriptors
         (tr ++ [x]) = 
       Some (trX, initial_descriptors)
       ∧ finite_constrained_trace_init_to Free isX
           final_stateX trX)
  -message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is, final_state: state
  (equivocators_no_equivocations_vlsm IM)
Htr: finite_constrained_trace_init_to FreeE is
  final_state []
final_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  final_descriptors final_state
∃ initial_descriptors : equivocator_descriptors IM,
  proper_equivocator_descriptors IM
    initial_descriptors is
  ∧ (let isX :=
       equivocators_state_project IM
         initial_descriptors is in
     let final_stateX :=
       equivocators_state_project IM final_descriptors
         final_state in
     ∃ trX : list (composite_transition_item IM),
       equivocators_trace_project final_descriptors [] =
       Some (trX, initial_descriptors)
       ∧ finite_constrained_trace_init_to Free isX
           final_stateX trX) red in Htr; apply valid_trace_get_last in Htr as Hfinal_state_eq.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is, final_state: state
  (equivocators_no_equivocations_vlsm IM)
Htr: finite_valid_trace_init_to
  (preloaded_with_all_messages_vlsm FreeE) is
  final_state []
final_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  final_descriptors final_state
Hfinal_state_eq: finite_trace_last is [] =
final_state
∃ initial_descriptors : equivocator_descriptors IM,
  proper_equivocator_descriptors IM
    initial_descriptors is
  ∧ (let isX :=
       equivocators_state_project IM
         initial_descriptors is in
     let final_stateX :=
       equivocators_state_project IM final_descriptors
         final_state in
     ∃ trX : list (composite_transition_item IM),
       equivocators_trace_project final_descriptors [] =
       Some (trX, initial_descriptors)
       ∧ finite_constrained_trace_init_to Free isX
           final_stateX trX)
    subst.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: state (equivocators_no_equivocations_vlsm IM)
Htr: finite_valid_trace_init_to
  (preloaded_with_all_messages_vlsm FreeE) is
  (finite_trace_last is []) []
final_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  final_descriptors
  (finite_trace_last is [])
∃ initial_descriptors : equivocator_descriptors IM,
  proper_equivocator_descriptors IM
    initial_descriptors is
  ∧ (let isX :=
       equivocators_state_project IM
         initial_descriptors is in
     let final_stateX :=
       equivocators_state_project IM final_descriptors
         (finite_trace_last is []) in
     ∃ trX : list (composite_transition_item IM),
       equivocators_trace_project final_descriptors [] =
       Some (trX, initial_descriptors)
       ∧ finite_constrained_trace_init_to Free isX
           final_stateX trX)
    exists final_descriptors.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: state (equivocators_no_equivocations_vlsm IM)
Htr: finite_valid_trace_init_to
  (preloaded_with_all_messages_vlsm FreeE) is
  (finite_trace_last is []) []
final_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  final_descriptors
  (finite_trace_last is [])
proper_equivocator_descriptors IM final_descriptors is
∧ (let isX :=
     equivocators_state_project IM final_descriptors
       is in
   let final_stateX :=
     equivocators_state_project IM final_descriptors
       (finite_trace_last is []) in
   ∃ trX : list (composite_transition_item IM),
     equivocators_trace_project final_descriptors [] =
     Some (trX, final_descriptors)
     ∧ finite_constrained_trace_init_to Free isX
         final_stateX trX) split; [done |].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: state (equivocators_no_equivocations_vlsm IM)
Htr: finite_valid_trace_init_to
  (preloaded_with_all_messages_vlsm FreeE) is
  (finite_trace_last is []) []
final_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  final_descriptors
  (finite_trace_last is [])
let isX :=
  equivocators_state_project IM final_descriptors is
  in
let final_stateX :=
  equivocators_state_project IM final_descriptors
    (finite_trace_last is []) in
∃ trX : list (composite_transition_item IM),
  equivocators_trace_project final_descriptors [] =
  Some (trX, final_descriptors)
  ∧ finite_constrained_trace_init_to Free isX
      final_stateX trX
    exists [].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: state (equivocators_no_equivocations_vlsm IM)
Htr: finite_valid_trace_init_to
  (preloaded_with_all_messages_vlsm FreeE) is
  (finite_trace_last is []) []
final_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  final_descriptors
  (finite_trace_last is [])
equivocators_trace_project final_descriptors [] =
Some ([], final_descriptors)
∧ finite_constrained_trace_init_to Free
    (equivocators_state_project IM final_descriptors
       is)
    (equivocators_state_project IM final_descriptors
       (finite_trace_last is [])) []
    repeat (split; [done |]).message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: state (equivocators_no_equivocations_vlsm IM)
Htr: finite_valid_trace_init_to
  (preloaded_with_all_messages_vlsm FreeE) is
  (finite_trace_last is []) []
final_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  final_descriptors
  (finite_trace_last is [])
finite_constrained_trace_init_to Free
  (equivocators_state_project IM final_descriptors is)
  (equivocators_state_project IM final_descriptors
     (finite_trace_last is [])) []
    cut (initial_state_prop (free_composite_vlsm IM)
      (equivocators_state_project IM final_descriptors is)).message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: state (equivocators_no_equivocations_vlsm IM)
Htr: finite_valid_trace_init_to
  (preloaded_with_all_messages_vlsm FreeE) is
  (finite_trace_last is []) []
final_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  final_descriptors
  (finite_trace_last is [])
initial_state_prop
  (equivocators_state_project IM final_descriptors is)
→ finite_constrained_trace_init_to Free
    (equivocators_state_project IM final_descriptors
       is)
    (equivocators_state_project IM final_descriptors
       (finite_trace_last is [])) []
message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: state (equivocators_no_equivocations_vlsm IM)
Htr: finite_valid_trace_init_to
  (preloaded_with_all_messages_vlsm FreeE) is
  (finite_trace_last is []) []
final_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  final_descriptors
  (finite_trace_last is [])
initial_state_prop
  (equivocators_state_project IM final_descriptors is)
    {message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: state (equivocators_no_equivocations_vlsm IM)
Htr: finite_valid_trace_init_to
  (preloaded_with_all_messages_vlsm FreeE) is
  (finite_trace_last is []) []
final_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  final_descriptors
  (finite_trace_last is [])
initial_state_prop
  (equivocators_state_project IM final_descriptors is)
→ finite_constrained_trace_init_to Free
    (equivocators_state_project IM final_descriptors
       is)
    (equivocators_state_project IM final_descriptors
       (finite_trace_last is [])) []
      intro Hinit; split; [| done].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: state (equivocators_no_equivocations_vlsm IM)
Htr: finite_valid_trace_init_to
  (preloaded_with_all_messages_vlsm FreeE) is
  (finite_trace_last is []) []
final_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  final_descriptors
  (finite_trace_last is [])
Hinit: initial_state_prop
  (equivocators_state_project IM
     final_descriptors is)
finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM final_descriptors is)
  (equivocators_state_project IM final_descriptors
     (finite_trace_last is [])) []
      by constructor; apply initial_state_is_valid.
    }message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: state (equivocators_no_equivocations_vlsm IM)
Htr: finite_valid_trace_init_to
  (preloaded_with_all_messages_vlsm FreeE) is
  (finite_trace_last is []) []
final_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  final_descriptors
  (finite_trace_last is [])
initial_state_prop
  (equivocators_state_project IM final_descriptors is)
    by apply (equivocators_initial_state_project IM); [apply Htr |].
  -message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: state (equivocators_no_equivocations_vlsm IM)
x: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
IHtr: ∀ final_state : state
                  (equivocators_no_equivocations_vlsm
                     IM),
  finite_constrained_trace_init_to FreeE is
    final_state tr
  → ∀ final_descriptors : 
       equivocator_descriptors IM,
      proper_equivocator_descriptors IM
        final_descriptors final_state
      → ∃ initial_descriptors : 
           equivocator_descriptors IM,
          proper_equivocator_descriptors IM
            initial_descriptors is
          ∧ (let isX :=
               equivocators_state_project IM
                 initial_descriptors is in
             let final_stateX :=
               equivocators_state_project IM
                 final_descriptors final_state
               in
             ∃ trX : list
                      (composite_transition_item
                      IM),
               equivocators_trace_project
                 final_descriptors tr =
               Some (trX, initial_descriptors)
               ∧ finite_constrained_trace_init_to
                   Free isX final_stateX trX)
final_state: state
  (equivocators_no_equivocations_vlsm IM)
Htr: finite_constrained_trace_init_to FreeE is
  final_state (tr ++ [x])
final_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  final_descriptors final_state
∃ initial_descriptors : equivocator_descriptors IM,
  proper_equivocator_descriptors IM
    initial_descriptors is
  ∧ (let isX :=
       equivocators_state_project IM
         initial_descriptors is in
     let final_stateX :=
       equivocators_state_project IM final_descriptors
         final_state in
     ∃ trX : list (composite_transition_item IM),
       equivocators_trace_project final_descriptors
         (tr ++ [x]) = Some (trX, initial_descriptors)
       ∧ finite_constrained_trace_init_to Free isX
           final_stateX trX) destruct Htr as [Htr Hinit].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: state (equivocators_no_equivocations_vlsm IM)
x: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
IHtr: ∀ final_state : state
                  (equivocators_no_equivocations_vlsm
                     IM),
  finite_constrained_trace_init_to FreeE is
    final_state tr
  → ∀ final_descriptors : 
       equivocator_descriptors IM,
      proper_equivocator_descriptors IM
        final_descriptors final_state
      → ∃ initial_descriptors : 
           equivocator_descriptors IM,
          proper_equivocator_descriptors IM
            initial_descriptors is
          ∧ (let isX :=
               equivocators_state_project IM
                 initial_descriptors is in
             let final_stateX :=
               equivocators_state_project IM
                 final_descriptors final_state
               in
             ∃ trX : list
                      (composite_transition_item
                      IM),
               equivocators_trace_project
                 final_descriptors tr =
               Some (trX, initial_descriptors)
               ∧ finite_constrained_trace_init_to
                   Free isX final_stateX trX)
final_state: state
  (equivocators_no_equivocations_vlsm IM)
Htr: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm FreeE) is
  final_state (tr ++ [x])
Hinit: initial_state_prop is
final_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  final_descriptors final_state
∃ initial_descriptors : equivocator_descriptors IM,
  proper_equivocator_descriptors IM
    initial_descriptors is
  ∧ (let isX :=
       equivocators_state_project IM
         initial_descriptors is in
     let final_stateX :=
       equivocators_state_project IM final_descriptors
         final_state in
     ∃ trX : list (composite_transition_item IM),
       equivocators_trace_project final_descriptors
         (tr ++ [x]) = Some (trX, initial_descriptors)
       ∧ finite_constrained_trace_init_to Free isX
           final_stateX trX)
    apply finite_valid_trace_from_to_app_split in Htr.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: state (equivocators_no_equivocations_vlsm IM)
x: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
IHtr: ∀ final_state : state
                  (equivocators_no_equivocations_vlsm
                     IM),
  finite_constrained_trace_init_to FreeE is
    final_state tr
  → ∀ final_descriptors : 
       equivocator_descriptors IM,
      proper_equivocator_descriptors IM
        final_descriptors final_state
      → ∃ initial_descriptors : 
           equivocator_descriptors IM,
          proper_equivocator_descriptors IM
            initial_descriptors is
          ∧ (let isX :=
               equivocators_state_project IM
                 initial_descriptors is in
             let final_stateX :=
               equivocators_state_project IM
                 final_descriptors final_state
               in
             ∃ trX : list
                      (composite_transition_item
                      IM),
               equivocators_trace_project
                 final_descriptors tr =
               Some (trX, initial_descriptors)
               ∧ finite_constrained_trace_init_to
                   Free isX final_stateX trX)
final_state: state
  (equivocators_no_equivocations_vlsm IM)
Htr: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm FreeE) is
  (finite_trace_last is tr) tr
∧ finite_valid_trace_from_to
    (preloaded_with_all_messages_vlsm FreeE)
    (finite_trace_last is tr) final_state [x]
Hinit: initial_state_prop is
final_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  final_descriptors final_state
∃ initial_descriptors : equivocator_descriptors IM,
  proper_equivocator_descriptors IM
    initial_descriptors is
  ∧ (let isX :=
       equivocators_state_project IM
         initial_descriptors is in
     let final_stateX :=
       equivocators_state_project IM final_descriptors
         final_state in
     ∃ trX : list (composite_transition_item IM),
       equivocators_trace_project final_descriptors
         (tr ++ [x]) = Some (trX, initial_descriptors)
       ∧ finite_constrained_trace_init_to Free isX
           final_stateX trX)
    destruct Htr as [Htr Hx].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: state (equivocators_no_equivocations_vlsm IM)
x: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
IHtr: ∀ final_state : state
                  (equivocators_no_equivocations_vlsm
                     IM),
  finite_constrained_trace_init_to FreeE is
    final_state tr
  → ∀ final_descriptors : 
       equivocator_descriptors IM,
      proper_equivocator_descriptors IM
        final_descriptors final_state
      → ∃ initial_descriptors : 
           equivocator_descriptors IM,
          proper_equivocator_descriptors IM
            initial_descriptors is
          ∧ (let isX :=
               equivocators_state_project IM
                 initial_descriptors is in
             let final_stateX :=
               equivocators_state_project IM
                 final_descriptors final_state
               in
             ∃ trX : list
                      (composite_transition_item
                      IM),
               equivocators_trace_project
                 final_descriptors tr =
               Some (trX, initial_descriptors)
               ∧ finite_constrained_trace_init_to
                   Free isX final_stateX trX)
final_state: state
  (equivocators_no_equivocations_vlsm IM)
Htr: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm FreeE) is
  (finite_trace_last is tr) tr
Hx: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm FreeE)
  (finite_trace_last is tr) final_state [x]
Hinit: initial_state_prop is
final_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  final_descriptors final_state
∃ initial_descriptors : equivocator_descriptors IM,
  proper_equivocator_descriptors IM
    initial_descriptors is
  ∧ (let isX :=
       equivocators_state_project IM
         initial_descriptors is in
     let final_stateX :=
       equivocators_state_project IM final_descriptors
         final_state in
     ∃ trX : list (composite_transition_item IM),
       equivocators_trace_project final_descriptors
         (tr ++ [x]) = Some (trX, initial_descriptors)
       ∧ finite_constrained_trace_init_to Free isX
           final_stateX trX)
    specialize (IHtr _ (conj Htr Hinit)).message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: state (equivocators_no_equivocations_vlsm IM)
x: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
IHtr: ∀ final_descriptors : equivocator_descriptors
                      IM,
  proper_equivocator_descriptors IM
    final_descriptors (finite_trace_last is tr)
  → ∃ initial_descriptors : 
       equivocator_descriptors IM,
      proper_equivocator_descriptors IM
        initial_descriptors is
      ∧ (let isX :=
           equivocators_state_project IM
             initial_descriptors is in
         let final_stateX :=
           equivocators_state_project IM
             final_descriptors
             (finite_trace_last is tr) in
         ∃ trX : list
                   (composite_transition_item
                      IM),
           equivocators_trace_project
             final_descriptors tr =
           Some (trX, initial_descriptors)
           ∧ finite_constrained_trace_init_to
               Free isX final_stateX trX)
final_state: state
  (equivocators_no_equivocations_vlsm IM)
Htr: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm FreeE) is
  (finite_trace_last is tr) tr
Hx: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm FreeE)
  (finite_trace_last is tr) final_state [x]
Hinit: initial_state_prop is
final_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  final_descriptors final_state
∃ initial_descriptors : equivocator_descriptors IM,
  proper_equivocator_descriptors IM
    initial_descriptors is
  ∧ (let isX :=
       equivocators_state_project IM
         initial_descriptors is in
     let final_stateX :=
       equivocators_state_project IM final_descriptors
         final_state in
     ∃ trX : list (composite_transition_item IM),
       equivocators_trace_project final_descriptors
         (tr ++ [x]) = Some (trX, initial_descriptors)
       ∧ finite_constrained_trace_init_to Free isX
           final_stateX trX)
    apply finite_valid_trace_from_to_last in Hx as Hfinal_state_eq.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: state (equivocators_no_equivocations_vlsm IM)
x: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
IHtr: ∀ final_descriptors : equivocator_descriptors
                      IM,
  proper_equivocator_descriptors IM
    final_descriptors (finite_trace_last is tr)
  → ∃ initial_descriptors : 
       equivocator_descriptors IM,
      proper_equivocator_descriptors IM
        initial_descriptors is
      ∧ (let isX :=
           equivocators_state_project IM
             initial_descriptors is in
         let final_stateX :=
           equivocators_state_project IM
             final_descriptors
             (finite_trace_last is tr) in
         ∃ trX : list
                   (composite_transition_item
                      IM),
           equivocators_trace_project
             final_descriptors tr =
           Some (trX, initial_descriptors)
           ∧ finite_constrained_trace_init_to
               Free isX final_stateX trX)
final_state: state
  (equivocators_no_equivocations_vlsm IM)
Htr: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm FreeE) is
  (finite_trace_last is tr) tr
Hx: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm FreeE)
  (finite_trace_last is tr) final_state [x]
Hinit: initial_state_prop is
final_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  final_descriptors final_state
Hfinal_state_eq: finite_trace_last
  (finite_trace_last is tr) [x] =
final_state
∃ initial_descriptors : equivocator_descriptors IM,
  proper_equivocator_descriptors IM
    initial_descriptors is
  ∧ (let isX :=
       equivocators_state_project IM
         initial_descriptors is in
     let final_stateX :=
       equivocators_state_project IM final_descriptors
         final_state in
     ∃ trX : list (composite_transition_item IM),
       equivocators_trace_project final_descriptors
         (tr ++ [x]) = Some (trX, initial_descriptors)
       ∧ finite_constrained_trace_init_to Free isX
           final_stateX trX)
    change [x] with ([] ++ [x]) in Hfinal_state_eq.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: state (equivocators_no_equivocations_vlsm IM)
x: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
IHtr: ∀ final_descriptors : equivocator_descriptors
                      IM,
  proper_equivocator_descriptors IM
    final_descriptors (finite_trace_last is tr)
  → ∃ initial_descriptors : 
       equivocator_descriptors IM,
      proper_equivocator_descriptors IM
        initial_descriptors is
      ∧ (let isX :=
           equivocators_state_project IM
             initial_descriptors is in
         let final_stateX :=
           equivocators_state_project IM
             final_descriptors
             (finite_trace_last is tr) in
         ∃ trX : list
                   (composite_transition_item
                      IM),
           equivocators_trace_project
             final_descriptors tr =
           Some (trX, initial_descriptors)
           ∧ finite_constrained_trace_init_to
               Free isX final_stateX trX)
final_state: state
  (equivocators_no_equivocations_vlsm IM)
Htr: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm FreeE) is
  (finite_trace_last is tr) tr
Hx: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm FreeE)
  (finite_trace_last is tr) final_state [x]
Hinit: initial_state_prop is
final_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  final_descriptors final_state
Hfinal_state_eq: finite_trace_last
  (finite_trace_last is tr)
  ([] ++ [x]) = final_state
∃ initial_descriptors : equivocator_descriptors IM,
  proper_equivocator_descriptors IM
    initial_descriptors is
  ∧ (let isX :=
       equivocators_state_project IM
         initial_descriptors is in
     let final_stateX :=
       equivocators_state_project IM final_descriptors
         final_state in
     ∃ trX : list (composite_transition_item IM),
       equivocators_trace_project final_descriptors
         (tr ++ [x]) = Some (trX, initial_descriptors)
       ∧ finite_constrained_trace_init_to Free isX
           final_stateX trX)
    rewrite finite_trace_last_is_last in Hfinal_state_eq.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: state (equivocators_no_equivocations_vlsm IM)
x: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
IHtr: ∀ final_descriptors : equivocator_descriptors
                      IM,
  proper_equivocator_descriptors IM
    final_descriptors (finite_trace_last is tr)
  → ∃ initial_descriptors : 
       equivocator_descriptors IM,
      proper_equivocator_descriptors IM
        initial_descriptors is
      ∧ (let isX :=
           equivocators_state_project IM
             initial_descriptors is in
         let final_stateX :=
           equivocators_state_project IM
             final_descriptors
             (finite_trace_last is tr) in
         ∃ trX : list
                   (composite_transition_item
                      IM),
           equivocators_trace_project
             final_descriptors tr =
           Some (trX, initial_descriptors)
           ∧ finite_constrained_trace_init_to
               Free isX final_stateX trX)
final_state: state
  (equivocators_no_equivocations_vlsm IM)
Htr: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm FreeE) is
  (finite_trace_last is tr) tr
Hx: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm FreeE)
  (finite_trace_last is tr) final_state [x]
Hinit: initial_state_prop is
final_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  final_descriptors final_state
Hfinal_state_eq: destination x = final_state
∃ initial_descriptors : equivocator_descriptors IM,
  proper_equivocator_descriptors IM
    initial_descriptors is
  ∧ (let isX :=
       equivocators_state_project IM
         initial_descriptors is in
     let final_stateX :=
       equivocators_state_project IM final_descriptors
         final_state in
     ∃ trX : list (composite_transition_item IM),
       equivocators_trace_project final_descriptors
         (tr ++ [x]) = Some (trX, initial_descriptors)
       ∧ finite_constrained_trace_init_to Free isX
           final_stateX trX)
    subst.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: state (equivocators_no_equivocations_vlsm IM)
x: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
IHtr: ∀ final_descriptors : equivocator_descriptors
                      IM,
  proper_equivocator_descriptors IM
    final_descriptors (finite_trace_last is tr)
  → ∃ initial_descriptors : 
       equivocator_descriptors IM,
      proper_equivocator_descriptors IM
        initial_descriptors is
      ∧ (let isX :=
           equivocators_state_project IM
             initial_descriptors is in
         let final_stateX :=
           equivocators_state_project IM
             final_descriptors
             (finite_trace_last is tr) in
         ∃ trX : list
                   (composite_transition_item
                      IM),
           equivocators_trace_project
             final_descriptors tr =
           Some (trX, initial_descriptors)
           ∧ finite_constrained_trace_init_to
               Free isX final_stateX trX)
Htr: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm FreeE) is
  (finite_trace_last is tr) tr
Hx: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm FreeE)
  (finite_trace_last is tr) 
  (destination x) [x]
Hinit: initial_state_prop is
final_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  final_descriptors (destination x)
∃ initial_descriptors : equivocator_descriptors IM,
  proper_equivocator_descriptors IM
    initial_descriptors is
  ∧ (let isX :=
       equivocators_state_project IM
         initial_descriptors is in
     let final_stateX :=
       equivocators_state_project IM final_descriptors
         (destination x) in
     ∃ trX : list (composite_transition_item IM),
       equivocators_trace_project final_descriptors
         (tr ++ [x]) = Some (trX, initial_descriptors)
       ∧ finite_constrained_trace_init_to Free isX
           final_stateX trX)
    destruct
      (equivocators_transition_item_project_proper_characterization _ x Hproper)
      as [oitem [final_descriptors' [Hpr_x [Hchar1 Hchar2]]]].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: state (equivocators_no_equivocations_vlsm IM)
x: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
IHtr: ∀ final_descriptors : equivocator_descriptors
                      IM,
  proper_equivocator_descriptors IM
    final_descriptors (finite_trace_last is tr)
  → ∃ initial_descriptors : 
       equivocator_descriptors IM,
      proper_equivocator_descriptors IM
        initial_descriptors is
      ∧ (let isX :=
           equivocators_state_project IM
             initial_descriptors is in
         let final_stateX :=
           equivocators_state_project IM
             final_descriptors
             (finite_trace_last is tr) in
         ∃ trX : list
                   (composite_transition_item
                      IM),
           equivocators_trace_project
             final_descriptors tr =
           Some (trX, initial_descriptors)
           ∧ finite_constrained_trace_init_to
               Free isX final_stateX trX)
Htr: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm FreeE) is
  (finite_trace_last is tr) tr
Hx: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm FreeE)
  (finite_trace_last is tr) 
  (destination x) [x]
Hinit: initial_state_prop is
final_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  final_descriptors (destination x)
oitem: option (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hpr_x: equivocators_transition_item_project
  final_descriptors x =
Some (oitem, final_descriptors')
Hchar1: match oitem with
| Some itemx =>
    (∃ Hex : existing_equivocator_label
               (IM (projT1 (l x)))
               (projT2 (l x)),
       existT (projT1 (l x))
         (existing_equivocator_label_extract
            (IM (projT1 (l x)))
            (projT2 (l x)) Hex) = 
       l itemx)
    ∧ input x = input itemx
      ∧ output x = output itemx
        ∧ equivocators_state_project IM
            final_descriptors 
            (destination x) =
          destination itemx
          ∧ final_descriptors' (projT1 (l x)) =
            equivocator_label_descriptor
              (projT2 (l x))
| None => True
end
Hchar2: ∀ s : composite_state (equivocator_IM IM),
  composite_valid (equivocator_IM IM) 
    (l x) (s, input x)
  → composite_transition 
      (equivocator_IM IM) 
      (l x) (s, input x) =
    (destination x, output x)
    → proper_equivocator_descriptors IM
        final_descriptors' s
      ∧ final_descriptors' =
        equivocator_descriptors_update IM
          final_descriptors 
          (projT1 (l x))
          (final_descriptors' (projT1 (l x)))
        ∧ s =
          state_update 
            (equivocator_IM IM)
            (destination x) 
            (projT1 (l x)) 
            (s (projT1 (l x)))
          ∧ previous_state_descriptor_prop
              (IM (projT1 (l x)))
              (final_descriptors
                 (projT1 (l x)))
              (s (projT1 (l x)))
              (final_descriptors'
                 (projT1 (l x)))
            ∧ match oitem with
              | Some itemx =>
                  ∀ sx : composite_state IM,
                    sx =
                    equivocators_state_project
                      IM final_descriptors' s
                    → composite_valid IM
                      (l itemx)
                      (sx, input itemx)
                      ∧ 
                      composite_transition IM
                      (l itemx)
                      (sx, input itemx) =
                      (
                      destination itemx,
                      output itemx)
              | None =>
                  equivocators_state_project
                    IM final_descriptors
                    (destination x) =
                  equivocators_state_project
                    IM final_descriptors' s
              end
∃ initial_descriptors : equivocator_descriptors IM,
  proper_equivocator_descriptors IM
    initial_descriptors is
  ∧ (let isX :=
       equivocators_state_project IM
         initial_descriptors is in
     let final_stateX :=
       equivocators_state_project IM final_descriptors
         (destination x) in
     ∃ trX : list (composite_transition_item IM),
       equivocators_trace_project final_descriptors
         (tr ++ [x]) = Some (trX, initial_descriptors)
       ∧ finite_constrained_trace_init_to Free isX
           final_stateX trX)
    specialize (equivocators_trace_project_app_iff tr [x]) as Hpr_app.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: state (equivocators_no_equivocations_vlsm IM)
x: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
IHtr: ∀ final_descriptors : equivocator_descriptors
                      IM,
  proper_equivocator_descriptors IM
    final_descriptors (finite_trace_last is tr)
  → ∃ initial_descriptors : 
       equivocator_descriptors IM,
      proper_equivocator_descriptors IM
        initial_descriptors is
      ∧ (let isX :=
           equivocators_state_project IM
             initial_descriptors is in
         let final_stateX :=
           equivocators_state_project IM
             final_descriptors
             (finite_trace_last is tr) in
         ∃ trX : list
                   (composite_transition_item
                      IM),
           equivocators_trace_project
             final_descriptors tr =
           Some (trX, initial_descriptors)
           ∧ finite_constrained_trace_init_to
               Free isX final_stateX trX)
Htr: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm FreeE) is
  (finite_trace_last is tr) tr
Hx: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm FreeE)
  (finite_trace_last is tr) 
  (destination x) [x]
Hinit: initial_state_prop is
final_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  final_descriptors (destination x)
oitem: option (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hpr_x: equivocators_transition_item_project
  final_descriptors x =
Some (oitem, final_descriptors')
Hchar1: match oitem with
| Some itemx =>
    (∃ Hex : existing_equivocator_label
               (IM (projT1 (l x)))
               (projT2 (l x)),
       existT (projT1 (l x))
         (existing_equivocator_label_extract
            (IM (projT1 (l x)))
            (projT2 (l x)) Hex) = 
       l itemx)
    ∧ input x = input itemx
      ∧ output x = output itemx
        ∧ equivocators_state_project IM
            final_descriptors 
            (destination x) =
          destination itemx
          ∧ final_descriptors' (projT1 (l x)) =
            equivocator_label_descriptor
              (projT2 (l x))
| None => True
end
Hchar2: ∀ s : composite_state (equivocator_IM IM),
  composite_valid (equivocator_IM IM) 
    (l x) (s, input x)
  → composite_transition 
      (equivocator_IM IM) 
      (l x) (s, input x) =
    (destination x, output x)
    → proper_equivocator_descriptors IM
        final_descriptors' s
      ∧ final_descriptors' =
        equivocator_descriptors_update IM
          final_descriptors 
          (projT1 (l x))
          (final_descriptors' (projT1 (l x)))
        ∧ s =
          state_update 
            (equivocator_IM IM)
            (destination x) 
            (projT1 (l x)) 
            (s (projT1 (l x)))
          ∧ previous_state_descriptor_prop
              (IM (projT1 (l x)))
              (final_descriptors
                 (projT1 (l x)))
              (s (projT1 (l x)))
              (final_descriptors'
                 (projT1 (l x)))
            ∧ match oitem with
              | Some itemx =>
                  ∀ sx : composite_state IM,
                    sx =
                    equivocators_state_project
                      IM final_descriptors' s
                    → composite_valid IM
                      (l itemx)
                      (sx, input itemx)
                      ∧ 
                      composite_transition IM
                      (l itemx)
                      (sx, input itemx) =
                      (
                      destination itemx,
                      output itemx)
              | None =>
                  equivocators_state_project
                    IM final_descriptors
                    (destination x) =
                  equivocators_state_project
                    IM final_descriptors' s
              end
Hpr_app: ∀ (ieqv_descriptors
   eqv_descriptors : equivocator_descriptors
                      IM) 
  (trX : list (composite_transition_item IM)),
  equivocators_trace_project eqv_descriptors
    (tr ++ [x]) =
  Some (trX, ieqv_descriptors)
  ↔ (∃ (preX
        sufX : list
                 (composite_transition_item
                    IM)) 
       (eqv_descriptors' : 
        equivocator_descriptors IM),
       equivocators_trace_project
         eqv_descriptors [x] =
       Some (sufX, eqv_descriptors')
       ∧ equivocators_trace_project
           eqv_descriptors' tr =
         Some (preX, ieqv_descriptors)
         ∧ trX = preX ++ sufX)
∃ initial_descriptors : equivocator_descriptors IM,
  proper_equivocator_descriptors IM
    initial_descriptors is
  ∧ (let isX :=
       equivocators_state_project IM
         initial_descriptors is in
     let final_stateX :=
       equivocators_state_project IM final_descriptors
         (destination x) in
     ∃ trX : list (composite_transition_item IM),
       equivocators_trace_project final_descriptors
         (tr ++ [x]) = Some (trX, initial_descriptors)
       ∧ finite_constrained_trace_init_to Free isX
           final_stateX trX)
    inversion Hx.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: state (equivocators_no_equivocations_vlsm IM)
x: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
IHtr: ∀ final_descriptors : equivocator_descriptors
                      IM,
  proper_equivocator_descriptors IM
    final_descriptors (finite_trace_last is tr)
  → ∃ initial_descriptors : 
       equivocator_descriptors IM,
      proper_equivocator_descriptors IM
        initial_descriptors is
      ∧ (let isX :=
           equivocators_state_project IM
             initial_descriptors is in
         let final_stateX :=
           equivocators_state_project IM
             final_descriptors
             (finite_trace_last is tr) in
         ∃ trX : list
                   (composite_transition_item
                      IM),
           equivocators_trace_project
             final_descriptors tr =
           Some (trX, initial_descriptors)
           ∧ finite_constrained_trace_init_to
               Free isX final_stateX trX)
Htr: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm FreeE) is
  (finite_trace_last is tr) tr
Hx: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm FreeE)
  (finite_trace_last is tr) 
  (destination x) [x]
Hinit: initial_state_prop is
final_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  final_descriptors (destination x)
oitem: option (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hpr_x: equivocators_transition_item_project
  final_descriptors x =
Some (oitem, final_descriptors')
Hchar1: match oitem with
| Some itemx =>
    (∃ Hex : existing_equivocator_label
               (IM (projT1 (VLSM.l x)))
               (projT2 (VLSM.l x)),
       existT (projT1 (VLSM.l x))
         (existing_equivocator_label_extract
            (IM (projT1 (VLSM.l x)))
            (projT2 (VLSM.l x)) Hex) =
       VLSM.l itemx)
    ∧ input x = input itemx
      ∧ output x = output itemx
        ∧ equivocators_state_project IM
            final_descriptors 
            (destination x) =
          destination itemx
          ∧ final_descriptors'
              (projT1 (VLSM.l x)) =
            equivocator_label_descriptor
              (projT2 (VLSM.l x))
| None => True
end
Hchar2: ∀ s : composite_state (equivocator_IM IM),
  composite_valid (equivocator_IM IM)
    (VLSM.l x) (s, input x)
  → composite_transition 
      (equivocator_IM IM) 
      (VLSM.l x) (s, input x) =
    (destination x, output x)
    → proper_equivocator_descriptors IM
        final_descriptors' s
      ∧ final_descriptors' =
        equivocator_descriptors_update IM
          final_descriptors
          (projT1 (VLSM.l x))
          (final_descriptors'
             (projT1 (VLSM.l x)))
        ∧ s =
          state_update 
            (equivocator_IM IM)
            (destination x)
            (projT1 (VLSM.l x))
            (s (projT1 (VLSM.l x)))
          ∧ previous_state_descriptor_prop
              (IM (projT1 (VLSM.l x)))
              (final_descriptors
                 (projT1 (VLSM.l x)))
              (s (projT1 (VLSM.l x)))
              (final_descriptors'
                 (projT1 (VLSM.l x)))
            ∧ match oitem with
              | Some itemx =>
                  ∀ sx : composite_state IM,
                    sx =
                    equivocators_state_project
                      IM final_descriptors' s
                    → composite_valid IM
                      (VLSM.l itemx)
                      (sx, input itemx)
                      ∧ 
                      composite_transition IM
                      (VLSM.l itemx)
                      (sx, input itemx) =
                      (
                      destination itemx,
                      output itemx)
              | None =>
                  equivocators_state_project
                    IM final_descriptors
                    (destination x) =
                  equivocators_state_project
                    IM final_descriptors' s
              end
Hpr_app: ∀ (ieqv_descriptors
   eqv_descriptors : equivocator_descriptors
                      IM) 
  (trX : list (composite_transition_item IM)),
  equivocators_trace_project eqv_descriptors
    (tr ++ [x]) =
  Some (trX, ieqv_descriptors)
  ↔ (∃ (preX
        sufX : list
                 (composite_transition_item
                    IM)) 
       (eqv_descriptors' : 
        equivocator_descriptors IM),
       equivocators_trace_project
         eqv_descriptors [x] =
       Some (sufX, eqv_descriptors')
       ∧ equivocators_trace_project
           eqv_descriptors' tr =
         Some (preX, ieqv_descriptors)
         ∧ trX = preX ++ sufX)
s, f: state (preloaded_with_all_messages_vlsm FreeE)
tl: list transition_item
s': state (preloaded_with_all_messages_vlsm FreeE)
iom, oom: option message
l: label (preloaded_with_all_messages_vlsm FreeE)
Htl: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm FreeE) s
  (destination
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |}) []
Ht: input_valid_transition
  (preloaded_with_all_messages_vlsm FreeE) l
  (finite_trace_last is tr, iom) (
  s, oom)
H1: s' = finite_trace_last is tr
H2: f = destination x
H4: {|
  l := l;
  input := iom;
  destination := s;
  output := oom
|} = x
H5: tl = []
∃ initial_descriptors : equivocator_descriptors IM,
  proper_equivocator_descriptors IM
    initial_descriptors is
  ∧ (let isX :=
       equivocators_state_project IM
         initial_descriptors is in
     let final_stateX :=
       equivocators_state_project IM final_descriptors
         (destination
            {|
              l := l;
              input := iom;
              destination := s;
              output := oom
            |}) in
     ∃ trX : list (composite_transition_item IM),
       equivocators_trace_project final_descriptors
         (tr ++
          [{|
             l := l;
             input := iom;
             destination := s;
             output := oom
           |}]) = Some (trX, initial_descriptors)
       ∧ finite_constrained_trace_init_to Free isX
           final_stateX trX) subst.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: state (equivocators_no_equivocations_vlsm IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
IHtr: ∀ final_descriptors : equivocator_descriptors
                      IM,
  proper_equivocator_descriptors IM
    final_descriptors (finite_trace_last is tr)
  → ∃ initial_descriptors : 
       equivocator_descriptors IM,
      proper_equivocator_descriptors IM
        initial_descriptors is
      ∧ (let isX :=
           equivocators_state_project IM
             initial_descriptors is in
         let final_stateX :=
           equivocators_state_project IM
             final_descriptors
             (finite_trace_last is tr) in
         ∃ trX : list
                   (composite_transition_item
                      IM),
           equivocators_trace_project
             final_descriptors tr =
           Some (trX, initial_descriptors)
           ∧ finite_constrained_trace_init_to
               Free isX final_stateX trX)
Htr: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm FreeE) is
  (finite_trace_last is tr) tr
s: state (preloaded_with_all_messages_vlsm FreeE)
iom, oom: option message
l: label (preloaded_with_all_messages_vlsm FreeE)
Hx: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm FreeE)
  (finite_trace_last is tr)
  (destination
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |})
  [{|
     l := l;
     input := iom;
     destination := s;
     output := oom
   |}]
Hinit: initial_state_prop is
final_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  final_descriptors
  (destination
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |})
oitem: option (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hpr_app: ∀ (ieqv_descriptors
   eqv_descriptors : equivocator_descriptors
                      IM) 
  (trX : list (composite_transition_item IM)),
  equivocators_trace_project eqv_descriptors
    (tr ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}]) = Some (trX, ieqv_descriptors)
  ↔ (∃ (preX
        sufX : list
                 (composite_transition_item
                    IM)) 
       (eqv_descriptors' : 
        equivocator_descriptors IM),
       equivocators_trace_project
         eqv_descriptors
         [{|
            l := l;
            input := iom;
            destination := s;
            output := oom
          |}] =
       Some (sufX, eqv_descriptors')
       ∧ equivocators_trace_project
           eqv_descriptors' tr =
         Some (preX, ieqv_descriptors)
         ∧ trX = preX ++ sufX)
Hchar2: ∀ s0 : composite_state (equivocator_IM IM),
  composite_valid (equivocator_IM IM)
    (VLSM.l
       {|
         l := l;
         input := iom;
         destination := s;
         output := oom
       |})
    (s0,
     input
       {|
         l := l;
         input := iom;
         destination := s;
         output := oom
       |})
  → composite_transition 
      (equivocator_IM IM)
      (VLSM.l
         {|
           l := l;
           input := iom;
           destination := s;
           output := oom
         |})
      (s0,
       input
         {|
           l := l;
           input := iom;
           destination := s;
           output := oom
         |}) =
    (destination
       {|
         l := l;
         input := iom;
         destination := s;
         output := oom
       |},
     output
       {|
         l := l;
         input := iom;
         destination := s;
         output := oom
       |})
    → proper_equivocator_descriptors IM
        final_descriptors' s0
      ∧ final_descriptors' =
        equivocator_descriptors_update IM
          final_descriptors
          (projT1
             (VLSM.l
                {|
                  l := l;
                  input := iom;
                  destination := s;
                  output := oom
                |}))
          (final_descriptors'
             (projT1
                (VLSM.l
                   {|
                     l := l;
                     input := iom;
                     destination := s;
                     output := oom
                   |})))
        ∧ s0 =
          state_update 
            (equivocator_IM IM)
            (destination
               {|
                 l := l;
                 input := iom;
                 destination := s;
                 output := oom
               |})
            (projT1
               (VLSM.l
                  {|
                    l := l;
                    input := iom;
                    destination := s;
                    output := oom
                  |}))
            (s0
               (projT1
                  (VLSM.l
                     {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                     |})))
          ∧ previous_state_descriptor_prop
              (IM
                 (projT1
                    (VLSM.l
                      {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                      |})))
              (final_descriptors
                 (projT1
                    (VLSM.l
                      {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                      |})))
              (s0
                 (projT1
                    (VLSM.l
                      {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                      |})))
              (final_descriptors'
                 (projT1
                    (VLSM.l
                      {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                      |})))
            ∧ match oitem with
              | Some itemx =>
                  ∀ sx : composite_state IM,
                    sx =
                    equivocators_state_project
                      IM final_descriptors'
                      s0
                    → composite_valid IM
                      (VLSM.l itemx)
                      (sx, input itemx)
                      ∧ 
                      composite_transition IM
                      (VLSM.l itemx)
                      (sx, input itemx) =
                      (
                      destination itemx,
                      output itemx)
              | None =>
                  equivocators_state_project
                    IM final_descriptors
                    (destination
                      {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                      |}) =
                  equivocators_state_project
                    IM final_descriptors' s0
              end
Hchar1: match oitem with
| Some itemx =>
    (∃ Hex : existing_equivocator_label
               (IM
                  (projT1
                     (VLSM.l
                      {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                      |})))
               (projT2
                  (VLSM.l
                     {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                     |})),
       existT
         (projT1
            (VLSM.l
               {|
                 l := l;
                 input := iom;
                 destination := s;
                 output := oom
               |}))
         (existing_equivocator_label_extract
            (IM
               (projT1
                  (VLSM.l
                     {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                     |})))
            (projT2
               (VLSM.l
                  {|
                    l := l;
                    input := iom;
                    destination := s;
                    output := oom
                  |})) Hex) = 
       VLSM.l itemx)
    ∧ input
        {|
          l := l;
          input := iom;
          destination := s;
          output := oom
        |} = input itemx
      ∧ output
          {|
            l := l;
            input := iom;
            destination := s;
            output := oom
          |} = output itemx
        ∧ equivocators_state_project IM
            final_descriptors
            (destination
               {|
                 l := l;
                 input := iom;
                 destination := s;
                 output := oom
               |}) = destination itemx
          ∧ final_descriptors'
              (projT1
                 (VLSM.l
                    {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                    |})) =
            equivocator_label_descriptor
              (projT2
                 (VLSM.l
                    {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                    |}))
| None => True
end
Hpr_x: equivocators_transition_item_project
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some (oitem, final_descriptors')
Htl: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm FreeE) s
  (destination
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |}) []
Ht: input_valid_transition
  (preloaded_with_all_messages_vlsm FreeE) l
  (finite_trace_last is tr, iom) (
  s, oom)
∃ initial_descriptors : equivocator_descriptors IM,
  proper_equivocator_descriptors IM
    initial_descriptors is
  ∧ (let isX :=
       equivocators_state_project IM
         initial_descriptors is in
     let final_stateX :=
       equivocators_state_project IM final_descriptors
         (destination
            {|
              l := l;
              input := iom;
              destination := s;
              output := oom
            |}) in
     ∃ trX : list (composite_transition_item IM),
       equivocators_trace_project final_descriptors
         (tr ++
          [{|
             l := l;
             input := iom;
             destination := s;
             output := oom
           |}]) = Some (trX, initial_descriptors)
       ∧ finite_constrained_trace_init_to Free isX
           final_stateX trX) clear Hx Htl.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: state (equivocators_no_equivocations_vlsm IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
IHtr: ∀ final_descriptors : equivocator_descriptors
                      IM,
  proper_equivocator_descriptors IM
    final_descriptors (finite_trace_last is tr)
  → ∃ initial_descriptors : 
       equivocator_descriptors IM,
      proper_equivocator_descriptors IM
        initial_descriptors is
      ∧ (let isX :=
           equivocators_state_project IM
             initial_descriptors is in
         let final_stateX :=
           equivocators_state_project IM
             final_descriptors
             (finite_trace_last is tr) in
         ∃ trX : list
                   (composite_transition_item
                      IM),
           equivocators_trace_project
             final_descriptors tr =
           Some (trX, initial_descriptors)
           ∧ finite_constrained_trace_init_to
               Free isX final_stateX trX)
Htr: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm FreeE) is
  (finite_trace_last is tr) tr
s: state (preloaded_with_all_messages_vlsm FreeE)
iom, oom: option message
l: label (preloaded_with_all_messages_vlsm FreeE)
Hinit: initial_state_prop is
final_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  final_descriptors
  (destination
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |})
oitem: option (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hpr_app: ∀ (ieqv_descriptors
   eqv_descriptors : equivocator_descriptors
                      IM) 
  (trX : list (composite_transition_item IM)),
  equivocators_trace_project eqv_descriptors
    (tr ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}]) = Some (trX, ieqv_descriptors)
  ↔ (∃ (preX
        sufX : list
                 (composite_transition_item
                    IM)) 
       (eqv_descriptors' : 
        equivocator_descriptors IM),
       equivocators_trace_project
         eqv_descriptors
         [{|
            l := l;
            input := iom;
            destination := s;
            output := oom
          |}] =
       Some (sufX, eqv_descriptors')
       ∧ equivocators_trace_project
           eqv_descriptors' tr =
         Some (preX, ieqv_descriptors)
         ∧ trX = preX ++ sufX)
Hchar2: ∀ s0 : composite_state (equivocator_IM IM),
  composite_valid (equivocator_IM IM)
    (VLSM.l
       {|
         l := l;
         input := iom;
         destination := s;
         output := oom
       |})
    (s0,
     input
       {|
         l := l;
         input := iom;
         destination := s;
         output := oom
       |})
  → composite_transition 
      (equivocator_IM IM)
      (VLSM.l
         {|
           l := l;
           input := iom;
           destination := s;
           output := oom
         |})
      (s0,
       input
         {|
           l := l;
           input := iom;
           destination := s;
           output := oom
         |}) =
    (destination
       {|
         l := l;
         input := iom;
         destination := s;
         output := oom
       |},
     output
       {|
         l := l;
         input := iom;
         destination := s;
         output := oom
       |})
    → proper_equivocator_descriptors IM
        final_descriptors' s0
      ∧ final_descriptors' =
        equivocator_descriptors_update IM
          final_descriptors
          (projT1
             (VLSM.l
                {|
                  l := l;
                  input := iom;
                  destination := s;
                  output := oom
                |}))
          (final_descriptors'
             (projT1
                (VLSM.l
                   {|
                     l := l;
                     input := iom;
                     destination := s;
                     output := oom
                   |})))
        ∧ s0 =
          state_update 
            (equivocator_IM IM)
            (destination
               {|
                 l := l;
                 input := iom;
                 destination := s;
                 output := oom
               |})
            (projT1
               (VLSM.l
                  {|
                    l := l;
                    input := iom;
                    destination := s;
                    output := oom
                  |}))
            (s0
               (projT1
                  (VLSM.l
                     {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                     |})))
          ∧ previous_state_descriptor_prop
              (IM
                 (projT1
                    (VLSM.l
                      {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                      |})))
              (final_descriptors
                 (projT1
                    (VLSM.l
                      {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                      |})))
              (s0
                 (projT1
                    (VLSM.l
                      {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                      |})))
              (final_descriptors'
                 (projT1
                    (VLSM.l
                      {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                      |})))
            ∧ match oitem with
              | Some itemx =>
                  ∀ sx : composite_state IM,
                    sx =
                    equivocators_state_project
                      IM final_descriptors'
                      s0
                    → composite_valid IM
                      (VLSM.l itemx)
                      (sx, input itemx)
                      ∧ 
                      composite_transition IM
                      (VLSM.l itemx)
                      (sx, input itemx) =
                      (
                      destination itemx,
                      output itemx)
              | None =>
                  equivocators_state_project
                    IM final_descriptors
                    (destination
                      {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                      |}) =
                  equivocators_state_project
                    IM final_descriptors' s0
              end
Hchar1: match oitem with
| Some itemx =>
    (∃ Hex : existing_equivocator_label
               (IM
                  (projT1
                     (VLSM.l
                      {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                      |})))
               (projT2
                  (VLSM.l
                     {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                     |})),
       existT
         (projT1
            (VLSM.l
               {|
                 l := l;
                 input := iom;
                 destination := s;
                 output := oom
               |}))
         (existing_equivocator_label_extract
            (IM
               (projT1
                  (VLSM.l
                     {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                     |})))
            (projT2
               (VLSM.l
                  {|
                    l := l;
                    input := iom;
                    destination := s;
                    output := oom
                  |})) Hex) = 
       VLSM.l itemx)
    ∧ input
        {|
          l := l;
          input := iom;
          destination := s;
          output := oom
        |} = input itemx
      ∧ output
          {|
            l := l;
            input := iom;
            destination := s;
            output := oom
          |} = output itemx
        ∧ equivocators_state_project IM
            final_descriptors
            (destination
               {|
                 l := l;
                 input := iom;
                 destination := s;
                 output := oom
               |}) = destination itemx
          ∧ final_descriptors'
              (projT1
                 (VLSM.l
                    {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                    |})) =
            equivocator_label_descriptor
              (projT2
                 (VLSM.l
                    {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                    |}))
| None => True
end
Hpr_x: equivocators_transition_item_project
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some (oitem, final_descriptors')
Ht: input_valid_transition
  (preloaded_with_all_messages_vlsm FreeE) l
  (finite_trace_last is tr, iom) (
  s, oom)
∃ initial_descriptors : equivocator_descriptors IM,
  proper_equivocator_descriptors IM
    initial_descriptors is
  ∧ (let isX :=
       equivocators_state_project IM
         initial_descriptors is in
     let final_stateX :=
       equivocators_state_project IM final_descriptors
         (destination
            {|
              l := l;
              input := iom;
              destination := s;
              output := oom
            |}) in
     ∃ trX : list (composite_transition_item IM),
       equivocators_trace_project final_descriptors
         (tr ++
          [{|
             l := l;
             input := iom;
             destination := s;
             output := oom
           |}]) = Some (trX, initial_descriptors)
       ∧ finite_constrained_trace_init_to Free isX
           final_stateX trX)
    destruct Ht as [[_ [_ Hvx]]  Htx].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: state (equivocators_no_equivocations_vlsm IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
IHtr: ∀ final_descriptors : equivocator_descriptors
                      IM,
  proper_equivocator_descriptors IM
    final_descriptors (finite_trace_last is tr)
  → ∃ initial_descriptors : 
       equivocator_descriptors IM,
      proper_equivocator_descriptors IM
        initial_descriptors is
      ∧ (let isX :=
           equivocators_state_project IM
             initial_descriptors is in
         let final_stateX :=
           equivocators_state_project IM
             final_descriptors
             (finite_trace_last is tr) in
         ∃ trX : list
                   (composite_transition_item
                      IM),
           equivocators_trace_project
             final_descriptors tr =
           Some (trX, initial_descriptors)
           ∧ finite_constrained_trace_init_to
               Free isX final_stateX trX)
Htr: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm FreeE) is
  (finite_trace_last is tr) tr
s: state (preloaded_with_all_messages_vlsm FreeE)
iom, oom: option message
l: label (preloaded_with_all_messages_vlsm FreeE)
Hinit: initial_state_prop is
final_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  final_descriptors
  (destination
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |})
oitem: option (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hpr_app: ∀ (ieqv_descriptors
   eqv_descriptors : equivocator_descriptors
                      IM) 
  (trX : list (composite_transition_item IM)),
  equivocators_trace_project eqv_descriptors
    (tr ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}]) = Some (trX, ieqv_descriptors)
  ↔ (∃ (preX
        sufX : list
                 (composite_transition_item
                    IM)) 
       (eqv_descriptors' : 
        equivocator_descriptors IM),
       equivocators_trace_project
         eqv_descriptors
         [{|
            l := l;
            input := iom;
            destination := s;
            output := oom
          |}] =
       Some (sufX, eqv_descriptors')
       ∧ equivocators_trace_project
           eqv_descriptors' tr =
         Some (preX, ieqv_descriptors)
         ∧ trX = preX ++ sufX)
Hchar2: ∀ s0 : composite_state (equivocator_IM IM),
  composite_valid (equivocator_IM IM)
    (VLSM.l
       {|
         l := l;
         input := iom;
         destination := s;
         output := oom
       |})
    (s0,
     input
       {|
         l := l;
         input := iom;
         destination := s;
         output := oom
       |})
  → composite_transition 
      (equivocator_IM IM)
      (VLSM.l
         {|
           l := l;
           input := iom;
           destination := s;
           output := oom
         |})
      (s0,
       input
         {|
           l := l;
           input := iom;
           destination := s;
           output := oom
         |}) =
    (destination
       {|
         l := l;
         input := iom;
         destination := s;
         output := oom
       |},
     output
       {|
         l := l;
         input := iom;
         destination := s;
         output := oom
       |})
    → proper_equivocator_descriptors IM
        final_descriptors' s0
      ∧ final_descriptors' =
        equivocator_descriptors_update IM
          final_descriptors
          (projT1
             (VLSM.l
                {|
                  l := l;
                  input := iom;
                  destination := s;
                  output := oom
                |}))
          (final_descriptors'
             (projT1
                (VLSM.l
                   {|
                     l := l;
                     input := iom;
                     destination := s;
                     output := oom
                   |})))
        ∧ s0 =
          state_update 
            (equivocator_IM IM)
            (destination
               {|
                 l := l;
                 input := iom;
                 destination := s;
                 output := oom
               |})
            (projT1
               (VLSM.l
                  {|
                    l := l;
                    input := iom;
                    destination := s;
                    output := oom
                  |}))
            (s0
               (projT1
                  (VLSM.l
                     {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                     |})))
          ∧ previous_state_descriptor_prop
              (IM
                 (projT1
                    (VLSM.l
                      {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                      |})))
              (final_descriptors
                 (projT1
                    (VLSM.l
                      {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                      |})))
              (s0
                 (projT1
                    (VLSM.l
                      {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                      |})))
              (final_descriptors'
                 (projT1
                    (VLSM.l
                      {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                      |})))
            ∧ match oitem with
              | Some itemx =>
                  ∀ sx : composite_state IM,
                    sx =
                    equivocators_state_project
                      IM final_descriptors'
                      s0
                    → composite_valid IM
                      (VLSM.l itemx)
                      (sx, input itemx)
                      ∧ 
                      composite_transition IM
                      (VLSM.l itemx)
                      (sx, input itemx) =
                      (
                      destination itemx,
                      output itemx)
              | None =>
                  equivocators_state_project
                    IM final_descriptors
                    (destination
                      {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                      |}) =
                  equivocators_state_project
                    IM final_descriptors' s0
              end
Hchar1: match oitem with
| Some itemx =>
    (∃ Hex : existing_equivocator_label
               (IM
                  (projT1
                     (VLSM.l
                      {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                      |})))
               (projT2
                  (VLSM.l
                     {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                     |})),
       existT
         (projT1
            (VLSM.l
               {|
                 l := l;
                 input := iom;
                 destination := s;
                 output := oom
               |}))
         (existing_equivocator_label_extract
            (IM
               (projT1
                  (VLSM.l
                     {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                     |})))
            (projT2
               (VLSM.l
                  {|
                    l := l;
                    input := iom;
                    destination := s;
                    output := oom
                  |})) Hex) = 
       VLSM.l itemx)
    ∧ input
        {|
          l := l;
          input := iom;
          destination := s;
          output := oom
        |} = input itemx
      ∧ output
          {|
            l := l;
            input := iom;
            destination := s;
            output := oom
          |} = output itemx
        ∧ equivocators_state_project IM
            final_descriptors
            (destination
               {|
                 l := l;
                 input := iom;
                 destination := s;
                 output := oom
               |}) = destination itemx
          ∧ final_descriptors'
              (projT1
                 (VLSM.l
                    {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                    |})) =
            equivocator_label_descriptor
              (projT2
                 (VLSM.l
                    {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                    |}))
| None => True
end
Hpr_x: equivocators_transition_item_project
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some (oitem, final_descriptors')
Hvx: valid l (finite_trace_last is tr, iom)
Htx: transition l (finite_trace_last is tr, iom) =
(s, oom)
∃ initial_descriptors : equivocator_descriptors IM,
  proper_equivocator_descriptors IM
    initial_descriptors is
  ∧ (let isX :=
       equivocators_state_project IM
         initial_descriptors is in
     let final_stateX :=
       equivocators_state_project IM final_descriptors
         (destination
            {|
              l := l;
              input := iom;
              destination := s;
              output := oom
            |}) in
     ∃ trX : list (composite_transition_item IM),
       equivocators_trace_project final_descriptors
         (tr ++
          [{|
             l := l;
             input := iom;
             destination := s;
             output := oom
           |}]) = Some (trX, initial_descriptors)
       ∧ finite_constrained_trace_init_to Free isX
           final_stateX trX)
    specialize (Hchar2 (finite_trace_last is tr) Hvx Htx).message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: state (equivocators_no_equivocations_vlsm IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
IHtr: ∀ final_descriptors : equivocator_descriptors
                      IM,
  proper_equivocator_descriptors IM
    final_descriptors (finite_trace_last is tr)
  → ∃ initial_descriptors : 
       equivocator_descriptors IM,
      proper_equivocator_descriptors IM
        initial_descriptors is
      ∧ (let isX :=
           equivocators_state_project IM
             initial_descriptors is in
         let final_stateX :=
           equivocators_state_project IM
             final_descriptors
             (finite_trace_last is tr) in
         ∃ trX : list
                   (composite_transition_item
                      IM),
           equivocators_trace_project
             final_descriptors tr =
           Some (trX, initial_descriptors)
           ∧ finite_constrained_trace_init_to
               Free isX final_stateX trX)
Htr: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm FreeE) is
  (finite_trace_last is tr) tr
s: state (preloaded_with_all_messages_vlsm FreeE)
iom, oom: option message
l: label (preloaded_with_all_messages_vlsm FreeE)
Hinit: initial_state_prop is
final_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  final_descriptors
  (destination
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |})
oitem: option (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hpr_app: ∀ (ieqv_descriptors
   eqv_descriptors : equivocator_descriptors
                      IM) 
  (trX : list (composite_transition_item IM)),
  equivocators_trace_project eqv_descriptors
    (tr ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}]) = Some (trX, ieqv_descriptors)
  ↔ (∃ (preX
        sufX : list
                 (composite_transition_item
                    IM)) 
       (eqv_descriptors' : 
        equivocator_descriptors IM),
       equivocators_trace_project
         eqv_descriptors
         [{|
            l := l;
            input := iom;
            destination := s;
            output := oom
          |}] =
       Some (sufX, eqv_descriptors')
       ∧ equivocators_trace_project
           eqv_descriptors' tr =
         Some (preX, ieqv_descriptors)
         ∧ trX = preX ++ sufX)
Hchar2: proper_equivocator_descriptors IM
  final_descriptors'
  (finite_trace_last is tr)
∧ final_descriptors' =
  equivocator_descriptors_update IM
    final_descriptors
    (projT1
       (VLSM.l
          {|
            l := l;
            input := iom;
            destination := s;
            output := oom
          |}))
    (final_descriptors'
       (projT1
          (VLSM.l
             {|
               l := l;
               input := iom;
               destination := s;
               output := oom
             |})))
  ∧ finite_trace_last is tr =
    state_update (equivocator_IM IM)
      (destination
         {|
           l := l;
           input := iom;
           destination := s;
           output := oom
         |})
      (projT1
         (VLSM.l
            {|
              l := l;
              input := iom;
              destination := s;
              output := oom
            |}))
      (finite_trace_last is tr
         (projT1
            (VLSM.l
               {|
                 l := l;
                 input := iom;
                 destination := s;
                 output := oom
               |})))
    ∧ previous_state_descriptor_prop
        (IM
           (projT1
              (VLSM.l
                 {|
                   l := l;
                   input := iom;
                   destination := s;
                   output := oom
                 |})))
        (final_descriptors
           (projT1
              (VLSM.l
                 {|
                   l := l;
                   input := iom;
                   destination := s;
                   output := oom
                 |})))
        (finite_trace_last is tr
           (projT1
              (VLSM.l
                 {|
                   l := l;
                   input := iom;
                   destination := s;
                   output := oom
                 |})))
        (final_descriptors'
           (projT1
              (VLSM.l
                 {|
                   l := l;
                   input := iom;
                   destination := s;
                   output := oom
                 |})))
      ∧ match oitem with
        | Some itemx =>
            ∀ sx : composite_state IM,
              sx =
              equivocators_state_project IM
                final_descriptors'
                (finite_trace_last is tr)
              → composite_valid IM
                  (VLSM.l itemx)
                  (sx, input itemx)
                ∧ composite_transition IM
                    (VLSM.l itemx)
                    (sx, input itemx) =
                  (destination itemx,
                   output itemx)
        | None =>
            equivocators_state_project IM
              final_descriptors
              (destination
                 {|
                   l := l;
                   input := iom;
                   destination := s;
                   output := oom
                 |}) =
            equivocators_state_project IM
              final_descriptors'
              (finite_trace_last is tr)
        end
Hchar1: match oitem with
| Some itemx =>
    (∃ Hex : existing_equivocator_label
               (IM
                  (projT1
                     (VLSM.l
                      {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                      |})))
               (projT2
                  (VLSM.l
                     {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                     |})),
       existT
         (projT1
            (VLSM.l
               {|
                 l := l;
                 input := iom;
                 destination := s;
                 output := oom
               |}))
         (existing_equivocator_label_extract
            (IM
               (projT1
                  (VLSM.l
                     {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                     |})))
            (projT2
               (VLSM.l
                  {|
                    l := l;
                    input := iom;
                    destination := s;
                    output := oom
                  |})) Hex) = 
       VLSM.l itemx)
    ∧ input
        {|
          l := l;
          input := iom;
          destination := s;
          output := oom
        |} = input itemx
      ∧ output
          {|
            l := l;
            input := iom;
            destination := s;
            output := oom
          |} = output itemx
        ∧ equivocators_state_project IM
            final_descriptors
            (destination
               {|
                 l := l;
                 input := iom;
                 destination := s;
                 output := oom
               |}) = destination itemx
          ∧ final_descriptors'
              (projT1
                 (VLSM.l
                    {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                    |})) =
            equivocator_label_descriptor
              (projT2
                 (VLSM.l
                    {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                    |}))
| None => True
end
Hpr_x: equivocators_transition_item_project
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some (oitem, final_descriptors')
Hvx: valid l (finite_trace_last is tr, iom)
Htx: transition l (finite_trace_last is tr, iom) =
(s, oom)
∃ initial_descriptors : equivocator_descriptors IM,
  proper_equivocator_descriptors IM
    initial_descriptors is
  ∧ (let isX :=
       equivocators_state_project IM
         initial_descriptors is in
     let final_stateX :=
       equivocators_state_project IM final_descriptors
         (destination
            {|
              l := l;
              input := iom;
              destination := s;
              output := oom
            |}) in
     ∃ trX : list (composite_transition_item IM),
       equivocators_trace_project final_descriptors
         (tr ++
          [{|
             l := l;
             input := iom;
             destination := s;
             output := oom
           |}]) = Some (trX, initial_descriptors)
       ∧ finite_constrained_trace_init_to Free isX
           final_stateX trX)
    simpl in *.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
IHtr: ∀ final_descriptors : equivocator_descriptors
                      IM,
  proper_equivocator_descriptors IM
    final_descriptors (finite_trace_last is tr)
  → ∃ initial_descriptors : 
       equivocator_descriptors IM,
      proper_equivocator_descriptors IM
        initial_descriptors is
      ∧ (∃ trX : list
                   (composite_transition_item
                      IM),
           equivocators_trace_project
             final_descriptors tr =
           Some (trX, initial_descriptors)
           ∧ finite_constrained_trace_init_to
               Free
               (equivocators_state_project IM
                  initial_descriptors is)
               (equivocators_state_project IM
                  final_descriptors
                  (finite_trace_last is tr))
               trX)
Htr: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm FreeE) is
  (finite_trace_last is tr) tr
s: composite_state (equivocator_IM IM)
iom, oom: option message
l: composite_label (equivocator_IM IM)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  final_descriptors s
oitem: option (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hpr_app: ∀ (ieqv_descriptors
   eqv_descriptors : equivocator_descriptors
                      IM) 
  (trX : list (composite_transition_item IM)),
  equivocators_trace_project eqv_descriptors
    (tr ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}]) = Some (trX, ieqv_descriptors)
  ↔ (∃ (preX
        sufX : list
                 (composite_transition_item
                    IM)) 
       (eqv_descriptors' : 
        equivocator_descriptors IM),
       match
         equivocators_transition_item_project
           eqv_descriptors
           {|
             l := l;
             input := iom;
             destination := s;
             output := oom
           |}
       with
       | Some (Some item', odescriptor) =>
           Some ([item'], odescriptor)
       | Some (None, odescriptor) =>
           Some ([], odescriptor)
       | None => None
       end = Some (sufX, eqv_descriptors')
       ∧ equivocators_trace_project
           eqv_descriptors' tr =
         Some (preX, ieqv_descriptors)
         ∧ trX = preX ++ sufX)
Hchar2: proper_equivocator_descriptors IM
  final_descriptors'
  (finite_trace_last is tr)
∧ final_descriptors' =
  equivocator_descriptors_update IM
    final_descriptors (projT1 l)
    (final_descriptors' (projT1 l))
  ∧ finite_trace_last is tr =
    state_update (equivocator_IM IM) s
      (projT1 l)
      (finite_trace_last is tr (projT1 l))
    ∧ previous_state_descriptor_prop
        (IM (projT1 l))
        (final_descriptors (projT1 l))
        (finite_trace_last is tr (projT1 l))
        (final_descriptors' (projT1 l))
      ∧ match oitem with
        | Some itemx =>
            ∀ sx : composite_state IM,
              sx =
              equivocators_state_project IM
                final_descriptors'
                (finite_trace_last is tr)
              → (let (i, li) :=
                   VLSM.l itemx in
                 valid li (sx i, input itemx))
                ∧ (let 
                     (i, li) :=
                     VLSM.l itemx in
                   let 
                     (si', om') :=
                     transition li
                      (sx i, input itemx) in
                   (state_update IM sx i si',
                    om')) =
                  (destination itemx,
                   output itemx)
        | None =>
            equivocators_state_project IM
              final_descriptors s =
            equivocators_state_project IM
              final_descriptors'
              (finite_trace_last is tr)
        end
Hchar1: match oitem with
| Some itemx =>
    (∃ Hex : existing_equivocator_label
               (IM (projT1 l)) 
               (projT2 l),
       existT (projT1 l)
         (existing_equivocator_label_extract
            (IM (projT1 l)) 
            (projT2 l) Hex) = 
       VLSM.l itemx)
    ∧ iom = input itemx
      ∧ oom = output itemx
        ∧ equivocators_state_project IM
            final_descriptors s =
          destination itemx
          ∧ final_descriptors' (projT1 l) =
            equivocator_label_descriptor
              (projT2 l)
| None => True
end
Hpr_x: equivocators_transition_item_project
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some (oitem, final_descriptors')
Hvx: let (i, li) := l in
equivocator_valid (IM i) li
  (finite_trace_last is tr i, iom)
Htx: (let (i, li) := l in
 let (si', om') :=
   equivocator_transition 
     (IM i) li (finite_trace_last is tr i, iom) in
 (state_update (equivocator_IM IM)
    (finite_trace_last is tr) i si', om')) =
(s, oom)
∃ initial_descriptors : equivocator_descriptors IM,
  proper_equivocator_descriptors IM
    initial_descriptors is
  ∧ (∃ trX : list (composite_transition_item IM),
       equivocators_trace_project final_descriptors
         (tr ++
          [{|
             l := l;
             input := iom;
             destination := s;
             output := oom
           |}]) = Some (trX, initial_descriptors)
       ∧ finite_constrained_trace_init_to Free
           (equivocators_state_project IM
              initial_descriptors is)
           (equivocators_state_project IM
              final_descriptors s) trX)
    destruct Hchar2 as [Hproper' [Heq_final_descriptors' [Heq_last_tr [Hex_new Hchar2]]]].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
IHtr: ∀ final_descriptors : equivocator_descriptors
                      IM,
  proper_equivocator_descriptors IM
    final_descriptors (finite_trace_last is tr)
  → ∃ initial_descriptors : 
       equivocator_descriptors IM,
      proper_equivocator_descriptors IM
        initial_descriptors is
      ∧ (∃ trX : list
                   (composite_transition_item
                      IM),
           equivocators_trace_project
             final_descriptors tr =
           Some (trX, initial_descriptors)
           ∧ finite_constrained_trace_init_to
               Free
               (equivocators_state_project IM
                  initial_descriptors is)
               (equivocators_state_project IM
                  final_descriptors
                  (finite_trace_last is tr))
               trX)
Htr: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm FreeE) is
  (finite_trace_last is tr) tr
s: composite_state (equivocator_IM IM)
iom, oom: option message
l: composite_label (equivocator_IM IM)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  final_descriptors s
oitem: option (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hpr_app: ∀ (ieqv_descriptors
   eqv_descriptors : equivocator_descriptors
                      IM) 
  (trX : list (composite_transition_item IM)),
  equivocators_trace_project eqv_descriptors
    (tr ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}]) = Some (trX, ieqv_descriptors)
  ↔ (∃ (preX
        sufX : list
                 (composite_transition_item
                    IM)) 
       (eqv_descriptors' : 
        equivocator_descriptors IM),
       match
         equivocators_transition_item_project
           eqv_descriptors
           {|
             l := l;
             input := iom;
             destination := s;
             output := oom
           |}
       with
       | Some (Some item', odescriptor) =>
           Some ([item'], odescriptor)
       | Some (None, odescriptor) =>
           Some ([], odescriptor)
       | None => None
       end = Some (sufX, eqv_descriptors')
       ∧ equivocators_trace_project
           eqv_descriptors' tr =
         Some (preX, ieqv_descriptors)
         ∧ trX = preX ++ sufX)
Hproper': proper_equivocator_descriptors IM
  final_descriptors'
  (finite_trace_last is tr)
Heq_final_descriptors': final_descriptors' =
equivocator_descriptors_update
  IM final_descriptors
  (projT1 l)
  (final_descriptors'
     (projT1 l))
Heq_last_tr: finite_trace_last is tr =
state_update (equivocator_IM IM) s
  (projT1 l)
  (finite_trace_last is tr (projT1 l))
Hex_new: previous_state_descriptor_prop
  (IM (projT1 l))
  (final_descriptors (projT1 l))
  (finite_trace_last is tr (projT1 l))
  (final_descriptors' (projT1 l))
Hchar2: match oitem with
| Some itemx =>
    ∀ sx : composite_state IM,
      sx =
      equivocators_state_project IM
        final_descriptors'
        (finite_trace_last is tr)
      → (let (i, li) := VLSM.l itemx in
         valid li (sx i, input itemx))
        ∧ (let (i, li) := VLSM.l itemx in
           let (si', om') :=
             transition li
               (sx i, input itemx) in
           (state_update IM sx i si', om')) =
          (destination itemx, output itemx)
| None =>
    equivocators_state_project IM
      final_descriptors s =
    equivocators_state_project IM
      final_descriptors'
      (finite_trace_last is tr)
end
Hchar1: match oitem with
| Some itemx =>
    (∃ Hex : existing_equivocator_label
               (IM (projT1 l)) 
               (projT2 l),
       existT (projT1 l)
         (existing_equivocator_label_extract
            (IM (projT1 l)) 
            (projT2 l) Hex) = 
       VLSM.l itemx)
    ∧ iom = input itemx
      ∧ oom = output itemx
        ∧ equivocators_state_project IM
            final_descriptors s =
          destination itemx
          ∧ final_descriptors' (projT1 l) =
            equivocator_label_descriptor
              (projT2 l)
| None => True
end
Hpr_x: equivocators_transition_item_project
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some (oitem, final_descriptors')
Hvx: let (i, li) := l in
equivocator_valid (IM i) li
  (finite_trace_last is tr i, iom)
Htx: (let (i, li) := l in
 let (si', om') :=
   equivocator_transition 
     (IM i) li (finite_trace_last is tr i, iom) in
 (state_update (equivocator_IM IM)
    (finite_trace_last is tr) i si', om')) =
(s, oom)
∃ initial_descriptors : equivocator_descriptors IM,
  proper_equivocator_descriptors IM
    initial_descriptors is
  ∧ (∃ trX : list (composite_transition_item IM),
       equivocators_trace_project final_descriptors
         (tr ++
          [{|
             l := l;
             input := iom;
             destination := s;
             output := oom
           |}]) = Some (trX, initial_descriptors)
       ∧ finite_constrained_trace_init_to Free
           (equivocators_state_project IM
              initial_descriptors is)
           (equivocators_state_project IM
              final_descriptors s) trX)
    specialize (IHtr _ Hproper').message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
final_descriptors': equivocator_descriptors IM
IHtr: ∃ initial_descriptors : 
   equivocator_descriptors IM,
  proper_equivocator_descriptors IM
    initial_descriptors is
  ∧ (∃ trX : list
               (composite_transition_item IM),
       equivocators_trace_project
         final_descriptors' tr =
       Some (trX, initial_descriptors)
       ∧ finite_constrained_trace_init_to Free
           (equivocators_state_project IM
              initial_descriptors is)
           (equivocators_state_project IM
              final_descriptors'
              (finite_trace_last is tr)) trX)
Htr: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm FreeE) is
  (finite_trace_last is tr) tr
s: composite_state (equivocator_IM IM)
iom, oom: option message
l: composite_label (equivocator_IM IM)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  final_descriptors s
oitem: option (composite_transition_item IM)
Hpr_app: ∀ (ieqv_descriptors
   eqv_descriptors : equivocator_descriptors
                      IM) 
  (trX : list (composite_transition_item IM)),
  equivocators_trace_project eqv_descriptors
    (tr ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}]) = Some (trX, ieqv_descriptors)
  ↔ (∃ (preX
        sufX : list
                 (composite_transition_item
                    IM)) 
       (eqv_descriptors' : 
        equivocator_descriptors IM),
       match
         equivocators_transition_item_project
           eqv_descriptors
           {|
             l := l;
             input := iom;
             destination := s;
             output := oom
           |}
       with
       | Some (Some item', odescriptor) =>
           Some ([item'], odescriptor)
       | Some (None, odescriptor) =>
           Some ([], odescriptor)
       | None => None
       end = Some (sufX, eqv_descriptors')
       ∧ equivocators_trace_project
           eqv_descriptors' tr =
         Some (preX, ieqv_descriptors)
         ∧ trX = preX ++ sufX)
Hproper': proper_equivocator_descriptors IM
  final_descriptors'
  (finite_trace_last is tr)
Heq_final_descriptors': final_descriptors' =
equivocator_descriptors_update
  IM final_descriptors
  (projT1 l)
  (final_descriptors'
     (projT1 l))
Heq_last_tr: finite_trace_last is tr =
state_update (equivocator_IM IM) s
  (projT1 l)
  (finite_trace_last is tr (projT1 l))
Hex_new: previous_state_descriptor_prop
  (IM (projT1 l))
  (final_descriptors (projT1 l))
  (finite_trace_last is tr (projT1 l))
  (final_descriptors' (projT1 l))
Hchar2: match oitem with
| Some itemx =>
    ∀ sx : composite_state IM,
      sx =
      equivocators_state_project IM
        final_descriptors'
        (finite_trace_last is tr)
      → (let (i, li) := VLSM.l itemx in
         valid li (sx i, input itemx))
        ∧ (let (i, li) := VLSM.l itemx in
           let (si', om') :=
             transition li
               (sx i, input itemx) in
           (state_update IM sx i si', om')) =
          (destination itemx, output itemx)
| None =>
    equivocators_state_project IM
      final_descriptors s =
    equivocators_state_project IM
      final_descriptors'
      (finite_trace_last is tr)
end
Hchar1: match oitem with
| Some itemx =>
    (∃ Hex : existing_equivocator_label
               (IM (projT1 l)) 
               (projT2 l),
       existT (projT1 l)
         (existing_equivocator_label_extract
            (IM (projT1 l)) 
            (projT2 l) Hex) = 
       VLSM.l itemx)
    ∧ iom = input itemx
      ∧ oom = output itemx
        ∧ equivocators_state_project IM
            final_descriptors s =
          destination itemx
          ∧ final_descriptors' (projT1 l) =
            equivocator_label_descriptor
              (projT2 l)
| None => True
end
Hpr_x: equivocators_transition_item_project
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some (oitem, final_descriptors')
Hvx: let (i, li) := l in
equivocator_valid (IM i) li
  (finite_trace_last is tr i, iom)
Htx: (let (i, li) := l in
 let (si', om') :=
   equivocator_transition 
     (IM i) li (finite_trace_last is tr i, iom) in
 (state_update (equivocator_IM IM)
    (finite_trace_last is tr) i si', om')) =
(s, oom)
∃ initial_descriptors : equivocator_descriptors IM,
  proper_equivocator_descriptors IM
    initial_descriptors is
  ∧ (∃ trX : list (composite_transition_item IM),
       equivocators_trace_project final_descriptors
         (tr ++
          [{|
             l := l;
             input := iom;
             destination := s;
             output := oom
           |}]) = Some (trX, initial_descriptors)
       ∧ finite_constrained_trace_init_to Free
           (equivocators_state_project IM
              initial_descriptors is)
           (equivocators_state_project IM
              final_descriptors s) trX)
    destruct IHtr as [initial_descriptors [Hproper_initial [trX [Hpr_tr HtrX]]]].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
final_descriptors', initial_descriptors: equivocator_descriptors IM
Hproper_initial: proper_equivocator_descriptors IM
  initial_descriptors is
trX: list (composite_transition_item IM)
Hpr_tr: equivocators_trace_project final_descriptors'
  tr = Some (trX, initial_descriptors)
HtrX: finite_constrained_trace_init_to Free
  (equivocators_state_project IM
     initial_descriptors is)
  (equivocators_state_project IM
     final_descriptors'
     (finite_trace_last is tr)) trX
Htr: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm FreeE) is
  (finite_trace_last is tr) tr
s: composite_state (equivocator_IM IM)
iom, oom: option message
l: composite_label (equivocator_IM IM)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  final_descriptors s
oitem: option (composite_transition_item IM)
Hpr_app: ∀ (ieqv_descriptors
   eqv_descriptors : equivocator_descriptors
                      IM) 
  (trX : list (composite_transition_item IM)),
  equivocators_trace_project eqv_descriptors
    (tr ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}]) = Some (trX, ieqv_descriptors)
  ↔ (∃ (preX
        sufX : list
                 (composite_transition_item
                    IM)) 
       (eqv_descriptors' : 
        equivocator_descriptors IM),
       match
         equivocators_transition_item_project
           eqv_descriptors
           {|
             l := l;
             input := iom;
             destination := s;
             output := oom
           |}
       with
       | Some (Some item', odescriptor) =>
           Some ([item'], odescriptor)
       | Some (None, odescriptor) =>
           Some ([], odescriptor)
       | None => None
       end = Some (sufX, eqv_descriptors')
       ∧ equivocators_trace_project
           eqv_descriptors' tr =
         Some (preX, ieqv_descriptors)
         ∧ trX = preX ++ sufX)
Hproper': proper_equivocator_descriptors IM
  final_descriptors'
  (finite_trace_last is tr)
Heq_final_descriptors': final_descriptors' =
equivocator_descriptors_update
  IM final_descriptors
  (projT1 l)
  (final_descriptors'
     (projT1 l))
Heq_last_tr: finite_trace_last is tr =
state_update (equivocator_IM IM) s
  (projT1 l)
  (finite_trace_last is tr (projT1 l))
Hex_new: previous_state_descriptor_prop
  (IM (projT1 l))
  (final_descriptors (projT1 l))
  (finite_trace_last is tr (projT1 l))
  (final_descriptors' (projT1 l))
Hchar2: match oitem with
| Some itemx =>
    ∀ sx : composite_state IM,
      sx =
      equivocators_state_project IM
        final_descriptors'
        (finite_trace_last is tr)
      → (let (i, li) := VLSM.l itemx in
         valid li (sx i, input itemx))
        ∧ (let (i, li) := VLSM.l itemx in
           let (si', om') :=
             transition li
               (sx i, input itemx) in
           (state_update IM sx i si', om')) =
          (destination itemx, output itemx)
| None =>
    equivocators_state_project IM
      final_descriptors s =
    equivocators_state_project IM
      final_descriptors'
      (finite_trace_last is tr)
end
Hchar1: match oitem with
| Some itemx =>
    (∃ Hex : existing_equivocator_label
               (IM (projT1 l)) 
               (projT2 l),
       existT (projT1 l)
         (existing_equivocator_label_extract
            (IM (projT1 l)) 
            (projT2 l) Hex) = 
       VLSM.l itemx)
    ∧ iom = input itemx
      ∧ oom = output itemx
        ∧ equivocators_state_project IM
            final_descriptors s =
          destination itemx
          ∧ final_descriptors' (projT1 l) =
            equivocator_label_descriptor
              (projT2 l)
| None => True
end
Hpr_x: equivocators_transition_item_project
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some (oitem, final_descriptors')
Hvx: let (i, li) := l in
equivocator_valid (IM i) li
  (finite_trace_last is tr i, iom)
Htx: (let (i, li) := l in
 let (si', om') :=
   equivocator_transition 
     (IM i) li (finite_trace_last is tr i, iom) in
 (state_update (equivocator_IM IM)
    (finite_trace_last is tr) i si', om')) =
(s, oom)
∃ initial_descriptors : equivocator_descriptors IM,
  proper_equivocator_descriptors IM
    initial_descriptors is
  ∧ (∃ trX : list (composite_transition_item IM),
       equivocators_trace_project final_descriptors
         (tr ++
          [{|
             l := l;
             input := iom;
             destination := s;
             output := oom
           |}]) = Some (trX, initial_descriptors)
       ∧ finite_constrained_trace_init_to Free
           (equivocators_state_project IM
              initial_descriptors is)
           (equivocators_state_project IM
              final_descriptors s) trX)
    exists initial_descriptors.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
final_descriptors', initial_descriptors: equivocator_descriptors IM
Hproper_initial: proper_equivocator_descriptors IM
  initial_descriptors is
trX: list (composite_transition_item IM)
Hpr_tr: equivocators_trace_project final_descriptors'
  tr = Some (trX, initial_descriptors)
HtrX: finite_constrained_trace_init_to Free
  (equivocators_state_project IM
     initial_descriptors is)
  (equivocators_state_project IM
     final_descriptors'
     (finite_trace_last is tr)) trX
Htr: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm FreeE) is
  (finite_trace_last is tr) tr
s: composite_state (equivocator_IM IM)
iom, oom: option message
l: composite_label (equivocator_IM IM)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  final_descriptors s
oitem: option (composite_transition_item IM)
Hpr_app: ∀ (ieqv_descriptors
   eqv_descriptors : equivocator_descriptors
                      IM) 
  (trX : list (composite_transition_item IM)),
  equivocators_trace_project eqv_descriptors
    (tr ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}]) = Some (trX, ieqv_descriptors)
  ↔ (∃ (preX
        sufX : list
                 (composite_transition_item
                    IM)) 
       (eqv_descriptors' : 
        equivocator_descriptors IM),
       match
         equivocators_transition_item_project
           eqv_descriptors
           {|
             l := l;
             input := iom;
             destination := s;
             output := oom
           |}
       with
       | Some (Some item', odescriptor) =>
           Some ([item'], odescriptor)
       | Some (None, odescriptor) =>
           Some ([], odescriptor)
       | None => None
       end = Some (sufX, eqv_descriptors')
       ∧ equivocators_trace_project
           eqv_descriptors' tr =
         Some (preX, ieqv_descriptors)
         ∧ trX = preX ++ sufX)
Hproper': proper_equivocator_descriptors IM
  final_descriptors'
  (finite_trace_last is tr)
Heq_final_descriptors': final_descriptors' =
equivocator_descriptors_update
  IM final_descriptors
  (projT1 l)
  (final_descriptors'
     (projT1 l))
Heq_last_tr: finite_trace_last is tr =
state_update (equivocator_IM IM) s
  (projT1 l)
  (finite_trace_last is tr (projT1 l))
Hex_new: previous_state_descriptor_prop
  (IM (projT1 l))
  (final_descriptors (projT1 l))
  (finite_trace_last is tr (projT1 l))
  (final_descriptors' (projT1 l))
Hchar2: match oitem with
| Some itemx =>
    ∀ sx : composite_state IM,
      sx =
      equivocators_state_project IM
        final_descriptors'
        (finite_trace_last is tr)
      → (let (i, li) := VLSM.l itemx in
         valid li (sx i, input itemx))
        ∧ (let (i, li) := VLSM.l itemx in
           let (si', om') :=
             transition li
               (sx i, input itemx) in
           (state_update IM sx i si', om')) =
          (destination itemx, output itemx)
| None =>
    equivocators_state_project IM
      final_descriptors s =
    equivocators_state_project IM
      final_descriptors'
      (finite_trace_last is tr)
end
Hchar1: match oitem with
| Some itemx =>
    (∃ Hex : existing_equivocator_label
               (IM (projT1 l)) 
               (projT2 l),
       existT (projT1 l)
         (existing_equivocator_label_extract
            (IM (projT1 l)) 
            (projT2 l) Hex) = 
       VLSM.l itemx)
    ∧ iom = input itemx
      ∧ oom = output itemx
        ∧ equivocators_state_project IM
            final_descriptors s =
          destination itemx
          ∧ final_descriptors' (projT1 l) =
            equivocator_label_descriptor
              (projT2 l)
| None => True
end
Hpr_x: equivocators_transition_item_project
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some (oitem, final_descriptors')
Hvx: let (i, li) := l in
equivocator_valid (IM i) li
  (finite_trace_last is tr i, iom)
Htx: (let (i, li) := l in
 let (si', om') :=
   equivocator_transition 
     (IM i) li (finite_trace_last is tr i, iom) in
 (state_update (equivocator_IM IM)
    (finite_trace_last is tr) i si', om')) =
(s, oom)
proper_equivocator_descriptors IM initial_descriptors
  is
∧ (∃ trX : list (composite_transition_item IM),
     equivocators_trace_project final_descriptors
       (tr ++
        [{|
           l := l;
           input := iom;
           destination := s;
           output := oom
         |}]) = Some (trX, initial_descriptors)
     ∧ finite_constrained_trace_init_to Free
         (equivocators_state_project IM
            initial_descriptors is)
         (equivocators_state_project IM
            final_descriptors s) trX)
    split; [done |].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
final_descriptors', initial_descriptors: equivocator_descriptors IM
Hproper_initial: proper_equivocator_descriptors IM
  initial_descriptors is
trX: list (composite_transition_item IM)
Hpr_tr: equivocators_trace_project final_descriptors'
  tr = Some (trX, initial_descriptors)
HtrX: finite_constrained_trace_init_to Free
  (equivocators_state_project IM
     initial_descriptors is)
  (equivocators_state_project IM
     final_descriptors'
     (finite_trace_last is tr)) trX
Htr: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm FreeE) is
  (finite_trace_last is tr) tr
s: composite_state (equivocator_IM IM)
iom, oom: option message
l: composite_label (equivocator_IM IM)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  final_descriptors s
oitem: option (composite_transition_item IM)
Hpr_app: ∀ (ieqv_descriptors
   eqv_descriptors : equivocator_descriptors
                      IM) 
  (trX : list (composite_transition_item IM)),
  equivocators_trace_project eqv_descriptors
    (tr ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}]) = Some (trX, ieqv_descriptors)
  ↔ (∃ (preX
        sufX : list
                 (composite_transition_item
                    IM)) 
       (eqv_descriptors' : 
        equivocator_descriptors IM),
       match
         equivocators_transition_item_project
           eqv_descriptors
           {|
             l := l;
             input := iom;
             destination := s;
             output := oom
           |}
       with
       | Some (Some item', odescriptor) =>
           Some ([item'], odescriptor)
       | Some (None, odescriptor) =>
           Some ([], odescriptor)
       | None => None
       end = Some (sufX, eqv_descriptors')
       ∧ equivocators_trace_project
           eqv_descriptors' tr =
         Some (preX, ieqv_descriptors)
         ∧ trX = preX ++ sufX)
Hproper': proper_equivocator_descriptors IM
  final_descriptors'
  (finite_trace_last is tr)
Heq_final_descriptors': final_descriptors' =
equivocator_descriptors_update
  IM final_descriptors
  (projT1 l)
  (final_descriptors'
     (projT1 l))
Heq_last_tr: finite_trace_last is tr =
state_update (equivocator_IM IM) s
  (projT1 l)
  (finite_trace_last is tr (projT1 l))
Hex_new: previous_state_descriptor_prop
  (IM (projT1 l))
  (final_descriptors (projT1 l))
  (finite_trace_last is tr (projT1 l))
  (final_descriptors' (projT1 l))
Hchar2: match oitem with
| Some itemx =>
    ∀ sx : composite_state IM,
      sx =
      equivocators_state_project IM
        final_descriptors'
        (finite_trace_last is tr)
      → (let (i, li) := VLSM.l itemx in
         valid li (sx i, input itemx))
        ∧ (let (i, li) := VLSM.l itemx in
           let (si', om') :=
             transition li
               (sx i, input itemx) in
           (state_update IM sx i si', om')) =
          (destination itemx, output itemx)
| None =>
    equivocators_state_project IM
      final_descriptors s =
    equivocators_state_project IM
      final_descriptors'
      (finite_trace_last is tr)
end
Hchar1: match oitem with
| Some itemx =>
    (∃ Hex : existing_equivocator_label
               (IM (projT1 l)) 
               (projT2 l),
       existT (projT1 l)
         (existing_equivocator_label_extract
            (IM (projT1 l)) 
            (projT2 l) Hex) = 
       VLSM.l itemx)
    ∧ iom = input itemx
      ∧ oom = output itemx
        ∧ equivocators_state_project IM
            final_descriptors s =
          destination itemx
          ∧ final_descriptors' (projT1 l) =
            equivocator_label_descriptor
              (projT2 l)
| None => True
end
Hpr_x: equivocators_transition_item_project
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some (oitem, final_descriptors')
Hvx: let (i, li) := l in
equivocator_valid (IM i) li
  (finite_trace_last is tr i, iom)
Htx: (let (i, li) := l in
 let (si', om') :=
   equivocator_transition 
     (IM i) li (finite_trace_last is tr i, iom) in
 (state_update (equivocator_IM IM)
    (finite_trace_last is tr) i si', om')) =
(s, oom)
∃ trX : list (composite_transition_item IM),
  equivocators_trace_project final_descriptors
    (tr ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}]) = Some (trX, initial_descriptors)
  ∧ finite_constrained_trace_init_to Free
      (equivocators_state_project IM
         initial_descriptors is)
      (equivocators_state_project IM final_descriptors
         s) trX
    specialize (Hpr_app initial_descriptors final_descriptors).message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
final_descriptors', initial_descriptors: equivocator_descriptors IM
Hproper_initial: proper_equivocator_descriptors IM
  initial_descriptors is
trX: list (composite_transition_item IM)
Hpr_tr: equivocators_trace_project final_descriptors'
  tr = Some (trX, initial_descriptors)
HtrX: finite_constrained_trace_init_to Free
  (equivocators_state_project IM
     initial_descriptors is)
  (equivocators_state_project IM
     final_descriptors'
     (finite_trace_last is tr)) trX
Htr: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm FreeE) is
  (finite_trace_last is tr) tr
s: composite_state (equivocator_IM IM)
iom, oom: option message
l: composite_label (equivocator_IM IM)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  final_descriptors s
oitem: option (composite_transition_item IM)
Hpr_app: ∀ trX : list (composite_transition_item IM),
  equivocators_trace_project
    final_descriptors
    (tr ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}]) = Some (trX, initial_descriptors)
  ↔ (∃ (preX
        sufX : list
                 (composite_transition_item
                    IM)) 
       (eqv_descriptors' : 
        equivocator_descriptors IM),
       match
         equivocators_transition_item_project
           final_descriptors
           {|
             l := l;
             input := iom;
             destination := s;
             output := oom
           |}
       with
       | Some (Some item', odescriptor) =>
           Some ([item'], odescriptor)
       | Some (None, odescriptor) =>
           Some ([], odescriptor)
       | None => None
       end = Some (sufX, eqv_descriptors')
       ∧ equivocators_trace_project
           eqv_descriptors' tr =
         Some (preX, initial_descriptors)
         ∧ trX = preX ++ sufX)
Hproper': proper_equivocator_descriptors IM
  final_descriptors'
  (finite_trace_last is tr)
Heq_final_descriptors': final_descriptors' =
equivocator_descriptors_update
  IM final_descriptors
  (projT1 l)
  (final_descriptors'
     (projT1 l))
Heq_last_tr: finite_trace_last is tr =
state_update (equivocator_IM IM) s
  (projT1 l)
  (finite_trace_last is tr (projT1 l))
Hex_new: previous_state_descriptor_prop
  (IM (projT1 l))
  (final_descriptors (projT1 l))
  (finite_trace_last is tr (projT1 l))
  (final_descriptors' (projT1 l))
Hchar2: match oitem with
| Some itemx =>
    ∀ sx : composite_state IM,
      sx =
      equivocators_state_project IM
        final_descriptors'
        (finite_trace_last is tr)
      → (let (i, li) := VLSM.l itemx in
         valid li (sx i, input itemx))
        ∧ (let (i, li) := VLSM.l itemx in
           let (si', om') :=
             transition li
               (sx i, input itemx) in
           (state_update IM sx i si', om')) =
          (destination itemx, output itemx)
| None =>
    equivocators_state_project IM
      final_descriptors s =
    equivocators_state_project IM
      final_descriptors'
      (finite_trace_last is tr)
end
Hchar1: match oitem with
| Some itemx =>
    (∃ Hex : existing_equivocator_label
               (IM (projT1 l)) 
               (projT2 l),
       existT (projT1 l)
         (existing_equivocator_label_extract
            (IM (projT1 l)) 
            (projT2 l) Hex) = 
       VLSM.l itemx)
    ∧ iom = input itemx
      ∧ oom = output itemx
        ∧ equivocators_state_project IM
            final_descriptors s =
          destination itemx
          ∧ final_descriptors' (projT1 l) =
            equivocator_label_descriptor
              (projT2 l)
| None => True
end
Hpr_x: equivocators_transition_item_project
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some (oitem, final_descriptors')
Hvx: let (i, li) := l in
equivocator_valid (IM i) li
  (finite_trace_last is tr i, iom)
Htx: (let (i, li) := l in
 let (si', om') :=
   equivocator_transition 
     (IM i) li (finite_trace_last is tr i, iom) in
 (state_update (equivocator_IM IM)
    (finite_trace_last is tr) i si', om')) =
(s, oom)
∃ trX : list (composite_transition_item IM),
  equivocators_trace_project final_descriptors
    (tr ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}]) = Some (trX, initial_descriptors)
  ∧ finite_constrained_trace_init_to Free
      (equivocators_state_project IM
         initial_descriptors is)
      (equivocators_state_project IM final_descriptors
         s) trX
    destruct oitem as [item |].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
final_descriptors', initial_descriptors: equivocator_descriptors IM
Hproper_initial: proper_equivocator_descriptors IM
  initial_descriptors is
trX: list (composite_transition_item IM)
Hpr_tr: equivocators_trace_project final_descriptors'
  tr = Some (trX, initial_descriptors)
HtrX: finite_constrained_trace_init_to Free
  (equivocators_state_project IM
     initial_descriptors is)
  (equivocators_state_project IM
     final_descriptors'
     (finite_trace_last is tr)) trX
Htr: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm FreeE) is
  (finite_trace_last is tr) tr
s: composite_state (equivocator_IM IM)
iom, oom: option message
l: composite_label (equivocator_IM IM)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  final_descriptors s
item: composite_transition_item IM
Hpr_app: ∀ trX : list (composite_transition_item IM),
  equivocators_trace_project
    final_descriptors
    (tr ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}]) = Some (trX, initial_descriptors)
  ↔ (∃ (preX
        sufX : list
                 (composite_transition_item
                    IM)) 
       (eqv_descriptors' : 
        equivocator_descriptors IM),
       match
         equivocators_transition_item_project
           final_descriptors
           {|
             l := l;
             input := iom;
             destination := s;
             output := oom
           |}
       with
       | Some (Some item', odescriptor) =>
           Some ([item'], odescriptor)
       | Some (None, odescriptor) =>
           Some ([], odescriptor)
       | None => None
       end = Some (sufX, eqv_descriptors')
       ∧ equivocators_trace_project
           eqv_descriptors' tr =
         Some (preX, initial_descriptors)
         ∧ trX = preX ++ sufX)
Hproper': proper_equivocator_descriptors IM
  final_descriptors'
  (finite_trace_last is tr)
Heq_final_descriptors': final_descriptors' =
equivocator_descriptors_update
  IM final_descriptors
  (projT1 l)
  (final_descriptors'
     (projT1 l))
Heq_last_tr: finite_trace_last is tr =
state_update (equivocator_IM IM) s
  (projT1 l)
  (finite_trace_last is tr (projT1 l))
Hex_new: previous_state_descriptor_prop
  (IM (projT1 l))
  (final_descriptors (projT1 l))
  (finite_trace_last is tr (projT1 l))
  (final_descriptors' (projT1 l))
Hchar2: ∀ sx : composite_state IM,
  sx =
  equivocators_state_project IM
    final_descriptors'
    (finite_trace_last is tr)
  → (let (i, li) := VLSM.l item in
     valid li (sx i, input item))
    ∧ (let (i, li) := VLSM.l item in
       let (si', om') :=
         transition li (sx i, input item) in
       (state_update IM sx i si', om')) =
      (destination item, output item)
Hchar1: (∃ Hex : existing_equivocator_label
           (IM (projT1 l)) 
           (projT2 l),
   existT (projT1 l)
     (existing_equivocator_label_extract
        (IM (projT1 l)) 
        (projT2 l) Hex) = 
   VLSM.l item)
∧ iom = input item
  ∧ oom = output item
    ∧ equivocators_state_project IM
        final_descriptors s =
      destination item
      ∧ final_descriptors' (projT1 l) =
        equivocator_label_descriptor
          (projT2 l)
Hpr_x: equivocators_transition_item_project
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some (Some item, final_descriptors')
Hvx: let (i, li) := l in
equivocator_valid (IM i) li
  (finite_trace_last is tr i, iom)
Htx: (let (i, li) := l in
 let (si', om') :=
   equivocator_transition 
     (IM i) li (finite_trace_last is tr i, iom) in
 (state_update (equivocator_IM IM)
    (finite_trace_last is tr) i si', om')) =
(s, oom)
∃ trX : list (composite_transition_item IM),
  equivocators_trace_project final_descriptors
    (tr ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}]) = Some (trX, initial_descriptors)
  ∧ finite_constrained_trace_init_to Free
      (equivocators_state_project IM
         initial_descriptors is)
      (equivocators_state_project IM final_descriptors
         s) trX
message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
final_descriptors', initial_descriptors: equivocator_descriptors IM
Hproper_initial: proper_equivocator_descriptors IM
  initial_descriptors is
trX: list (composite_transition_item IM)
Hpr_tr: equivocators_trace_project final_descriptors'
  tr = Some (trX, initial_descriptors)
HtrX: finite_constrained_trace_init_to Free
  (equivocators_state_project IM
     initial_descriptors is)
  (equivocators_state_project IM
     final_descriptors'
     (finite_trace_last is tr)) trX
Htr: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm FreeE) is
  (finite_trace_last is tr) tr
s: composite_state (equivocator_IM IM)
iom, oom: option message
l: composite_label (equivocator_IM IM)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  final_descriptors s
Hpr_app: ∀ trX : list (composite_transition_item IM),
  equivocators_trace_project
    final_descriptors
    (tr ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}]) = Some (trX, initial_descriptors)
  ↔ (∃ (preX
        sufX : list
                 (composite_transition_item
                    IM)) 
       (eqv_descriptors' : 
        equivocator_descriptors IM),
       match
         equivocators_transition_item_project
           final_descriptors
           {|
             l := l;
             input := iom;
             destination := s;
             output := oom
           |}
       with
       | Some (Some item', odescriptor) =>
           Some ([item'], odescriptor)
       | Some (None, odescriptor) =>
           Some ([], odescriptor)
       | None => None
       end = Some (sufX, eqv_descriptors')
       ∧ equivocators_trace_project
           eqv_descriptors' tr =
         Some (preX, initial_descriptors)
         ∧ trX = preX ++ sufX)
Hproper': proper_equivocator_descriptors IM
  final_descriptors'
  (finite_trace_last is tr)
Heq_final_descriptors': final_descriptors' =
equivocator_descriptors_update
  IM final_descriptors
  (projT1 l)
  (final_descriptors'
     (projT1 l))
Heq_last_tr: finite_trace_last is tr =
state_update (equivocator_IM IM) s
  (projT1 l)
  (finite_trace_last is tr (projT1 l))
Hex_new: previous_state_descriptor_prop
  (IM (projT1 l))
  (final_descriptors (projT1 l))
  (finite_trace_last is tr (projT1 l))
  (final_descriptors' (projT1 l))
Hchar2: equivocators_state_project IM
  final_descriptors s =
equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr)
Hchar1: True
Hpr_x: equivocators_transition_item_project
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some (None, final_descriptors')
Hvx: let (i, li) := l in
equivocator_valid (IM i) li
  (finite_trace_last is tr i, iom)
Htx: (let (i, li) := l in
 let (si', om') :=
   equivocator_transition 
     (IM i) li (finite_trace_last is tr i, iom) in
 (state_update (equivocator_IM IM)
    (finite_trace_last is tr) i si', om')) =
(s, oom)
∃ trX : list (composite_transition_item IM),
  equivocators_trace_project final_descriptors
    (tr ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}]) = Some (trX, initial_descriptors)
  ∧ finite_constrained_trace_init_to Free
      (equivocators_state_project IM
         initial_descriptors is)
      (equivocators_state_project IM final_descriptors
         s) trX
    +message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
final_descriptors', initial_descriptors: equivocator_descriptors IM
Hproper_initial: proper_equivocator_descriptors IM
  initial_descriptors is
trX: list (composite_transition_item IM)
Hpr_tr: equivocators_trace_project final_descriptors'
  tr = Some (trX, initial_descriptors)
HtrX: finite_constrained_trace_init_to Free
  (equivocators_state_project IM
     initial_descriptors is)
  (equivocators_state_project IM
     final_descriptors'
     (finite_trace_last is tr)) trX
Htr: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm FreeE) is
  (finite_trace_last is tr) tr
s: composite_state (equivocator_IM IM)
iom, oom: option message
l: composite_label (equivocator_IM IM)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  final_descriptors s
item: composite_transition_item IM
Hpr_app: ∀ trX : list (composite_transition_item IM),
  equivocators_trace_project
    final_descriptors
    (tr ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}]) = Some (trX, initial_descriptors)
  ↔ (∃ (preX
        sufX : list
                 (composite_transition_item
                    IM)) 
       (eqv_descriptors' : 
        equivocator_descriptors IM),
       match
         equivocators_transition_item_project
           final_descriptors
           {|
             l := l;
             input := iom;
             destination := s;
             output := oom
           |}
       with
       | Some (Some item', odescriptor) =>
           Some ([item'], odescriptor)
       | Some (None, odescriptor) =>
           Some ([], odescriptor)
       | None => None
       end = Some (sufX, eqv_descriptors')
       ∧ equivocators_trace_project
           eqv_descriptors' tr =
         Some (preX, initial_descriptors)
         ∧ trX = preX ++ sufX)
Hproper': proper_equivocator_descriptors IM
  final_descriptors'
  (finite_trace_last is tr)
Heq_final_descriptors': final_descriptors' =
equivocator_descriptors_update
  IM final_descriptors
  (projT1 l)
  (final_descriptors'
     (projT1 l))
Heq_last_tr: finite_trace_last is tr =
state_update (equivocator_IM IM) s
  (projT1 l)
  (finite_trace_last is tr (projT1 l))
Hex_new: previous_state_descriptor_prop
  (IM (projT1 l))
  (final_descriptors (projT1 l))
  (finite_trace_last is tr (projT1 l))
  (final_descriptors' (projT1 l))
Hchar2: ∀ sx : composite_state IM,
  sx =
  equivocators_state_project IM
    final_descriptors'
    (finite_trace_last is tr)
  → (let (i, li) := VLSM.l item in
     valid li (sx i, input item))
    ∧ (let (i, li) := VLSM.l item in
       let (si', om') :=
         transition li (sx i, input item) in
       (state_update IM sx i si', om')) =
      (destination item, output item)
Hchar1: (∃ Hex : existing_equivocator_label
           (IM (projT1 l)) 
           (projT2 l),
   existT (projT1 l)
     (existing_equivocator_label_extract
        (IM (projT1 l)) 
        (projT2 l) Hex) = 
   VLSM.l item)
∧ iom = input item
  ∧ oom = output item
    ∧ equivocators_state_project IM
        final_descriptors s =
      destination item
      ∧ final_descriptors' (projT1 l) =
        equivocator_label_descriptor
          (projT2 l)
Hpr_x: equivocators_transition_item_project
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some (Some item, final_descriptors')
Hvx: let (i, li) := l in
equivocator_valid (IM i) li
  (finite_trace_last is tr i, iom)
Htx: (let (i, li) := l in
 let (si', om') :=
   equivocator_transition 
     (IM i) li (finite_trace_last is tr i, iom) in
 (state_update (equivocator_IM IM)
    (finite_trace_last is tr) i si', om')) =
(s, oom)
∃ trX : list (composite_transition_item IM),
  equivocators_trace_project final_descriptors
    (tr ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}]) = Some (trX, initial_descriptors)
  ∧ finite_constrained_trace_init_to Free
      (equivocators_state_project IM
         initial_descriptors is)
      (equivocators_state_project IM final_descriptors
         s) trX exists (trX ++ [item]).message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
final_descriptors', initial_descriptors: equivocator_descriptors IM
Hproper_initial: proper_equivocator_descriptors IM
  initial_descriptors is
trX: list (composite_transition_item IM)
Hpr_tr: equivocators_trace_project final_descriptors'
  tr = Some (trX, initial_descriptors)
HtrX: finite_constrained_trace_init_to Free
  (equivocators_state_project IM
     initial_descriptors is)
  (equivocators_state_project IM
     final_descriptors'
     (finite_trace_last is tr)) trX
Htr: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm FreeE) is
  (finite_trace_last is tr) tr
s: composite_state (equivocator_IM IM)
iom, oom: option message
l: composite_label (equivocator_IM IM)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  final_descriptors s
item: composite_transition_item IM
Hpr_app: ∀ trX : list (composite_transition_item IM),
  equivocators_trace_project
    final_descriptors
    (tr ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}]) = Some (trX, initial_descriptors)
  ↔ (∃ (preX
        sufX : list
                 (composite_transition_item
                    IM)) 
       (eqv_descriptors' : 
        equivocator_descriptors IM),
       match
         equivocators_transition_item_project
           final_descriptors
           {|
             l := l;
             input := iom;
             destination := s;
             output := oom
           |}
       with
       | Some (Some item', odescriptor) =>
           Some ([item'], odescriptor)
       | Some (None, odescriptor) =>
           Some ([], odescriptor)
       | None => None
       end = Some (sufX, eqv_descriptors')
       ∧ equivocators_trace_project
           eqv_descriptors' tr =
         Some (preX, initial_descriptors)
         ∧ trX = preX ++ sufX)
Hproper': proper_equivocator_descriptors IM
  final_descriptors'
  (finite_trace_last is tr)
Heq_final_descriptors': final_descriptors' =
equivocator_descriptors_update
  IM final_descriptors
  (projT1 l)
  (final_descriptors'
     (projT1 l))
Heq_last_tr: finite_trace_last is tr =
state_update (equivocator_IM IM) s
  (projT1 l)
  (finite_trace_last is tr (projT1 l))
Hex_new: previous_state_descriptor_prop
  (IM (projT1 l))
  (final_descriptors (projT1 l))
  (finite_trace_last is tr (projT1 l))
  (final_descriptors' (projT1 l))
Hchar2: ∀ sx : composite_state IM,
  sx =
  equivocators_state_project IM
    final_descriptors'
    (finite_trace_last is tr)
  → (let (i, li) := VLSM.l item in
     valid li (sx i, input item))
    ∧ (let (i, li) := VLSM.l item in
       let (si', om') :=
         transition li (sx i, input item) in
       (state_update IM sx i si', om')) =
      (destination item, output item)
Hchar1: (∃ Hex : existing_equivocator_label
           (IM (projT1 l)) 
           (projT2 l),
   existT (projT1 l)
     (existing_equivocator_label_extract
        (IM (projT1 l)) 
        (projT2 l) Hex) = 
   VLSM.l item)
∧ iom = input item
  ∧ oom = output item
    ∧ equivocators_state_project IM
        final_descriptors s =
      destination item
      ∧ final_descriptors' (projT1 l) =
        equivocator_label_descriptor
          (projT2 l)
Hpr_x: equivocators_transition_item_project
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some (Some item, final_descriptors')
Hvx: let (i, li) := l in
equivocator_valid (IM i) li
  (finite_trace_last is tr i, iom)
Htx: (let (i, li) := l in
 let (si', om') :=
   equivocator_transition 
     (IM i) li (finite_trace_last is tr i, iom) in
 (state_update (equivocator_IM IM)
    (finite_trace_last is tr) i si', om')) =
(s, oom)
equivocators_trace_project final_descriptors
  (tr ++
   [{|
      l := l;
      input := iom;
      destination := s;
      output := oom
    |}]) = Some (trX ++ [item], initial_descriptors)
∧ finite_constrained_trace_init_to Free
    (equivocators_state_project IM initial_descriptors
       is)
    (equivocators_state_project IM final_descriptors s)
    (trX ++ [item])
      destruct HtrX as [HtrX HinitX].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
final_descriptors', initial_descriptors: equivocator_descriptors IM
Hproper_initial: proper_equivocator_descriptors IM
  initial_descriptors is
trX: list (composite_transition_item IM)
Hpr_tr: equivocators_trace_project final_descriptors'
  tr = Some (trX, initial_descriptors)
HtrX: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is)
  (equivocators_state_project IM
     final_descriptors'
     (finite_trace_last is tr)) trX
HinitX: initial_state_prop
  (equivocators_state_project IM
     initial_descriptors is)
Htr: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm FreeE) is
  (finite_trace_last is tr) tr
s: composite_state (equivocator_IM IM)
iom, oom: option message
l: composite_label (equivocator_IM IM)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  final_descriptors s
item: composite_transition_item IM
Hpr_app: ∀ trX : list (composite_transition_item IM),
  equivocators_trace_project
    final_descriptors
    (tr ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}]) = Some (trX, initial_descriptors)
  ↔ (∃ (preX
        sufX : list
                 (composite_transition_item
                    IM)) 
       (eqv_descriptors' : 
        equivocator_descriptors IM),
       match
         equivocators_transition_item_project
           final_descriptors
           {|
             l := l;
             input := iom;
             destination := s;
             output := oom
           |}
       with
       | Some (Some item', odescriptor) =>
           Some ([item'], odescriptor)
       | Some (None, odescriptor) =>
           Some ([], odescriptor)
       | None => None
       end = Some (sufX, eqv_descriptors')
       ∧ equivocators_trace_project
           eqv_descriptors' tr =
         Some (preX, initial_descriptors)
         ∧ trX = preX ++ sufX)
Hproper': proper_equivocator_descriptors IM
  final_descriptors'
  (finite_trace_last is tr)
Heq_final_descriptors': final_descriptors' =
equivocator_descriptors_update
  IM final_descriptors
  (projT1 l)
  (final_descriptors'
     (projT1 l))
Heq_last_tr: finite_trace_last is tr =
state_update (equivocator_IM IM) s
  (projT1 l)
  (finite_trace_last is tr (projT1 l))
Hex_new: previous_state_descriptor_prop
  (IM (projT1 l))
  (final_descriptors (projT1 l))
  (finite_trace_last is tr (projT1 l))
  (final_descriptors' (projT1 l))
Hchar2: ∀ sx : composite_state IM,
  sx =
  equivocators_state_project IM
    final_descriptors'
    (finite_trace_last is tr)
  → (let (i, li) := VLSM.l item in
     valid li (sx i, input item))
    ∧ (let (i, li) := VLSM.l item in
       let (si', om') :=
         transition li (sx i, input item) in
       (state_update IM sx i si', om')) =
      (destination item, output item)
Hchar1: (∃ Hex : existing_equivocator_label
           (IM (projT1 l)) 
           (projT2 l),
   existT (projT1 l)
     (existing_equivocator_label_extract
        (IM (projT1 l)) 
        (projT2 l) Hex) = 
   VLSM.l item)
∧ iom = input item
  ∧ oom = output item
    ∧ equivocators_state_project IM
        final_descriptors s =
      destination item
      ∧ final_descriptors' (projT1 l) =
        equivocator_label_descriptor
          (projT2 l)
Hpr_x: equivocators_transition_item_project
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some (Some item, final_descriptors')
Hvx: let (i, li) := l in
equivocator_valid (IM i) li
  (finite_trace_last is tr i, iom)
Htx: (let (i, li) := l in
 let (si', om') :=
   equivocator_transition 
     (IM i) li (finite_trace_last is tr i, iom) in
 (state_update (equivocator_IM IM)
    (finite_trace_last is tr) i si', om')) =
(s, oom)
equivocators_trace_project final_descriptors
  (tr ++
   [{|
      l := l;
      input := iom;
      destination := s;
      output := oom
    |}]) = Some (trX ++ [item], initial_descriptors)
∧ finite_constrained_trace_init_to Free
    (equivocators_state_project IM initial_descriptors
       is)
    (equivocators_state_project IM final_descriptors s)
    (trX ++ [item])
      repeat split; [.. | done].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
final_descriptors', initial_descriptors: equivocator_descriptors IM
Hproper_initial: proper_equivocator_descriptors IM
  initial_descriptors is
trX: list (composite_transition_item IM)
Hpr_tr: equivocators_trace_project final_descriptors'
  tr = Some (trX, initial_descriptors)
HtrX: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is)
  (equivocators_state_project IM
     final_descriptors'
     (finite_trace_last is tr)) trX
HinitX: initial_state_prop
  (equivocators_state_project IM
     initial_descriptors is)
Htr: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm FreeE) is
  (finite_trace_last is tr) tr
s: composite_state (equivocator_IM IM)
iom, oom: option message
l: composite_label (equivocator_IM IM)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  final_descriptors s
item: composite_transition_item IM
Hpr_app: ∀ trX : list (composite_transition_item IM),
  equivocators_trace_project
    final_descriptors
    (tr ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}]) = Some (trX, initial_descriptors)
  ↔ (∃ (preX
        sufX : list
                 (composite_transition_item
                    IM)) 
       (eqv_descriptors' : 
        equivocator_descriptors IM),
       match
         equivocators_transition_item_project
           final_descriptors
           {|
             l := l;
             input := iom;
             destination := s;
             output := oom
           |}
       with
       | Some (Some item', odescriptor) =>
           Some ([item'], odescriptor)
       | Some (None, odescriptor) =>
           Some ([], odescriptor)
       | None => None
       end = Some (sufX, eqv_descriptors')
       ∧ equivocators_trace_project
           eqv_descriptors' tr =
         Some (preX, initial_descriptors)
         ∧ trX = preX ++ sufX)
Hproper': proper_equivocator_descriptors IM
  final_descriptors'
  (finite_trace_last is tr)
Heq_final_descriptors': final_descriptors' =
equivocator_descriptors_update
  IM final_descriptors
  (projT1 l)
  (final_descriptors'
     (projT1 l))
Heq_last_tr: finite_trace_last is tr =
state_update (equivocator_IM IM) s
  (projT1 l)
  (finite_trace_last is tr (projT1 l))
Hex_new: previous_state_descriptor_prop
  (IM (projT1 l))
  (final_descriptors (projT1 l))
  (finite_trace_last is tr (projT1 l))
  (final_descriptors' (projT1 l))
Hchar2: ∀ sx : composite_state IM,
  sx =
  equivocators_state_project IM
    final_descriptors'
    (finite_trace_last is tr)
  → (let (i, li) := VLSM.l item in
     valid li (sx i, input item))
    ∧ (let (i, li) := VLSM.l item in
       let (si', om') :=
         transition li (sx i, input item) in
       (state_update IM sx i si', om')) =
      (destination item, output item)
Hchar1: (∃ Hex : existing_equivocator_label
           (IM (projT1 l)) 
           (projT2 l),
   existT (projT1 l)
     (existing_equivocator_label_extract
        (IM (projT1 l)) 
        (projT2 l) Hex) = 
   VLSM.l item)
∧ iom = input item
  ∧ oom = output item
    ∧ equivocators_state_project IM
        final_descriptors s =
      destination item
      ∧ final_descriptors' (projT1 l) =
        equivocator_label_descriptor
          (projT2 l)
Hpr_x: equivocators_transition_item_project
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some (Some item, final_descriptors')
Hvx: let (i, li) := l in
equivocator_valid (IM i) li
  (finite_trace_last is tr i, iom)
Htx: (let (i, li) := l in
 let (si', om') :=
   equivocator_transition 
     (IM i) li (finite_trace_last is tr i, iom) in
 (state_update (equivocator_IM IM)
    (finite_trace_last is tr) i si', om')) =
(s, oom)
equivocators_trace_project final_descriptors
  (tr ++
   [{|
      l := l;
      input := iom;
      destination := s;
      output := oom
    |}]) = Some (trX ++ [item], initial_descriptors)
message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
final_descriptors', initial_descriptors: equivocator_descriptors IM
Hproper_initial: proper_equivocator_descriptors IM
  initial_descriptors is
trX: list (composite_transition_item IM)
Hpr_tr: equivocators_trace_project final_descriptors'
  tr = Some (trX, initial_descriptors)
HtrX: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is)
  (equivocators_state_project IM
     final_descriptors'
     (finite_trace_last is tr)) trX
HinitX: initial_state_prop
  (equivocators_state_project IM
     initial_descriptors is)
Htr: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm FreeE) is
  (finite_trace_last is tr) tr
s: composite_state (equivocator_IM IM)
iom, oom: option message
l: composite_label (equivocator_IM IM)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  final_descriptors s
item: composite_transition_item IM
Hpr_app: ∀ trX : list (composite_transition_item IM),
  equivocators_trace_project
    final_descriptors
    (tr ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}]) = Some (trX, initial_descriptors)
  ↔ (∃ (preX
        sufX : list
                 (composite_transition_item
                    IM)) 
       (eqv_descriptors' : 
        equivocator_descriptors IM),
       match
         equivocators_transition_item_project
           final_descriptors
           {|
             l := l;
             input := iom;
             destination := s;
             output := oom
           |}
       with
       | Some (Some item', odescriptor) =>
           Some ([item'], odescriptor)
       | Some (None, odescriptor) =>
           Some ([], odescriptor)
       | None => None
       end = Some (sufX, eqv_descriptors')
       ∧ equivocators_trace_project
           eqv_descriptors' tr =
         Some (preX, initial_descriptors)
         ∧ trX = preX ++ sufX)
Hproper': proper_equivocator_descriptors IM
  final_descriptors'
  (finite_trace_last is tr)
Heq_final_descriptors': final_descriptors' =
equivocator_descriptors_update
  IM final_descriptors
  (projT1 l)
  (final_descriptors'
     (projT1 l))
Heq_last_tr: finite_trace_last is tr =
state_update (equivocator_IM IM) s
  (projT1 l)
  (finite_trace_last is tr (projT1 l))
Hex_new: previous_state_descriptor_prop
  (IM (projT1 l))
  (final_descriptors (projT1 l))
  (finite_trace_last is tr (projT1 l))
  (final_descriptors' (projT1 l))
Hchar2: ∀ sx : composite_state IM,
  sx =
  equivocators_state_project IM
    final_descriptors'
    (finite_trace_last is tr)
  → (let (i, li) := VLSM.l item in
     valid li (sx i, input item))
    ∧ (let (i, li) := VLSM.l item in
       let (si', om') :=
         transition li (sx i, input item) in
       (state_update IM sx i si', om')) =
      (destination item, output item)
Hchar1: (∃ Hex : existing_equivocator_label
           (IM (projT1 l)) 
           (projT2 l),
   existT (projT1 l)
     (existing_equivocator_label_extract
        (IM (projT1 l)) 
        (projT2 l) Hex) = 
   VLSM.l item)
∧ iom = input item
  ∧ oom = output item
    ∧ equivocators_state_project IM
        final_descriptors s =
      destination item
      ∧ final_descriptors' (projT1 l) =
        equivocator_label_descriptor
          (projT2 l)
Hpr_x: equivocators_transition_item_project
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some (Some item, final_descriptors')
Hvx: let (i, li) := l in
equivocator_valid (IM i) li
  (finite_trace_last is tr i, iom)
Htx: (let (i, li) := l in
 let (si', om') :=
   equivocator_transition 
     (IM i) li (finite_trace_last is tr i, iom) in
 (state_update (equivocator_IM IM)
    (finite_trace_last is tr) i si', om')) =
(s, oom)
finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM initial_descriptors
     is)
  (equivocators_state_project IM final_descriptors s)
  (trX ++ [item])
      *message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
final_descriptors', initial_descriptors: equivocator_descriptors IM
Hproper_initial: proper_equivocator_descriptors IM
  initial_descriptors is
trX: list (composite_transition_item IM)
Hpr_tr: equivocators_trace_project final_descriptors'
  tr = Some (trX, initial_descriptors)
HtrX: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is)
  (equivocators_state_project IM
     final_descriptors'
     (finite_trace_last is tr)) trX
HinitX: initial_state_prop
  (equivocators_state_project IM
     initial_descriptors is)
Htr: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm FreeE) is
  (finite_trace_last is tr) tr
s: composite_state (equivocator_IM IM)
iom, oom: option message
l: composite_label (equivocator_IM IM)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  final_descriptors s
item: composite_transition_item IM
Hpr_app: ∀ trX : list (composite_transition_item IM),
  equivocators_trace_project
    final_descriptors
    (tr ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}]) = Some (trX, initial_descriptors)
  ↔ (∃ (preX
        sufX : list
                 (composite_transition_item
                    IM)) 
       (eqv_descriptors' : 
        equivocator_descriptors IM),
       match
         equivocators_transition_item_project
           final_descriptors
           {|
             l := l;
             input := iom;
             destination := s;
             output := oom
           |}
       with
       | Some (Some item', odescriptor) =>
           Some ([item'], odescriptor)
       | Some (None, odescriptor) =>
           Some ([], odescriptor)
       | None => None
       end = Some (sufX, eqv_descriptors')
       ∧ equivocators_trace_project
           eqv_descriptors' tr =
         Some (preX, initial_descriptors)
         ∧ trX = preX ++ sufX)
Hproper': proper_equivocator_descriptors IM
  final_descriptors'
  (finite_trace_last is tr)
Heq_final_descriptors': final_descriptors' =
equivocator_descriptors_update
  IM final_descriptors
  (projT1 l)
  (final_descriptors'
     (projT1 l))
Heq_last_tr: finite_trace_last is tr =
state_update (equivocator_IM IM) s
  (projT1 l)
  (finite_trace_last is tr (projT1 l))
Hex_new: previous_state_descriptor_prop
  (IM (projT1 l))
  (final_descriptors (projT1 l))
  (finite_trace_last is tr (projT1 l))
  (final_descriptors' (projT1 l))
Hchar2: ∀ sx : composite_state IM,
  sx =
  equivocators_state_project IM
    final_descriptors'
    (finite_trace_last is tr)
  → (let (i, li) := VLSM.l item in
     valid li (sx i, input item))
    ∧ (let (i, li) := VLSM.l item in
       let (si', om') :=
         transition li (sx i, input item) in
       (state_update IM sx i si', om')) =
      (destination item, output item)
Hchar1: (∃ Hex : existing_equivocator_label
           (IM (projT1 l)) 
           (projT2 l),
   existT (projT1 l)
     (existing_equivocator_label_extract
        (IM (projT1 l)) 
        (projT2 l) Hex) = 
   VLSM.l item)
∧ iom = input item
  ∧ oom = output item
    ∧ equivocators_state_project IM
        final_descriptors s =
      destination item
      ∧ final_descriptors' (projT1 l) =
        equivocator_label_descriptor
          (projT2 l)
Hpr_x: equivocators_transition_item_project
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some (Some item, final_descriptors')
Hvx: let (i, li) := l in
equivocator_valid (IM i) li
  (finite_trace_last is tr i, iom)
Htx: (let (i, li) := l in
 let (si', om') :=
   equivocator_transition 
     (IM i) li (finite_trace_last is tr i, iom) in
 (state_update (equivocator_IM IM)
    (finite_trace_last is tr) i si', om')) =
(s, oom)
equivocators_trace_project final_descriptors
  (tr ++
   [{|
      l := l;
      input := iom;
      destination := s;
      output := oom
    |}]) = Some (trX ++ [item], initial_descriptors) apply (Hpr_app (trX ++ [item])).message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
final_descriptors', initial_descriptors: equivocator_descriptors IM
Hproper_initial: proper_equivocator_descriptors IM
  initial_descriptors is
trX: list (composite_transition_item IM)
Hpr_tr: equivocators_trace_project final_descriptors'
  tr = Some (trX, initial_descriptors)
HtrX: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is)
  (equivocators_state_project IM
     final_descriptors'
     (finite_trace_last is tr)) trX
HinitX: initial_state_prop
  (equivocators_state_project IM
     initial_descriptors is)
Htr: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm FreeE) is
  (finite_trace_last is tr) tr
s: composite_state (equivocator_IM IM)
iom, oom: option message
l: composite_label (equivocator_IM IM)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  final_descriptors s
item: composite_transition_item IM
Hpr_app: ∀ trX : list (composite_transition_item IM),
  equivocators_trace_project
    final_descriptors
    (tr ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}]) = Some (trX, initial_descriptors)
  ↔ (∃ (preX
        sufX : list
                 (composite_transition_item
                    IM)) 
       (eqv_descriptors' : 
        equivocator_descriptors IM),
       match
         equivocators_transition_item_project
           final_descriptors
           {|
             l := l;
             input := iom;
             destination := s;
             output := oom
           |}
       with
       | Some (Some item', odescriptor) =>
           Some ([item'], odescriptor)
       | Some (None, odescriptor) =>
           Some ([], odescriptor)
       | None => None
       end = Some (sufX, eqv_descriptors')
       ∧ equivocators_trace_project
           eqv_descriptors' tr =
         Some (preX, initial_descriptors)
         ∧ trX = preX ++ sufX)
Hproper': proper_equivocator_descriptors IM
  final_descriptors'
  (finite_trace_last is tr)
Heq_final_descriptors': final_descriptors' =
equivocator_descriptors_update
  IM final_descriptors
  (projT1 l)
  (final_descriptors'
     (projT1 l))
Heq_last_tr: finite_trace_last is tr =
state_update (equivocator_IM IM) s
  (projT1 l)
  (finite_trace_last is tr (projT1 l))
Hex_new: previous_state_descriptor_prop
  (IM (projT1 l))
  (final_descriptors (projT1 l))
  (finite_trace_last is tr (projT1 l))
  (final_descriptors' (projT1 l))
Hchar2: ∀ sx : composite_state IM,
  sx =
  equivocators_state_project IM
    final_descriptors'
    (finite_trace_last is tr)
  → (let (i, li) := VLSM.l item in
     valid li (sx i, input item))
    ∧ (let (i, li) := VLSM.l item in
       let (si', om') :=
         transition li (sx i, input item) in
       (state_update IM sx i si', om')) =
      (destination item, output item)
Hchar1: (∃ Hex : existing_equivocator_label
           (IM (projT1 l)) 
           (projT2 l),
   existT (projT1 l)
     (existing_equivocator_label_extract
        (IM (projT1 l)) 
        (projT2 l) Hex) = 
   VLSM.l item)
∧ iom = input item
  ∧ oom = output item
    ∧ equivocators_state_project IM
        final_descriptors s =
      destination item
      ∧ final_descriptors' (projT1 l) =
        equivocator_label_descriptor
          (projT2 l)
Hpr_x: equivocators_transition_item_project
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some (Some item, final_descriptors')
Hvx: let (i, li) := l in
equivocator_valid (IM i) li
  (finite_trace_last is tr i, iom)
Htx: (let (i, li) := l in
 let (si', om') :=
   equivocator_transition 
     (IM i) li (finite_trace_last is tr i, iom) in
 (state_update (equivocator_IM IM)
    (finite_trace_last is tr) i si', om')) =
(s, oom)
∃ (preX sufX : list (composite_transition_item IM)) 
  (eqv_descriptors' : equivocator_descriptors IM),
  match
    equivocators_transition_item_project
      final_descriptors
      {|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}
  with
  | Some (Some item', odescriptor) =>
      Some ([item'], odescriptor)
  | Some (None, odescriptor) => Some ([], odescriptor)
  | None => None
  end = Some (sufX, eqv_descriptors')
  ∧ equivocators_trace_project eqv_descriptors' tr =
    Some (preX, initial_descriptors)
    ∧ trX ++ [item] = preX ++ sufX
        exists trX, [item], final_descriptors'.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
final_descriptors', initial_descriptors: equivocator_descriptors IM
Hproper_initial: proper_equivocator_descriptors IM
  initial_descriptors is
trX: list (composite_transition_item IM)
Hpr_tr: equivocators_trace_project final_descriptors'
  tr = Some (trX, initial_descriptors)
HtrX: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is)
  (equivocators_state_project IM
     final_descriptors'
     (finite_trace_last is tr)) trX
HinitX: initial_state_prop
  (equivocators_state_project IM
     initial_descriptors is)
Htr: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm FreeE) is
  (finite_trace_last is tr) tr
s: composite_state (equivocator_IM IM)
iom, oom: option message
l: composite_label (equivocator_IM IM)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  final_descriptors s
item: composite_transition_item IM
Hpr_app: ∀ trX : list (composite_transition_item IM),
  equivocators_trace_project
    final_descriptors
    (tr ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}]) = Some (trX, initial_descriptors)
  ↔ (∃ (preX
        sufX : list
                 (composite_transition_item
                    IM)) 
       (eqv_descriptors' : 
        equivocator_descriptors IM),
       match
         equivocators_transition_item_project
           final_descriptors
           {|
             l := l;
             input := iom;
             destination := s;
             output := oom
           |}
       with
       | Some (Some item', odescriptor) =>
           Some ([item'], odescriptor)
       | Some (None, odescriptor) =>
           Some ([], odescriptor)
       | None => None
       end = Some (sufX, eqv_descriptors')
       ∧ equivocators_trace_project
           eqv_descriptors' tr =
         Some (preX, initial_descriptors)
         ∧ trX = preX ++ sufX)
Hproper': proper_equivocator_descriptors IM
  final_descriptors'
  (finite_trace_last is tr)
Heq_final_descriptors': final_descriptors' =
equivocator_descriptors_update
  IM final_descriptors
  (projT1 l)
  (final_descriptors'
     (projT1 l))
Heq_last_tr: finite_trace_last is tr =
state_update (equivocator_IM IM) s
  (projT1 l)
  (finite_trace_last is tr (projT1 l))
Hex_new: previous_state_descriptor_prop
  (IM (projT1 l))
  (final_descriptors (projT1 l))
  (finite_trace_last is tr (projT1 l))
  (final_descriptors' (projT1 l))
Hchar2: ∀ sx : composite_state IM,
  sx =
  equivocators_state_project IM
    final_descriptors'
    (finite_trace_last is tr)
  → (let (i, li) := VLSM.l item in
     valid li (sx i, input item))
    ∧ (let (i, li) := VLSM.l item in
       let (si', om') :=
         transition li (sx i, input item) in
       (state_update IM sx i si', om')) =
      (destination item, output item)
Hchar1: (∃ Hex : existing_equivocator_label
           (IM (projT1 l)) 
           (projT2 l),
   existT (projT1 l)
     (existing_equivocator_label_extract
        (IM (projT1 l)) 
        (projT2 l) Hex) = 
   VLSM.l item)
∧ iom = input item
  ∧ oom = output item
    ∧ equivocators_state_project IM
        final_descriptors s =
      destination item
      ∧ final_descriptors' (projT1 l) =
        equivocator_label_descriptor
          (projT2 l)
Hpr_x: equivocators_transition_item_project
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some (Some item, final_descriptors')
Hvx: let (i, li) := l in
equivocator_valid (IM i) li
  (finite_trace_last is tr i, iom)
Htx: (let (i, li) := l in
 let (si', om') :=
   equivocator_transition 
     (IM i) li (finite_trace_last is tr i, iom) in
 (state_update (equivocator_IM IM)
    (finite_trace_last is tr) i si', om')) =
(s, oom)
match
  equivocators_transition_item_project
    final_descriptors
    {|
      l := l;
      input := iom;
      destination := s;
      output := oom
    |}
with
| Some (Some item', odescriptor) =>
    Some ([item'], odescriptor)
| Some (None, odescriptor) => Some ([], odescriptor)
| None => None
end = Some ([item], final_descriptors')
∧ equivocators_trace_project final_descriptors' tr =
  Some (trX, initial_descriptors)
  ∧ trX ++ [item] = trX ++ [item]
        by rewrite Hpr_x.
      *message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
final_descriptors', initial_descriptors: equivocator_descriptors IM
Hproper_initial: proper_equivocator_descriptors IM
  initial_descriptors is
trX: list (composite_transition_item IM)
Hpr_tr: equivocators_trace_project final_descriptors'
  tr = Some (trX, initial_descriptors)
HtrX: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is)
  (equivocators_state_project IM
     final_descriptors'
     (finite_trace_last is tr)) trX
HinitX: initial_state_prop
  (equivocators_state_project IM
     initial_descriptors is)
Htr: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm FreeE) is
  (finite_trace_last is tr) tr
s: composite_state (equivocator_IM IM)
iom, oom: option message
l: composite_label (equivocator_IM IM)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  final_descriptors s
item: composite_transition_item IM
Hpr_app: ∀ trX : list (composite_transition_item IM),
  equivocators_trace_project
    final_descriptors
    (tr ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}]) = Some (trX, initial_descriptors)
  ↔ (∃ (preX
        sufX : list
                 (composite_transition_item
                    IM)) 
       (eqv_descriptors' : 
        equivocator_descriptors IM),
       match
         equivocators_transition_item_project
           final_descriptors
           {|
             l := l;
             input := iom;
             destination := s;
             output := oom
           |}
       with
       | Some (Some item', odescriptor) =>
           Some ([item'], odescriptor)
       | Some (None, odescriptor) =>
           Some ([], odescriptor)
       | None => None
       end = Some (sufX, eqv_descriptors')
       ∧ equivocators_trace_project
           eqv_descriptors' tr =
         Some (preX, initial_descriptors)
         ∧ trX = preX ++ sufX)
Hproper': proper_equivocator_descriptors IM
  final_descriptors'
  (finite_trace_last is tr)
Heq_final_descriptors': final_descriptors' =
equivocator_descriptors_update
  IM final_descriptors
  (projT1 l)
  (final_descriptors'
     (projT1 l))
Heq_last_tr: finite_trace_last is tr =
state_update (equivocator_IM IM) s
  (projT1 l)
  (finite_trace_last is tr (projT1 l))
Hex_new: previous_state_descriptor_prop
  (IM (projT1 l))
  (final_descriptors (projT1 l))
  (finite_trace_last is tr (projT1 l))
  (final_descriptors' (projT1 l))
Hchar2: ∀ sx : composite_state IM,
  sx =
  equivocators_state_project IM
    final_descriptors'
    (finite_trace_last is tr)
  → (let (i, li) := VLSM.l item in
     valid li (sx i, input item))
    ∧ (let (i, li) := VLSM.l item in
       let (si', om') :=
         transition li (sx i, input item) in
       (state_update IM sx i si', om')) =
      (destination item, output item)
Hchar1: (∃ Hex : existing_equivocator_label
           (IM (projT1 l)) 
           (projT2 l),
   existT (projT1 l)
     (existing_equivocator_label_extract
        (IM (projT1 l)) 
        (projT2 l) Hex) = 
   VLSM.l item)
∧ iom = input item
  ∧ oom = output item
    ∧ equivocators_state_project IM
        final_descriptors s =
      destination item
      ∧ final_descriptors' (projT1 l) =
        equivocator_label_descriptor
          (projT2 l)
Hpr_x: equivocators_transition_item_project
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some (Some item, final_descriptors')
Hvx: let (i, li) := l in
equivocator_valid (IM i) li
  (finite_trace_last is tr i, iom)
Htx: (let (i, li) := l in
 let (si', om') :=
   equivocator_transition 
     (IM i) li (finite_trace_last is tr i, iom) in
 (state_update (equivocator_IM IM)
    (finite_trace_last is tr) i si', om')) =
(s, oom)
finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM initial_descriptors
     is)
  (equivocators_state_project IM final_descriptors s)
  (trX ++ [item]) apply
          (finite_valid_trace_from_to_app PreFree
            (equivocators_state_project IM final_descriptors' (finite_trace_last is tr)))
        ; [done |].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
final_descriptors', initial_descriptors: equivocator_descriptors IM
Hproper_initial: proper_equivocator_descriptors IM
  initial_descriptors is
trX: list (composite_transition_item IM)
Hpr_tr: equivocators_trace_project final_descriptors'
  tr = Some (trX, initial_descriptors)
HtrX: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is)
  (equivocators_state_project IM
     final_descriptors'
     (finite_trace_last is tr)) trX
HinitX: initial_state_prop
  (equivocators_state_project IM
     initial_descriptors is)
Htr: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm FreeE) is
  (finite_trace_last is tr) tr
s: composite_state (equivocator_IM IM)
iom, oom: option message
l: composite_label (equivocator_IM IM)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  final_descriptors s
item: composite_transition_item IM
Hpr_app: ∀ trX : list (composite_transition_item IM),
  equivocators_trace_project
    final_descriptors
    (tr ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}]) = Some (trX, initial_descriptors)
  ↔ (∃ (preX
        sufX : list
                 (composite_transition_item
                    IM)) 
       (eqv_descriptors' : 
        equivocator_descriptors IM),
       match
         equivocators_transition_item_project
           final_descriptors
           {|
             l := l;
             input := iom;
             destination := s;
             output := oom
           |}
       with
       | Some (Some item', odescriptor) =>
           Some ([item'], odescriptor)
       | Some (None, odescriptor) =>
           Some ([], odescriptor)
       | None => None
       end = Some (sufX, eqv_descriptors')
       ∧ equivocators_trace_project
           eqv_descriptors' tr =
         Some (preX, initial_descriptors)
         ∧ trX = preX ++ sufX)
Hproper': proper_equivocator_descriptors IM
  final_descriptors'
  (finite_trace_last is tr)
Heq_final_descriptors': final_descriptors' =
equivocator_descriptors_update
  IM final_descriptors
  (projT1 l)
  (final_descriptors'
     (projT1 l))
Heq_last_tr: finite_trace_last is tr =
state_update (equivocator_IM IM) s
  (projT1 l)
  (finite_trace_last is tr (projT1 l))
Hex_new: previous_state_descriptor_prop
  (IM (projT1 l))
  (final_descriptors (projT1 l))
  (finite_trace_last is tr (projT1 l))
  (final_descriptors' (projT1 l))
Hchar2: ∀ sx : composite_state IM,
  sx =
  equivocators_state_project IM
    final_descriptors'
    (finite_trace_last is tr)
  → (let (i, li) := VLSM.l item in
     valid li (sx i, input item))
    ∧ (let (i, li) := VLSM.l item in
       let (si', om') :=
         transition li (sx i, input item) in
       (state_update IM sx i si', om')) =
      (destination item, output item)
Hchar1: (∃ Hex : existing_equivocator_label
           (IM (projT1 l)) 
           (projT2 l),
   existT (projT1 l)
     (existing_equivocator_label_extract
        (IM (projT1 l)) 
        (projT2 l) Hex) = 
   VLSM.l item)
∧ iom = input item
  ∧ oom = output item
    ∧ equivocators_state_project IM
        final_descriptors s =
      destination item
      ∧ final_descriptors' (projT1 l) =
        equivocator_label_descriptor
          (projT2 l)
Hpr_x: equivocators_transition_item_project
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some (Some item, final_descriptors')
Hvx: let (i, li) := l in
equivocator_valid (IM i) li
  (finite_trace_last is tr i, iom)
Htx: (let (i, li) := l in
 let (si', om') :=
   equivocator_transition 
     (IM i) li (finite_trace_last is tr i, iom) in
 (state_update (equivocator_IM IM)
    (finite_trace_last is tr) i si', om')) =
(s, oom)
finite_valid_trace_from_to PreFree
  (equivocators_state_project IM final_descriptors'
     (finite_trace_last is tr))
  (equivocators_state_project IM final_descriptors s)
  [item]
        specialize (Hchar2 _ eq_refl).message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
final_descriptors', initial_descriptors: equivocator_descriptors IM
Hproper_initial: proper_equivocator_descriptors IM
  initial_descriptors is
trX: list (composite_transition_item IM)
Hpr_tr: equivocators_trace_project final_descriptors'
  tr = Some (trX, initial_descriptors)
HtrX: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is)
  (equivocators_state_project IM
     final_descriptors'
     (finite_trace_last is tr)) trX
HinitX: initial_state_prop
  (equivocators_state_project IM
     initial_descriptors is)
Htr: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm FreeE) is
  (finite_trace_last is tr) tr
s: composite_state (equivocator_IM IM)
iom, oom: option message
l: composite_label (equivocator_IM IM)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  final_descriptors s
item: composite_transition_item IM
Hpr_app: ∀ trX : list (composite_transition_item IM),
  equivocators_trace_project
    final_descriptors
    (tr ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}]) = Some (trX, initial_descriptors)
  ↔ (∃ (preX
        sufX : list
                 (composite_transition_item
                    IM)) 
       (eqv_descriptors' : 
        equivocator_descriptors IM),
       match
         equivocators_transition_item_project
           final_descriptors
           {|
             l := l;
             input := iom;
             destination := s;
             output := oom
           |}
       with
       | Some (Some item', odescriptor) =>
           Some ([item'], odescriptor)
       | Some (None, odescriptor) =>
           Some ([], odescriptor)
       | None => None
       end = Some (sufX, eqv_descriptors')
       ∧ equivocators_trace_project
           eqv_descriptors' tr =
         Some (preX, initial_descriptors)
         ∧ trX = preX ++ sufX)
Hproper': proper_equivocator_descriptors IM
  final_descriptors'
  (finite_trace_last is tr)
Heq_final_descriptors': final_descriptors' =
equivocator_descriptors_update
  IM final_descriptors
  (projT1 l)
  (final_descriptors'
     (projT1 l))
Heq_last_tr: finite_trace_last is tr =
state_update (equivocator_IM IM) s
  (projT1 l)
  (finite_trace_last is tr (projT1 l))
Hex_new: previous_state_descriptor_prop
  (IM (projT1 l))
  (final_descriptors (projT1 l))
  (finite_trace_last is tr (projT1 l))
  (final_descriptors' (projT1 l))
Hchar2: (let (i, li) := VLSM.l item in
 valid li
   (equivocators_state_project IM
      final_descriptors'
      (finite_trace_last is tr) i, 
    input item))
∧ (let (i, li) := VLSM.l item in
   let (si', om') :=
     transition li
       (equivocators_state_project IM
          final_descriptors'
          (finite_trace_last is tr) i,
        input item) in
   (state_update IM
      (equivocators_state_project IM
         final_descriptors'
         (finite_trace_last is tr)) i si',
    om')) = (destination item, output item)
Hchar1: (∃ Hex : existing_equivocator_label
           (IM (projT1 l)) 
           (projT2 l),
   existT (projT1 l)
     (existing_equivocator_label_extract
        (IM (projT1 l)) 
        (projT2 l) Hex) = 
   VLSM.l item)
∧ iom = input item
  ∧ oom = output item
    ∧ equivocators_state_project IM
        final_descriptors s =
      destination item
      ∧ final_descriptors' (projT1 l) =
        equivocator_label_descriptor
          (projT2 l)
Hpr_x: equivocators_transition_item_project
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some (Some item, final_descriptors')
Hvx: let (i, li) := l in
equivocator_valid (IM i) li
  (finite_trace_last is tr i, iom)
Htx: (let (i, li) := l in
 let (si', om') :=
   equivocator_transition 
     (IM i) li (finite_trace_last is tr i, iom) in
 (state_update (equivocator_IM IM)
    (finite_trace_last is tr) i si', om')) =
(s, oom)
finite_valid_trace_from_to PreFree
  (equivocators_state_project IM final_descriptors'
     (finite_trace_last is tr))
  (equivocators_state_project IM final_descriptors s)
  [item]
        destruct item.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
final_descriptors', initial_descriptors: equivocator_descriptors IM
Hproper_initial: proper_equivocator_descriptors IM
  initial_descriptors is
trX: list (composite_transition_item IM)
Hpr_tr: equivocators_trace_project final_descriptors'
  tr = Some (trX, initial_descriptors)
HtrX: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is)
  (equivocators_state_project IM
     final_descriptors'
     (finite_trace_last is tr)) trX
HinitX: initial_state_prop
  (equivocators_state_project IM
     initial_descriptors is)
Htr: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm FreeE) is
  (finite_trace_last is tr) tr
s: composite_state (equivocator_IM IM)
iom, oom: option message
l: composite_label (equivocator_IM IM)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  final_descriptors s
l0: label (composite_type IM)
input: option message
destination: state (composite_type IM)
output: option message
Hpr_app: ∀ trX : list (composite_transition_item IM),
  equivocators_trace_project
    final_descriptors
    (tr ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}]) = Some (trX, initial_descriptors)
  ↔ (∃ (preX
        sufX : list
                 (composite_transition_item
                    IM)) 
       (eqv_descriptors' : 
        equivocator_descriptors IM),
       match
         equivocators_transition_item_project
           final_descriptors
           {|
             l := l;
             input := iom;
             destination := s;
             output := oom
           |}
       with
       | Some (Some item', odescriptor) =>
           Some ([item'], odescriptor)
       | Some (None, odescriptor) =>
           Some ([], odescriptor)
       | None => None
       end = Some (sufX, eqv_descriptors')
       ∧ equivocators_trace_project
           eqv_descriptors' tr =
         Some (preX, initial_descriptors)
         ∧ trX = preX ++ sufX)
Hproper': proper_equivocator_descriptors IM
  final_descriptors'
  (finite_trace_last is tr)
Heq_final_descriptors': final_descriptors' =
equivocator_descriptors_update
  IM final_descriptors
  (projT1 l)
  (final_descriptors'
     (projT1 l))
Heq_last_tr: finite_trace_last is tr =
state_update (equivocator_IM IM) s
  (projT1 l)
  (finite_trace_last is tr (projT1 l))
Hex_new: previous_state_descriptor_prop
  (IM (projT1 l))
  (final_descriptors (projT1 l))
  (finite_trace_last is tr (projT1 l))
  (final_descriptors' (projT1 l))
Hchar2: (let (i, li) :=
   VLSM.l
     {|
       l := l0;
       input := input;
       destination := destination;
       output := output
     |} in
 valid li
   (equivocators_state_project IM
      final_descriptors'
      (finite_trace_last is tr) i,
    VLSM.input
      {|
        l := l0;
        input := input;
        destination := destination;
        output := output
      |}))
∧ (let (i, li) :=
     VLSM.l
       {|
         l := l0;
         input := input;
         destination := destination;
         output := output
       |} in
   let (si', om') :=
     transition li
       (equivocators_state_project IM
          final_descriptors'
          (finite_trace_last is tr) i,
        VLSM.input
          {|
            l := l0;
            input := input;
            destination := destination;
            output := output
          |}) in
   (state_update IM
      (equivocators_state_project IM
         final_descriptors'
         (finite_trace_last is tr)) i si',
    om')) =
  (VLSM.destination
     {|
       l := l0;
       input := input;
       destination := destination;
       output := output
     |},
   VLSM.output
     {|
       l := l0;
       input := input;
       destination := destination;
       output := output
     |})
Hchar1: (∃ Hex : existing_equivocator_label
           (IM (projT1 l)) 
           (projT2 l),
   existT (projT1 l)
     (existing_equivocator_label_extract
        (IM (projT1 l)) 
        (projT2 l) Hex) =
   VLSM.l
     {|
       l := l0;
       input := input;
       destination := destination;
       output := output
     |})
∧ iom =
  VLSM.input
    {|
      l := l0;
      input := input;
      destination := destination;
      output := output
    |}
  ∧ oom =
    VLSM.output
      {|
        l := l0;
        input := input;
        destination := destination;
        output := output
      |}
    ∧ equivocators_state_project IM
        final_descriptors s =
      VLSM.destination
        {|
          l := l0;
          input := input;
          destination := destination;
          output := output
        |}
      ∧ final_descriptors' (projT1 l) =
        equivocator_label_descriptor
          (projT2 l)
Hpr_x: equivocators_transition_item_project
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} =
Some
  (Some
     {|
       l := l0;
       input := input;
       destination := destination;
       output := output
     |}, final_descriptors')
Hvx: let (i, li) := l in
equivocator_valid (IM i) li
  (finite_trace_last is tr i, iom)
Htx: (let (i, li) := l in
 let (si', om') :=
   equivocator_transition 
     (IM i) li (finite_trace_last is tr i, iom) in
 (state_update (equivocator_IM IM)
    (finite_trace_last is tr) i si', om')) =
(s, oom)
finite_valid_trace_from_to PreFree
  (equivocators_state_project IM final_descriptors'
     (finite_trace_last is tr))
  (equivocators_state_project IM final_descriptors s)
  [{|
     l := l0;
     input := input;
     destination := destination;
     output := output
   |}] destruct l0 as (ix, lix).message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
final_descriptors', initial_descriptors: equivocator_descriptors IM
Hproper_initial: proper_equivocator_descriptors IM
  initial_descriptors is
trX: list (composite_transition_item IM)
Hpr_tr: equivocators_trace_project final_descriptors'
  tr = Some (trX, initial_descriptors)
HtrX: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is)
  (equivocators_state_project IM
     final_descriptors'
     (finite_trace_last is tr)) trX
HinitX: initial_state_prop
  (equivocators_state_project IM
     initial_descriptors is)
Htr: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm FreeE) is
  (finite_trace_last is tr) tr
s: composite_state (equivocator_IM IM)
iom, oom: option message
l: composite_label (equivocator_IM IM)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  final_descriptors s
ix: index
lix: label (IM ix)
input: option message
destination: state (composite_type IM)
output: option message
Hpr_app: ∀ trX : list (composite_transition_item IM),
  equivocators_trace_project
    final_descriptors
    (tr ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}]) = Some (trX, initial_descriptors)
  ↔ (∃ (preX
        sufX : list
                 (composite_transition_item
                    IM)) 
       (eqv_descriptors' : 
        equivocator_descriptors IM),
       match
         equivocators_transition_item_project
           final_descriptors
           {|
             l := l;
             input := iom;
             destination := s;
             output := oom
           |}
       with
       | Some (Some item', odescriptor) =>
           Some ([item'], odescriptor)
       | Some (None, odescriptor) =>
           Some ([], odescriptor)
       | None => None
       end = Some (sufX, eqv_descriptors')
       ∧ equivocators_trace_project
           eqv_descriptors' tr =
         Some (preX, initial_descriptors)
         ∧ trX = preX ++ sufX)
Hproper': proper_equivocator_descriptors IM
  final_descriptors'
  (finite_trace_last is tr)
Heq_final_descriptors': final_descriptors' =
equivocator_descriptors_update
  IM final_descriptors
  (projT1 l)
  (final_descriptors'
     (projT1 l))
Heq_last_tr: finite_trace_last is tr =
state_update (equivocator_IM IM) s
  (projT1 l)
  (finite_trace_last is tr (projT1 l))
Hex_new: previous_state_descriptor_prop
  (IM (projT1 l))
  (final_descriptors (projT1 l))
  (finite_trace_last is tr (projT1 l))
  (final_descriptors' (projT1 l))
Hchar2: (let (i, li) :=
   VLSM.l
     {|
       l := existT ix lix;
       input := input;
       destination := destination;
       output := output
     |} in
 valid li
   (equivocators_state_project IM
      final_descriptors'
      (finite_trace_last is tr) i,
    VLSM.input
      {|
        l := existT ix lix;
        input := input;
        destination := destination;
        output := output
      |}))
∧ (let (i, li) :=
     VLSM.l
       {|
         l := existT ix lix;
         input := input;
         destination := destination;
         output := output
       |} in
   let (si', om') :=
     transition li
       (equivocators_state_project IM
          final_descriptors'
          (finite_trace_last is tr) i,
        VLSM.input
          {|
            l := existT ix lix;
            input := input;
            destination := destination;
            output := output
          |}) in
   (state_update IM
      (equivocators_state_project IM
         final_descriptors'
         (finite_trace_last is tr)) i si',
    om')) =
  (VLSM.destination
     {|
       l := existT ix lix;
       input := input;
       destination := destination;
       output := output
     |},
   VLSM.output
     {|
       l := existT ix lix;
       input := input;
       destination := destination;
       output := output
     |})
Hchar1: (∃ Hex : existing_equivocator_label
           (IM (projT1 l)) 
           (projT2 l),
   existT (projT1 l)
     (existing_equivocator_label_extract
        (IM (projT1 l)) 
        (projT2 l) Hex) =
   VLSM.l
     {|
       l := existT ix lix;
       input := input;
       destination := destination;
       output := output
     |})
∧ iom =
  VLSM.input
    {|
      l := existT ix lix;
      input := input;
      destination := destination;
      output := output
    |}
  ∧ oom =
    VLSM.output
      {|
        l := existT ix lix;
        input := input;
        destination := destination;
        output := output
      |}
    ∧ equivocators_state_project IM
        final_descriptors s =
      VLSM.destination
        {|
          l := existT ix lix;
          input := input;
          destination := destination;
          output := output
        |}
      ∧ final_descriptors' (projT1 l) =
        equivocator_label_descriptor
          (projT2 l)
Hpr_x: equivocators_transition_item_project
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} =
Some
  (Some
     {|
       l := existT ix lix;
       input := input;
       destination := destination;
       output := output
     |}, final_descriptors')
Hvx: let (i, li) := l in
equivocator_valid (IM i) li
  (finite_trace_last is tr i, iom)
Htx: (let (i, li) := l in
 let (si', om') :=
   equivocator_transition 
     (IM i) li (finite_trace_last is tr i, iom) in
 (state_update (equivocator_IM IM)
    (finite_trace_last is tr) i si', om')) =
(s, oom)
finite_valid_trace_from_to PreFree
  (equivocators_state_project IM final_descriptors'
     (finite_trace_last is tr))
  (equivocators_state_project IM final_descriptors s)
  [{|
     l := existT ix lix;
     input := input;
     destination := destination;
     output := output
   |}]
        destruct l as (i, li).message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
final_descriptors', initial_descriptors: equivocator_descriptors IM
Hproper_initial: proper_equivocator_descriptors IM
  initial_descriptors is
trX: list (composite_transition_item IM)
Hpr_tr: equivocators_trace_project final_descriptors'
  tr = Some (trX, initial_descriptors)
HtrX: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is)
  (equivocators_state_project IM
     final_descriptors'
     (finite_trace_last is tr)) trX
HinitX: initial_state_prop
  (equivocators_state_project IM
     initial_descriptors is)
Htr: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm FreeE) is
  (finite_trace_last is tr) tr
s: composite_state (equivocator_IM IM)
iom, oom: option message
i: index
li: label (equivocator_IM IM i)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  final_descriptors s
ix: index
lix: label (IM ix)
input: option message
destination: state (composite_type IM)
output: option message
Hpr_app: ∀ trX : list (composite_transition_item IM),
  equivocators_trace_project
    final_descriptors
    (tr ++
     [{|
        l := existT i li;
        input := iom;
        destination := s;
        output := oom
      |}]) = Some (trX, initial_descriptors)
  ↔ (∃ (preX
        sufX : list
                 (composite_transition_item
                    IM)) 
       (eqv_descriptors' : 
        equivocator_descriptors IM),
       match
         equivocators_transition_item_project
           final_descriptors
           {|
             l := existT i li;
             input := iom;
             destination := s;
             output := oom
           |}
       with
       | Some (Some item', odescriptor) =>
           Some ([item'], odescriptor)
       | Some (None, odescriptor) =>
           Some ([], odescriptor)
       | None => None
       end = Some (sufX, eqv_descriptors')
       ∧ equivocators_trace_project
           eqv_descriptors' tr =
         Some (preX, initial_descriptors)
         ∧ trX = preX ++ sufX)
Hproper': proper_equivocator_descriptors IM
  final_descriptors'
  (finite_trace_last is tr)
Heq_final_descriptors': final_descriptors' =
equivocator_descriptors_update
  IM final_descriptors
  (projT1 (existT i li))
  (final_descriptors'
     (projT1 (existT i li)))
Heq_last_tr: finite_trace_last is tr =
state_update (equivocator_IM IM) s
  (projT1 (existT i li))
  (finite_trace_last is tr
     (projT1 (existT i li)))
Hex_new: previous_state_descriptor_prop
  (IM (projT1 (existT i li)))
  (final_descriptors (projT1 (existT i li)))
  (finite_trace_last is tr
     (projT1 (existT i li)))
  (final_descriptors' (projT1 (existT i li)))
Hchar2: (let (i, li) :=
   l
     {|
       l := existT ix lix;
       input := input;
       destination := destination;
       output := output
     |} in
 valid li
   (equivocators_state_project IM
      final_descriptors'
      (finite_trace_last is tr) i,
    VLSM.input
      {|
        l := existT ix lix;
        input := input;
        destination := destination;
        output := output
      |}))
∧ (let (i, li) :=
     l
       {|
         l := existT ix lix;
         input := input;
         destination := destination;
         output := output
       |} in
   let (si', om') :=
     transition li
       (equivocators_state_project IM
          final_descriptors'
          (finite_trace_last is tr) i,
        VLSM.input
          {|
            l := existT ix lix;
            input := input;
            destination := destination;
            output := output
          |}) in
   (state_update IM
      (equivocators_state_project IM
         final_descriptors'
         (finite_trace_last is tr)) i si',
    om')) =
  (VLSM.destination
     {|
       l := existT ix lix;
       input := input;
       destination := destination;
       output := output
     |},
   VLSM.output
     {|
       l := existT ix lix;
       input := input;
       destination := destination;
       output := output
     |})
Hchar1: (∃ Hex : existing_equivocator_label
           (IM (projT1 (existT i li)))
           (projT2 (existT i li)),
   existT (projT1 (existT i li))
     (existing_equivocator_label_extract
        (IM (projT1 (existT i li)))
        (projT2 (existT i li)) Hex) =
   l
     {|
       l := existT ix lix;
       input := input;
       destination := destination;
       output := output
     |})
∧ iom =
  VLSM.input
    {|
      l := existT ix lix;
      input := input;
      destination := destination;
      output := output
    |}
  ∧ oom =
    VLSM.output
      {|
        l := existT ix lix;
        input := input;
        destination := destination;
        output := output
      |}
    ∧ equivocators_state_project IM
        final_descriptors s =
      VLSM.destination
        {|
          l := existT ix lix;
          input := input;
          destination := destination;
          output := output
        |}
      ∧ final_descriptors'
          (projT1 (existT i li)) =
        equivocator_label_descriptor
          (projT2 (existT i li))
Hpr_x: equivocators_transition_item_project
  final_descriptors
  {|
    l := existT i li;
    input := iom;
    destination := s;
    output := oom
  |} =
Some
  (Some
     {|
       l := existT ix lix;
       input := input;
       destination := destination;
       output := output
     |}, final_descriptors')
Hvx: equivocator_valid (IM i) li
  (finite_trace_last is tr i, iom)
Htx: (let (si', om') :=
   equivocator_transition 
     (IM i) li (finite_trace_last is tr i, iom) in
 (state_update (equivocator_IM IM)
    (finite_trace_last is tr) i si', om')) =
(s, oom)
finite_valid_trace_from_to PreFree
  (equivocators_state_project IM final_descriptors'
     (finite_trace_last is tr))
  (equivocators_state_project IM final_descriptors s)
  [{|
     l := existT ix lix;
     input := input;
     destination := destination;
     output := output
   |}]
        simpl in *.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
final_descriptors', initial_descriptors: equivocator_descriptors IM
Hproper_initial: proper_equivocator_descriptors IM
  initial_descriptors is
trX: list (composite_transition_item IM)
Hpr_tr: equivocators_trace_project final_descriptors'
  tr = Some (trX, initial_descriptors)
HtrX: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is)
  (equivocators_state_project IM
     final_descriptors'
     (finite_trace_last is tr)) trX
HinitX: composite_initial_state_prop IM
  (equivocators_state_project IM
     initial_descriptors is)
Htr: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm FreeE) is
  (finite_trace_last is tr) tr
s: composite_state (equivocator_IM IM)
iom, oom: option message
i: index
li: EquivocatorLabel (IM i)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  final_descriptors s
ix: index
lix: label (IM ix)
input: option message
destination: composite_state IM
output: option message
Hpr_app: ∀ trX : list (composite_transition_item IM),
  equivocators_trace_project
    final_descriptors
    (tr ++
     [{|
        l := existT i li;
        input := iom;
        destination := s;
        output := oom
      |}]) = Some (trX, initial_descriptors)
  ↔ (∃ (preX
        sufX : list
                 (composite_transition_item
                    IM)) 
       (eqv_descriptors' : 
        equivocator_descriptors IM),
       match
         equivocators_transition_item_project
           final_descriptors
           {|
             l := existT i li;
             input := iom;
             destination := s;
             output := oom
           |}
       with
       | Some (Some item', odescriptor) =>
           Some ([item'], odescriptor)
       | Some (None, odescriptor) =>
           Some ([], odescriptor)
       | None => None
       end = Some (sufX, eqv_descriptors')
       ∧ equivocators_trace_project
           eqv_descriptors' tr =
         Some (preX, initial_descriptors)
         ∧ trX = preX ++ sufX)
Hproper': proper_equivocator_descriptors IM
  final_descriptors'
  (finite_trace_last is tr)
Heq_final_descriptors': final_descriptors' =
equivocator_descriptors_update
  IM final_descriptors i
  (final_descriptors' i)
Heq_last_tr: finite_trace_last is tr =
state_update (equivocator_IM IM) s i
  (finite_trace_last is tr i)
Hex_new: previous_state_descriptor_prop 
  (IM i) (final_descriptors i)
  (finite_trace_last is tr i)
  (final_descriptors' i)
Hchar2: valid lix
  (equivocators_state_project IM
     final_descriptors'
     (finite_trace_last is tr) ix, input)
∧ (let (si', om') :=
     transition lix
       (equivocators_state_project IM
          final_descriptors'
          (finite_trace_last is tr) ix, input) in
   (state_update IM
      (equivocators_state_project IM
         final_descriptors'
         (finite_trace_last is tr)) ix si',
    om')) = (destination, output)
Hchar1: (∃ Hex : existing_equivocator_label (IM i) li,
   existT i
     (existing_equivocator_label_extract
        (IM i) li Hex) = 
   existT ix lix)
∧ iom = input
  ∧ oom = output
    ∧ equivocators_state_project IM
        final_descriptors s = destination
      ∧ final_descriptors' i =
        equivocator_label_descriptor li
Hpr_x: equivocators_transition_item_project
  final_descriptors
  {|
    l := existT i li;
    input := iom;
    destination := s;
    output := oom
  |} =
Some
  (Some
     {|
       l := existT ix lix;
       input := input;
       destination := destination;
       output := output
     |}, final_descriptors')
Hvx: equivocator_valid (IM i) li
  (finite_trace_last is tr i, iom)
Htx: (let (si', om') :=
   equivocator_transition 
     (IM i) li (finite_trace_last is tr i, iom) in
 (state_update (equivocator_IM IM)
    (finite_trace_last is tr) i si', om')) =
(s, oom)
finite_valid_trace_from_to PreFree
  (equivocators_state_project IM final_descriptors'
     (finite_trace_last is tr))
  (equivocators_state_project IM final_descriptors s)
  [{|
     l := existT ix lix;
     input := input;
     destination := destination;
     output := output
   |}]
        destruct Hchar1 as [[Hex Heq_l] [Heq_input [Heq_output [Hpr_s Heq_descli]]]].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
final_descriptors', initial_descriptors: equivocator_descriptors IM
Hproper_initial: proper_equivocator_descriptors IM
  initial_descriptors is
trX: list (composite_transition_item IM)
Hpr_tr: equivocators_trace_project final_descriptors'
  tr = Some (trX, initial_descriptors)
HtrX: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is)
  (equivocators_state_project IM
     final_descriptors'
     (finite_trace_last is tr)) trX
HinitX: composite_initial_state_prop IM
  (equivocators_state_project IM
     initial_descriptors is)
Htr: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm FreeE) is
  (finite_trace_last is tr) tr
s: composite_state (equivocator_IM IM)
iom, oom: option message
i: index
li: EquivocatorLabel (IM i)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  final_descriptors s
ix: index
lix: label (IM ix)
input: option message
destination: composite_state IM
output: option message
Hpr_app: ∀ trX : list (composite_transition_item IM),
  equivocators_trace_project
    final_descriptors
    (tr ++
     [{|
        l := existT i li;
        input := iom;
        destination := s;
        output := oom
      |}]) = Some (trX, initial_descriptors)
  ↔ (∃ (preX
        sufX : list
                 (composite_transition_item
                    IM)) 
       (eqv_descriptors' : 
        equivocator_descriptors IM),
       match
         equivocators_transition_item_project
           final_descriptors
           {|
             l := existT i li;
             input := iom;
             destination := s;
             output := oom
           |}
       with
       | Some (Some item', odescriptor) =>
           Some ([item'], odescriptor)
       | Some (None, odescriptor) =>
           Some ([], odescriptor)
       | None => None
       end = Some (sufX, eqv_descriptors')
       ∧ equivocators_trace_project
           eqv_descriptors' tr =
         Some (preX, initial_descriptors)
         ∧ trX = preX ++ sufX)
Hproper': proper_equivocator_descriptors IM
  final_descriptors'
  (finite_trace_last is tr)
Heq_final_descriptors': final_descriptors' =
equivocator_descriptors_update
  IM final_descriptors i
  (final_descriptors' i)
Heq_last_tr: finite_trace_last is tr =
state_update (equivocator_IM IM) s i
  (finite_trace_last is tr i)
Hex_new: previous_state_descriptor_prop 
  (IM i) (final_descriptors i)
  (finite_trace_last is tr i)
  (final_descriptors' i)
Hchar2: valid lix
  (equivocators_state_project IM
     final_descriptors'
     (finite_trace_last is tr) ix, input)
∧ (let (si', om') :=
     transition lix
       (equivocators_state_project IM
          final_descriptors'
          (finite_trace_last is tr) ix, input) in
   (state_update IM
      (equivocators_state_project IM
         final_descriptors'
         (finite_trace_last is tr)) ix si',
    om')) = (destination, output)
Hex: existing_equivocator_label (IM i) li
Heq_l: existT i
  (existing_equivocator_label_extract 
     (IM i) li Hex) = existT ix lix
Heq_input: iom = input
Heq_output: oom = output
Hpr_s: equivocators_state_project IM
  final_descriptors s = destination
Heq_descli: final_descriptors' i =
equivocator_label_descriptor li
Hpr_x: equivocators_transition_item_project
  final_descriptors
  {|
    l := existT i li;
    input := iom;
    destination := s;
    output := oom
  |} =
Some
  (Some
     {|
       l := existT ix lix;
       input := input;
       destination := destination;
       output := output
     |}, final_descriptors')
Hvx: equivocator_valid (IM i) li
  (finite_trace_last is tr i, iom)
Htx: (let (si', om') :=
   equivocator_transition 
     (IM i) li (finite_trace_last is tr i, iom) in
 (state_update (equivocator_IM IM)
    (finite_trace_last is tr) i si', om')) =
(s, oom)
finite_valid_trace_from_to PreFree
  (equivocators_state_project IM final_descriptors'
     (finite_trace_last is tr))
  (equivocators_state_project IM final_descriptors s)
  [{|
     l := existT ix lix;
     input := input;
     destination := destination;
     output := output
   |}]
        simplify_eq.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
final_descriptors', initial_descriptors: equivocator_descriptors IM
Hproper_initial: proper_equivocator_descriptors IM
  initial_descriptors is
trX: list (composite_transition_item IM)
Hpr_tr: equivocators_trace_project final_descriptors'
  tr = Some (trX, initial_descriptors)
HtrX: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is)
  (equivocators_state_project IM
     final_descriptors'
     (finite_trace_last is tr)) trX
HinitX: composite_initial_state_prop IM
  (equivocators_state_project IM
     initial_descriptors is)
Htr: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm FreeE) is
  (finite_trace_last is tr) tr
s: composite_state (equivocator_IM IM)
ix: index
li: EquivocatorLabel (IM ix)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  final_descriptors s
input, output: option message
Hpr_app: ∀ trX : list (composite_transition_item IM),
  equivocators_trace_project
    final_descriptors
    (tr ++
     [{|
        l := existT ix li;
        input := input;
        destination := s;
        output := output
      |}]) = Some (trX, initial_descriptors)
  ↔ (∃ (preX
        sufX : list
                 (composite_transition_item
                    IM)) 
       (eqv_descriptors' : 
        equivocator_descriptors IM),
       match
         equivocators_transition_item_project
           final_descriptors
           {|
             l := existT ix li;
             input := input;
             destination := s;
             output := output
           |}
       with
       | Some (Some item', odescriptor) =>
           Some ([item'], odescriptor)
       | Some (None, odescriptor) =>
           Some ([], odescriptor)
       | None => None
       end = Some (sufX, eqv_descriptors')
       ∧ equivocators_trace_project
           eqv_descriptors' tr =
         Some (preX, initial_descriptors)
         ∧ trX = preX ++ sufX)
Hproper': proper_equivocator_descriptors IM
  final_descriptors'
  (finite_trace_last is tr)
Hex_new: previous_state_descriptor_prop 
  (IM ix) (final_descriptors ix)
  (finite_trace_last is tr ix)
  (final_descriptors' ix)
Heq_last_tr: finite_trace_last is tr =
state_update (equivocator_IM IM) s ix
  (finite_trace_last is tr ix)
Heq_final_descriptors': final_descriptors' =
equivocator_descriptors_update
  IM final_descriptors ix
  (final_descriptors' ix)
Hex: existing_equivocator_label (IM ix) li
Hpr_x: equivocators_transition_item_project
  final_descriptors
  {|
    l := existT ix li;
    input := input;
    destination := s;
    output := output
  |} =
Some
  (Some
     {|
       l :=
         existT ix
           (existing_equivocator_label_extract
              (IM ix) li Hex);
       input := input;
       destination :=
         equivocators_state_project IM
           final_descriptors s;
       output := output
     |}, final_descriptors')
Hchar2: valid
  (existing_equivocator_label_extract 
     (IM ix) li Hex)
  (equivocators_state_project IM
     final_descriptors'
     (finite_trace_last is tr) ix, input)
∧ (let (si', om') :=
     transition
       (existing_equivocator_label_extract
          (IM ix) li Hex)
       (equivocators_state_project IM
          final_descriptors'
          (finite_trace_last is tr) ix, input) in
   (state_update IM
      (equivocators_state_project IM
         final_descriptors'
         (finite_trace_last is tr)) ix si',
    om')) =
  (equivocators_state_project IM
     final_descriptors s, output)
Hvx: equivocator_valid (IM ix) li
  (finite_trace_last is tr ix, input)
Htx: (let (si', om') :=
   equivocator_transition 
     (IM ix) li
     (finite_trace_last is tr ix, input) in
 (state_update (equivocator_IM IM)
    (finite_trace_last is tr) ix si', om')) =
(s, output)
Heq_descli: final_descriptors' ix =
equivocator_label_descriptor li
finite_valid_trace_from_to PreFree
  (equivocators_state_project IM final_descriptors'
     (finite_trace_last is tr))
  (equivocators_state_project IM final_descriptors s)
  [{|
     l :=
       existT ix
         (existing_equivocator_label_extract (IM ix)
            li Hex);
     input := input;
     destination :=
       equivocators_state_project IM final_descriptors
         s;
     output := output
   |}]
        destruct Hchar2 as [Hvx_pr Htx_pr].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
final_descriptors', initial_descriptors: equivocator_descriptors IM
Hproper_initial: proper_equivocator_descriptors IM
  initial_descriptors is
trX: list (composite_transition_item IM)
Hpr_tr: equivocators_trace_project final_descriptors'
  tr = Some (trX, initial_descriptors)
HtrX: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is)
  (equivocators_state_project IM
     final_descriptors'
     (finite_trace_last is tr)) trX
HinitX: composite_initial_state_prop IM
  (equivocators_state_project IM
     initial_descriptors is)
Htr: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm FreeE) is
  (finite_trace_last is tr) tr
s: composite_state (equivocator_IM IM)
ix: index
li: EquivocatorLabel (IM ix)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  final_descriptors s
input, output: option message
Hpr_app: ∀ trX : list (composite_transition_item IM),
  equivocators_trace_project
    final_descriptors
    (tr ++
     [{|
        l := existT ix li;
        input := input;
        destination := s;
        output := output
      |}]) = Some (trX, initial_descriptors)
  ↔ (∃ (preX
        sufX : list
                 (composite_transition_item
                    IM)) 
       (eqv_descriptors' : 
        equivocator_descriptors IM),
       match
         equivocators_transition_item_project
           final_descriptors
           {|
             l := existT ix li;
             input := input;
             destination := s;
             output := output
           |}
       with
       | Some (Some item', odescriptor) =>
           Some ([item'], odescriptor)
       | Some (None, odescriptor) =>
           Some ([], odescriptor)
       | None => None
       end = Some (sufX, eqv_descriptors')
       ∧ equivocators_trace_project
           eqv_descriptors' tr =
         Some (preX, initial_descriptors)
         ∧ trX = preX ++ sufX)
Hproper': proper_equivocator_descriptors IM
  final_descriptors'
  (finite_trace_last is tr)
Hex_new: previous_state_descriptor_prop 
  (IM ix) (final_descriptors ix)
  (finite_trace_last is tr ix)
  (final_descriptors' ix)
Heq_last_tr: finite_trace_last is tr =
state_update (equivocator_IM IM) s ix
  (finite_trace_last is tr ix)
Heq_final_descriptors': final_descriptors' =
equivocator_descriptors_update
  IM final_descriptors ix
  (final_descriptors' ix)
Hex: existing_equivocator_label (IM ix) li
Hpr_x: equivocators_transition_item_project
  final_descriptors
  {|
    l := existT ix li;
    input := input;
    destination := s;
    output := output
  |} =
Some
  (Some
     {|
       l :=
         existT ix
           (existing_equivocator_label_extract
              (IM ix) li Hex);
       input := input;
       destination :=
         equivocators_state_project IM
           final_descriptors s;
       output := output
     |}, final_descriptors')
Hvx_pr: valid
  (existing_equivocator_label_extract 
     (IM ix) li Hex)
  (equivocators_state_project IM
     final_descriptors'
     (finite_trace_last is tr) ix, input)
Htx_pr: (let (si', om') :=
   transition
     (existing_equivocator_label_extract
        (IM ix) li Hex)
     (equivocators_state_project IM
        final_descriptors'
        (finite_trace_last is tr) ix, input) in
 (state_update IM
    (equivocators_state_project IM
       final_descriptors'
       (finite_trace_last is tr)) ix si', om')) =
(equivocators_state_project IM
   final_descriptors s, output)
Hvx: equivocator_valid (IM ix) li
  (finite_trace_last is tr ix, input)
Htx: (let (si', om') :=
   equivocator_transition 
     (IM ix) li
     (finite_trace_last is tr ix, input) in
 (state_update (equivocator_IM IM)
    (finite_trace_last is tr) ix si', om')) =
(s, output)
Heq_descli: final_descriptors' ix =
equivocator_label_descriptor li
finite_valid_trace_from_to PreFree
  (equivocators_state_project IM final_descriptors'
     (finite_trace_last is tr))
  (equivocators_state_project IM final_descriptors s)
  [{|
     l :=
       existT ix
         (existing_equivocator_label_extract (IM ix)
            li Hex);
     input := input;
     destination :=
       equivocators_state_project IM final_descriptors
         s;
     output := output
   |}]
        apply finite_valid_trace_from_to_singleton.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
final_descriptors', initial_descriptors: equivocator_descriptors IM
Hproper_initial: proper_equivocator_descriptors IM
  initial_descriptors is
trX: list (composite_transition_item IM)
Hpr_tr: equivocators_trace_project final_descriptors'
  tr = Some (trX, initial_descriptors)
HtrX: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is)
  (equivocators_state_project IM
     final_descriptors'
     (finite_trace_last is tr)) trX
HinitX: composite_initial_state_prop IM
  (equivocators_state_project IM
     initial_descriptors is)
Htr: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm FreeE) is
  (finite_trace_last is tr) tr
s: composite_state (equivocator_IM IM)
ix: index
li: EquivocatorLabel (IM ix)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  final_descriptors s
input, output: option message
Hpr_app: ∀ trX : list (composite_transition_item IM),
  equivocators_trace_project
    final_descriptors
    (tr ++
     [{|
        l := existT ix li;
        input := input;
        destination := s;
        output := output
      |}]) = Some (trX, initial_descriptors)
  ↔ (∃ (preX
        sufX : list
                 (composite_transition_item
                    IM)) 
       (eqv_descriptors' : 
        equivocator_descriptors IM),
       match
         equivocators_transition_item_project
           final_descriptors
           {|
             l := existT ix li;
             input := input;
             destination := s;
             output := output
           |}
       with
       | Some (Some item', odescriptor) =>
           Some ([item'], odescriptor)
       | Some (None, odescriptor) =>
           Some ([], odescriptor)
       | None => None
       end = Some (sufX, eqv_descriptors')
       ∧ equivocators_trace_project
           eqv_descriptors' tr =
         Some (preX, initial_descriptors)
         ∧ trX = preX ++ sufX)
Hproper': proper_equivocator_descriptors IM
  final_descriptors'
  (finite_trace_last is tr)
Hex_new: previous_state_descriptor_prop 
  (IM ix) (final_descriptors ix)
  (finite_trace_last is tr ix)
  (final_descriptors' ix)
Heq_last_tr: finite_trace_last is tr =
state_update (equivocator_IM IM) s ix
  (finite_trace_last is tr ix)
Heq_final_descriptors': final_descriptors' =
equivocator_descriptors_update
  IM final_descriptors ix
  (final_descriptors' ix)
Hex: existing_equivocator_label (IM ix) li
Hpr_x: equivocators_transition_item_project
  final_descriptors
  {|
    l := existT ix li;
    input := input;
    destination := s;
    output := output
  |} =
Some
  (Some
     {|
       l :=
         existT ix
           (existing_equivocator_label_extract
              (IM ix) li Hex);
       input := input;
       destination :=
         equivocators_state_project IM
           final_descriptors s;
       output := output
     |}, final_descriptors')
Hvx_pr: valid
  (existing_equivocator_label_extract 
     (IM ix) li Hex)
  (equivocators_state_project IM
     final_descriptors'
     (finite_trace_last is tr) ix, input)
Htx_pr: (let (si', om') :=
   transition
     (existing_equivocator_label_extract
        (IM ix) li Hex)
     (equivocators_state_project IM
        final_descriptors'
        (finite_trace_last is tr) ix, input) in
 (state_update IM
    (equivocators_state_project IM
       final_descriptors'
       (finite_trace_last is tr)) ix si', om')) =
(equivocators_state_project IM
   final_descriptors s, output)
Hvx: equivocator_valid (IM ix) li
  (finite_trace_last is tr ix, input)
Htx: (let (si', om') :=
   equivocator_transition 
     (IM ix) li
     (finite_trace_last is tr ix, input) in
 (state_update (equivocator_IM IM)
    (finite_trace_last is tr) ix si', om')) =
(s, output)
Heq_descli: final_descriptors' ix =
equivocator_label_descriptor li
input_valid_transition PreFree
  (existT ix
     (existing_equivocator_label_extract (IM ix) li
        Hex))
  (equivocators_state_project IM final_descriptors'
     (finite_trace_last is tr), input)
  (equivocators_state_project IM final_descriptors s,
   output)
        repeat split
        ; [| apply any_message_is_valid_in_preloaded | done | done].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
final_descriptors', initial_descriptors: equivocator_descriptors IM
Hproper_initial: proper_equivocator_descriptors IM
  initial_descriptors is
trX: list (composite_transition_item IM)
Hpr_tr: equivocators_trace_project final_descriptors'
  tr = Some (trX, initial_descriptors)
HtrX: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is)
  (equivocators_state_project IM
     final_descriptors'
     (finite_trace_last is tr)) trX
HinitX: composite_initial_state_prop IM
  (equivocators_state_project IM
     initial_descriptors is)
Htr: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm FreeE) is
  (finite_trace_last is tr) tr
s: composite_state (equivocator_IM IM)
ix: index
li: EquivocatorLabel (IM ix)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  final_descriptors s
input, output: option message
Hpr_app: ∀ trX : list (composite_transition_item IM),
  equivocators_trace_project
    final_descriptors
    (tr ++
     [{|
        l := existT ix li;
        input := input;
        destination := s;
        output := output
      |}]) = Some (trX, initial_descriptors)
  ↔ (∃ (preX
        sufX : list
                 (composite_transition_item
                    IM)) 
       (eqv_descriptors' : 
        equivocator_descriptors IM),
       match
         equivocators_transition_item_project
           final_descriptors
           {|
             l := existT ix li;
             input := input;
             destination := s;
             output := output
           |}
       with
       | Some (Some item', odescriptor) =>
           Some ([item'], odescriptor)
       | Some (None, odescriptor) =>
           Some ([], odescriptor)
       | None => None
       end = Some (sufX, eqv_descriptors')
       ∧ equivocators_trace_project
           eqv_descriptors' tr =
         Some (preX, initial_descriptors)
         ∧ trX = preX ++ sufX)
Hproper': proper_equivocator_descriptors IM
  final_descriptors'
  (finite_trace_last is tr)
Hex_new: previous_state_descriptor_prop 
  (IM ix) (final_descriptors ix)
  (finite_trace_last is tr ix)
  (final_descriptors' ix)
Heq_last_tr: finite_trace_last is tr =
state_update (equivocator_IM IM) s ix
  (finite_trace_last is tr ix)
Heq_final_descriptors': final_descriptors' =
equivocator_descriptors_update
  IM final_descriptors ix
  (final_descriptors' ix)
Hex: existing_equivocator_label (IM ix) li
Hpr_x: equivocators_transition_item_project
  final_descriptors
  {|
    l := existT ix li;
    input := input;
    destination := s;
    output := output
  |} =
Some
  (Some
     {|
       l :=
         existT ix
           (existing_equivocator_label_extract
              (IM ix) li Hex);
       input := input;
       destination :=
         equivocators_state_project IM
           final_descriptors s;
       output := output
     |}, final_descriptors')
Hvx_pr: valid
  (existing_equivocator_label_extract 
     (IM ix) li Hex)
  (equivocators_state_project IM
     final_descriptors'
     (finite_trace_last is tr) ix, input)
Htx_pr: (let (si', om') :=
   transition
     (existing_equivocator_label_extract
        (IM ix) li Hex)
     (equivocators_state_project IM
        final_descriptors'
        (finite_trace_last is tr) ix, input) in
 (state_update IM
    (equivocators_state_project IM
       final_descriptors'
       (finite_trace_last is tr)) ix si', om')) =
(equivocators_state_project IM
   final_descriptors s, output)
Hvx: equivocator_valid (IM ix) li
  (finite_trace_last is tr ix, input)
Htx: (let (si', om') :=
   equivocator_transition 
     (IM ix) li
     (finite_trace_last is tr ix, input) in
 (state_update (equivocator_IM IM)
    (finite_trace_last is tr) ix si', om')) =
(s, output)
Heq_descli: final_descriptors' ix =
equivocator_label_descriptor li
valid_state_prop PreFree
  (equivocators_state_project IM final_descriptors'
     (finite_trace_last is tr))
        by apply finite_valid_trace_from_to_last_pstate in HtrX.
    +message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
final_descriptors', initial_descriptors: equivocator_descriptors IM
Hproper_initial: proper_equivocator_descriptors IM
  initial_descriptors is
trX: list (composite_transition_item IM)
Hpr_tr: equivocators_trace_project final_descriptors'
  tr = Some (trX, initial_descriptors)
HtrX: finite_constrained_trace_init_to Free
  (equivocators_state_project IM
     initial_descriptors is)
  (equivocators_state_project IM
     final_descriptors'
     (finite_trace_last is tr)) trX
Htr: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm FreeE) is
  (finite_trace_last is tr) tr
s: composite_state (equivocator_IM IM)
iom, oom: option message
l: composite_label (equivocator_IM IM)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  final_descriptors s
Hpr_app: ∀ trX : list (composite_transition_item IM),
  equivocators_trace_project
    final_descriptors
    (tr ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}]) = Some (trX, initial_descriptors)
  ↔ (∃ (preX
        sufX : list
                 (composite_transition_item
                    IM)) 
       (eqv_descriptors' : 
        equivocator_descriptors IM),
       match
         equivocators_transition_item_project
           final_descriptors
           {|
             l := l;
             input := iom;
             destination := s;
             output := oom
           |}
       with
       | Some (Some item', odescriptor) =>
           Some ([item'], odescriptor)
       | Some (None, odescriptor) =>
           Some ([], odescriptor)
       | None => None
       end = Some (sufX, eqv_descriptors')
       ∧ equivocators_trace_project
           eqv_descriptors' tr =
         Some (preX, initial_descriptors)
         ∧ trX = preX ++ sufX)
Hproper': proper_equivocator_descriptors IM
  final_descriptors'
  (finite_trace_last is tr)
Heq_final_descriptors': final_descriptors' =
equivocator_descriptors_update
  IM final_descriptors
  (projT1 l)
  (final_descriptors'
     (projT1 l))
Heq_last_tr: finite_trace_last is tr =
state_update (equivocator_IM IM) s
  (projT1 l)
  (finite_trace_last is tr (projT1 l))
Hex_new: previous_state_descriptor_prop
  (IM (projT1 l))
  (final_descriptors (projT1 l))
  (finite_trace_last is tr (projT1 l))
  (final_descriptors' (projT1 l))
Hchar2: equivocators_state_project IM
  final_descriptors s =
equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr)
Hchar1: True
Hpr_x: equivocators_transition_item_project
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some (None, final_descriptors')
Hvx: let (i, li) := l in
equivocator_valid (IM i) li
  (finite_trace_last is tr i, iom)
Htx: (let (i, li) := l in
 let (si', om') :=
   equivocator_transition 
     (IM i) li (finite_trace_last is tr i, iom) in
 (state_update (equivocator_IM IM)
    (finite_trace_last is tr) i si', om')) =
(s, oom)
∃ trX : list (composite_transition_item IM),
  equivocators_trace_project final_descriptors
    (tr ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}]) = Some (trX, initial_descriptors)
  ∧ finite_constrained_trace_init_to Free
      (equivocators_state_project IM
         initial_descriptors is)
      (equivocators_state_project IM final_descriptors
         s) trX exists trX.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
final_descriptors', initial_descriptors: equivocator_descriptors IM
Hproper_initial: proper_equivocator_descriptors IM
  initial_descriptors is
trX: list (composite_transition_item IM)
Hpr_tr: equivocators_trace_project final_descriptors'
  tr = Some (trX, initial_descriptors)
HtrX: finite_constrained_trace_init_to Free
  (equivocators_state_project IM
     initial_descriptors is)
  (equivocators_state_project IM
     final_descriptors'
     (finite_trace_last is tr)) trX
Htr: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm FreeE) is
  (finite_trace_last is tr) tr
s: composite_state (equivocator_IM IM)
iom, oom: option message
l: composite_label (equivocator_IM IM)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  final_descriptors s
Hpr_app: ∀ trX : list (composite_transition_item IM),
  equivocators_trace_project
    final_descriptors
    (tr ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}]) = Some (trX, initial_descriptors)
  ↔ (∃ (preX
        sufX : list
                 (composite_transition_item
                    IM)) 
       (eqv_descriptors' : 
        equivocator_descriptors IM),
       match
         equivocators_transition_item_project
           final_descriptors
           {|
             l := l;
             input := iom;
             destination := s;
             output := oom
           |}
       with
       | Some (Some item', odescriptor) =>
           Some ([item'], odescriptor)
       | Some (None, odescriptor) =>
           Some ([], odescriptor)
       | None => None
       end = Some (sufX, eqv_descriptors')
       ∧ equivocators_trace_project
           eqv_descriptors' tr =
         Some (preX, initial_descriptors)
         ∧ trX = preX ++ sufX)
Hproper': proper_equivocator_descriptors IM
  final_descriptors'
  (finite_trace_last is tr)
Heq_final_descriptors': final_descriptors' =
equivocator_descriptors_update
  IM final_descriptors
  (projT1 l)
  (final_descriptors'
     (projT1 l))
Heq_last_tr: finite_trace_last is tr =
state_update (equivocator_IM IM) s
  (projT1 l)
  (finite_trace_last is tr (projT1 l))
Hex_new: previous_state_descriptor_prop
  (IM (projT1 l))
  (final_descriptors (projT1 l))
  (finite_trace_last is tr (projT1 l))
  (final_descriptors' (projT1 l))
Hchar2: equivocators_state_project IM
  final_descriptors s =
equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr)
Hchar1: True
Hpr_x: equivocators_transition_item_project
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some (None, final_descriptors')
Hvx: let (i, li) := l in
equivocator_valid (IM i) li
  (finite_trace_last is tr i, iom)
Htx: (let (i, li) := l in
 let (si', om') :=
   equivocator_transition 
     (IM i) li (finite_trace_last is tr i, iom) in
 (state_update (equivocator_IM IM)
    (finite_trace_last is tr) i si', om')) =
(s, oom)
equivocators_trace_project final_descriptors
  (tr ++
   [{|
      l := l;
      input := iom;
      destination := s;
      output := oom
    |}]) = Some (trX, initial_descriptors)
∧ finite_constrained_trace_init_to Free
    (equivocators_state_project IM initial_descriptors
       is)
    (equivocators_state_project IM final_descriptors s)
    trX clear Hchar1.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
final_descriptors', initial_descriptors: equivocator_descriptors IM
Hproper_initial: proper_equivocator_descriptors IM
  initial_descriptors is
trX: list (composite_transition_item IM)
Hpr_tr: equivocators_trace_project final_descriptors'
  tr = Some (trX, initial_descriptors)
HtrX: finite_constrained_trace_init_to Free
  (equivocators_state_project IM
     initial_descriptors is)
  (equivocators_state_project IM
     final_descriptors'
     (finite_trace_last is tr)) trX
Htr: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm FreeE) is
  (finite_trace_last is tr) tr
s: composite_state (equivocator_IM IM)
iom, oom: option message
l: composite_label (equivocator_IM IM)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  final_descriptors s
Hpr_app: ∀ trX : list (composite_transition_item IM),
  equivocators_trace_project
    final_descriptors
    (tr ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}]) = Some (trX, initial_descriptors)
  ↔ (∃ (preX
        sufX : list
                 (composite_transition_item
                    IM)) 
       (eqv_descriptors' : 
        equivocator_descriptors IM),
       match
         equivocators_transition_item_project
           final_descriptors
           {|
             l := l;
             input := iom;
             destination := s;
             output := oom
           |}
       with
       | Some (Some item', odescriptor) =>
           Some ([item'], odescriptor)
       | Some (None, odescriptor) =>
           Some ([], odescriptor)
       | None => None
       end = Some (sufX, eqv_descriptors')
       ∧ equivocators_trace_project
           eqv_descriptors' tr =
         Some (preX, initial_descriptors)
         ∧ trX = preX ++ sufX)
Hproper': proper_equivocator_descriptors IM
  final_descriptors'
  (finite_trace_last is tr)
Heq_final_descriptors': final_descriptors' =
equivocator_descriptors_update
  IM final_descriptors
  (projT1 l)
  (final_descriptors'
     (projT1 l))
Heq_last_tr: finite_trace_last is tr =
state_update (equivocator_IM IM) s
  (projT1 l)
  (finite_trace_last is tr (projT1 l))
Hex_new: previous_state_descriptor_prop
  (IM (projT1 l))
  (final_descriptors (projT1 l))
  (finite_trace_last is tr (projT1 l))
  (final_descriptors' (projT1 l))
Hchar2: equivocators_state_project IM
  final_descriptors s =
equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr)
Hpr_x: equivocators_transition_item_project
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some (None, final_descriptors')
Hvx: let (i, li) := l in
equivocator_valid (IM i) li
  (finite_trace_last is tr i, iom)
Htx: (let (i, li) := l in
 let (si', om') :=
   equivocator_transition 
     (IM i) li (finite_trace_last is tr i, iom) in
 (state_update (equivocator_IM IM)
    (finite_trace_last is tr) i si', om')) =
(s, oom)
equivocators_trace_project final_descriptors
  (tr ++
   [{|
      l := l;
      input := iom;
      destination := s;
      output := oom
    |}]) = Some (trX, initial_descriptors)
∧ finite_constrained_trace_init_to Free
    (equivocators_state_project IM initial_descriptors
       is)
    (equivocators_state_project IM final_descriptors s)
    trX rewrite Hchar2.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
final_descriptors', initial_descriptors: equivocator_descriptors IM
Hproper_initial: proper_equivocator_descriptors IM
  initial_descriptors is
trX: list (composite_transition_item IM)
Hpr_tr: equivocators_trace_project final_descriptors'
  tr = Some (trX, initial_descriptors)
HtrX: finite_constrained_trace_init_to Free
  (equivocators_state_project IM
     initial_descriptors is)
  (equivocators_state_project IM
     final_descriptors'
     (finite_trace_last is tr)) trX
Htr: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm FreeE) is
  (finite_trace_last is tr) tr
s: composite_state (equivocator_IM IM)
iom, oom: option message
l: composite_label (equivocator_IM IM)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  final_descriptors s
Hpr_app: ∀ trX : list (composite_transition_item IM),
  equivocators_trace_project
    final_descriptors
    (tr ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}]) = Some (trX, initial_descriptors)
  ↔ (∃ (preX
        sufX : list
                 (composite_transition_item
                    IM)) 
       (eqv_descriptors' : 
        equivocator_descriptors IM),
       match
         equivocators_transition_item_project
           final_descriptors
           {|
             l := l;
             input := iom;
             destination := s;
             output := oom
           |}
       with
       | Some (Some item', odescriptor) =>
           Some ([item'], odescriptor)
       | Some (None, odescriptor) =>
           Some ([], odescriptor)
       | None => None
       end = Some (sufX, eqv_descriptors')
       ∧ equivocators_trace_project
           eqv_descriptors' tr =
         Some (preX, initial_descriptors)
         ∧ trX = preX ++ sufX)
Hproper': proper_equivocator_descriptors IM
  final_descriptors'
  (finite_trace_last is tr)
Heq_final_descriptors': final_descriptors' =
equivocator_descriptors_update
  IM final_descriptors
  (projT1 l)
  (final_descriptors'
     (projT1 l))
Heq_last_tr: finite_trace_last is tr =
state_update (equivocator_IM IM) s
  (projT1 l)
  (finite_trace_last is tr (projT1 l))
Hex_new: previous_state_descriptor_prop
  (IM (projT1 l))
  (final_descriptors (projT1 l))
  (finite_trace_last is tr (projT1 l))
  (final_descriptors' (projT1 l))
Hchar2: equivocators_state_project IM
  final_descriptors s =
equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr)
Hpr_x: equivocators_transition_item_project
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some (None, final_descriptors')
Hvx: let (i, li) := l in
equivocator_valid (IM i) li
  (finite_trace_last is tr i, iom)
Htx: (let (i, li) := l in
 let (si', om') :=
   equivocator_transition 
     (IM i) li (finite_trace_last is tr i, iom) in
 (state_update (equivocator_IM IM)
    (finite_trace_last is tr) i si', om')) =
(s, oom)
equivocators_trace_project final_descriptors
  (tr ++
   [{|
      l := l;
      input := iom;
      destination := s;
      output := oom
    |}]) = Some (trX, initial_descriptors)
∧ finite_constrained_trace_init_to Free
    (equivocators_state_project IM initial_descriptors
       is)
    (equivocators_state_project IM final_descriptors'
       (finite_trace_last is tr)) trX
      split; [| done].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
final_descriptors', initial_descriptors: equivocator_descriptors IM
Hproper_initial: proper_equivocator_descriptors IM
  initial_descriptors is
trX: list (composite_transition_item IM)
Hpr_tr: equivocators_trace_project final_descriptors'
  tr = Some (trX, initial_descriptors)
HtrX: finite_constrained_trace_init_to Free
  (equivocators_state_project IM
     initial_descriptors is)
  (equivocators_state_project IM
     final_descriptors'
     (finite_trace_last is tr)) trX
Htr: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm FreeE) is
  (finite_trace_last is tr) tr
s: composite_state (equivocator_IM IM)
iom, oom: option message
l: composite_label (equivocator_IM IM)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  final_descriptors s
Hpr_app: ∀ trX : list (composite_transition_item IM),
  equivocators_trace_project
    final_descriptors
    (tr ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}]) = Some (trX, initial_descriptors)
  ↔ (∃ (preX
        sufX : list
                 (composite_transition_item
                    IM)) 
       (eqv_descriptors' : 
        equivocator_descriptors IM),
       match
         equivocators_transition_item_project
           final_descriptors
           {|
             l := l;
             input := iom;
             destination := s;
             output := oom
           |}
       with
       | Some (Some item', odescriptor) =>
           Some ([item'], odescriptor)
       | Some (None, odescriptor) =>
           Some ([], odescriptor)
       | None => None
       end = Some (sufX, eqv_descriptors')
       ∧ equivocators_trace_project
           eqv_descriptors' tr =
         Some (preX, initial_descriptors)
         ∧ trX = preX ++ sufX)
Hproper': proper_equivocator_descriptors IM
  final_descriptors'
  (finite_trace_last is tr)
Heq_final_descriptors': final_descriptors' =
equivocator_descriptors_update
  IM final_descriptors
  (projT1 l)
  (final_descriptors'
     (projT1 l))
Heq_last_tr: finite_trace_last is tr =
state_update (equivocator_IM IM) s
  (projT1 l)
  (finite_trace_last is tr (projT1 l))
Hex_new: previous_state_descriptor_prop
  (IM (projT1 l))
  (final_descriptors (projT1 l))
  (finite_trace_last is tr (projT1 l))
  (final_descriptors' (projT1 l))
Hchar2: equivocators_state_project IM
  final_descriptors s =
equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr)
Hpr_x: equivocators_transition_item_project
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some (None, final_descriptors')
Hvx: let (i, li) := l in
equivocator_valid (IM i) li
  (finite_trace_last is tr i, iom)
Htx: (let (i, li) := l in
 let (si', om') :=
   equivocator_transition 
     (IM i) li (finite_trace_last is tr i, iom) in
 (state_update (equivocator_IM IM)
    (finite_trace_last is tr) i si', om')) =
(s, oom)
equivocators_trace_project final_descriptors
  (tr ++
   [{|
      l := l;
      input := iom;
      destination := s;
      output := oom
    |}]) = Some (trX, initial_descriptors)
      apply (Hpr_app trX).message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
final_descriptors', initial_descriptors: equivocator_descriptors IM
Hproper_initial: proper_equivocator_descriptors IM
  initial_descriptors is
trX: list (composite_transition_item IM)
Hpr_tr: equivocators_trace_project final_descriptors'
  tr = Some (trX, initial_descriptors)
HtrX: finite_constrained_trace_init_to Free
  (equivocators_state_project IM
     initial_descriptors is)
  (equivocators_state_project IM
     final_descriptors'
     (finite_trace_last is tr)) trX
Htr: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm FreeE) is
  (finite_trace_last is tr) tr
s: composite_state (equivocator_IM IM)
iom, oom: option message
l: composite_label (equivocator_IM IM)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  final_descriptors s
Hpr_app: ∀ trX : list (composite_transition_item IM),
  equivocators_trace_project
    final_descriptors
    (tr ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}]) = Some (trX, initial_descriptors)
  ↔ (∃ (preX
        sufX : list
                 (composite_transition_item
                    IM)) 
       (eqv_descriptors' : 
        equivocator_descriptors IM),
       match
         equivocators_transition_item_project
           final_descriptors
           {|
             l := l;
             input := iom;
             destination := s;
             output := oom
           |}
       with
       | Some (Some item', odescriptor) =>
           Some ([item'], odescriptor)
       | Some (None, odescriptor) =>
           Some ([], odescriptor)
       | None => None
       end = Some (sufX, eqv_descriptors')
       ∧ equivocators_trace_project
           eqv_descriptors' tr =
         Some (preX, initial_descriptors)
         ∧ trX = preX ++ sufX)
Hproper': proper_equivocator_descriptors IM
  final_descriptors'
  (finite_trace_last is tr)
Heq_final_descriptors': final_descriptors' =
equivocator_descriptors_update
  IM final_descriptors
  (projT1 l)
  (final_descriptors'
     (projT1 l))
Heq_last_tr: finite_trace_last is tr =
state_update (equivocator_IM IM) s
  (projT1 l)
  (finite_trace_last is tr (projT1 l))
Hex_new: previous_state_descriptor_prop
  (IM (projT1 l))
  (final_descriptors (projT1 l))
  (finite_trace_last is tr (projT1 l))
  (final_descriptors' (projT1 l))
Hchar2: equivocators_state_project IM
  final_descriptors s =
equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr)
Hpr_x: equivocators_transition_item_project
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some (None, final_descriptors')
Hvx: let (i, li) := l in
equivocator_valid (IM i) li
  (finite_trace_last is tr i, iom)
Htx: (let (i, li) := l in
 let (si', om') :=
   equivocator_transition 
     (IM i) li (finite_trace_last is tr i, iom) in
 (state_update (equivocator_IM IM)
    (finite_trace_last is tr) i si', om')) =
(s, oom)
∃ (preX sufX : list (composite_transition_item IM)) 
  (eqv_descriptors' : equivocator_descriptors IM),
  match
    equivocators_transition_item_project
      final_descriptors
      {|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}
  with
  | Some (Some item', odescriptor) =>
      Some ([item'], odescriptor)
  | Some (None, odescriptor) => Some ([], odescriptor)
  | None => None
  end = Some (sufX, eqv_descriptors')
  ∧ equivocators_trace_project eqv_descriptors' tr =
    Some (preX, initial_descriptors)
    ∧ trX = preX ++ sufX
      exists trX, [], final_descriptors'.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
final_descriptors', initial_descriptors: equivocator_descriptors IM
Hproper_initial: proper_equivocator_descriptors IM
  initial_descriptors is
trX: list (composite_transition_item IM)
Hpr_tr: equivocators_trace_project final_descriptors'
  tr = Some (trX, initial_descriptors)
HtrX: finite_constrained_trace_init_to Free
  (equivocators_state_project IM
     initial_descriptors is)
  (equivocators_state_project IM
     final_descriptors'
     (finite_trace_last is tr)) trX
Htr: finite_valid_trace_from_to
  (preloaded_with_all_messages_vlsm FreeE) is
  (finite_trace_last is tr) tr
s: composite_state (equivocator_IM IM)
iom, oom: option message
l: composite_label (equivocator_IM IM)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  final_descriptors s
Hpr_app: ∀ trX : list (composite_transition_item IM),
  equivocators_trace_project
    final_descriptors
    (tr ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}]) = Some (trX, initial_descriptors)
  ↔ (∃ (preX
        sufX : list
                 (composite_transition_item
                    IM)) 
       (eqv_descriptors' : 
        equivocator_descriptors IM),
       match
         equivocators_transition_item_project
           final_descriptors
           {|
             l := l;
             input := iom;
             destination := s;
             output := oom
           |}
       with
       | Some (Some item', odescriptor) =>
           Some ([item'], odescriptor)
       | Some (None, odescriptor) =>
           Some ([], odescriptor)
       | None => None
       end = Some (sufX, eqv_descriptors')
       ∧ equivocators_trace_project
           eqv_descriptors' tr =
         Some (preX, initial_descriptors)
         ∧ trX = preX ++ sufX)
Hproper': proper_equivocator_descriptors IM
  final_descriptors'
  (finite_trace_last is tr)
Heq_final_descriptors': final_descriptors' =
equivocator_descriptors_update
  IM final_descriptors
  (projT1 l)
  (final_descriptors'
     (projT1 l))
Heq_last_tr: finite_trace_last is tr =
state_update (equivocator_IM IM) s
  (projT1 l)
  (finite_trace_last is tr (projT1 l))
Hex_new: previous_state_descriptor_prop
  (IM (projT1 l))
  (final_descriptors (projT1 l))
  (finite_trace_last is tr (projT1 l))
  (final_descriptors' (projT1 l))
Hchar2: equivocators_state_project IM
  final_descriptors s =
equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr)
Hpr_x: equivocators_transition_item_project
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some (None, final_descriptors')
Hvx: let (i, li) := l in
equivocator_valid (IM i) li
  (finite_trace_last is tr i, iom)
Htx: (let (i, li) := l in
 let (si', om') :=
   equivocator_transition 
     (IM i) li (finite_trace_last is tr i, iom) in
 (state_update (equivocator_IM IM)
    (finite_trace_last is tr) i si', om')) =
(s, oom)
match
  equivocators_transition_item_project
    final_descriptors
    {|
      l := l;
      input := iom;
      destination := s;
      output := oom
    |}
with
| Some (Some item', odescriptor) =>
    Some ([item'], odescriptor)
| Some (None, odescriptor) => Some ([], odescriptor)
| None => None
end = Some ([], final_descriptors')
∧ equivocators_trace_project final_descriptors' tr =
  Some (trX, initial_descriptors) ∧ trX = trX ++ []
      by rewrite Hpr_x, app_nil_r.
Qed.

Definition equivocators_partial_trace_project
  (final_descriptors : equivocator_descriptors IM)
  (str : composite_state (equivocator_IM IM) * list (composite_transition_item (equivocator_IM IM)))
  : option (composite_state IM * list (composite_transition_item IM))
  :=
  let (s, tr) := str in
  if
    (decide (not_equivocating_equivocator_descriptors IM final_descriptors
      (finite_trace_last s tr)))
  then
    match equivocators_trace_project final_descriptors tr with
    | None => None
    | Some (trX, initial_descriptors) =>
        Some (equivocators_state_project IM initial_descriptors s, trX)
    end
    else None.

Lemma equivocators_partial_trace_project_characterization
  (final_descriptors : equivocator_descriptors IM)
  (X := free_composite_vlsm (equivocator_IM IM))
  (partial_trace_project := equivocators_partial_trace_project final_descriptors)
  sX trX sY trY
  : partial_trace_project (sX, trX) = Some (sY, trY) <->
    not_equivocating_equivocator_descriptors IM final_descriptors (finite_trace_last sX trX) /\
    exists initial_descriptors,
      equivocators_trace_project final_descriptors trX = Some (trY, initial_descriptors) /\
      equivocators_state_project IM initial_descriptors sX = sY.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
final_descriptors: equivocator_descriptors IM
X:= free_composite_vlsm (equivocator_IM IM): VLSM message
partial_trace_project:= equivocators_partial_trace_project
  final_descriptors: composite_state
  (equivocator_IM IM) *
list
  (composite_transition_item
     (equivocator_IM IM))
→ option
    (composite_state IM *
     list
       (composite_transition_item
          IM))
sX: composite_state (equivocator_IM IM)
trX: list
  (composite_transition_item (equivocator_IM IM))
sY: composite_state IM
trY: list (composite_transition_item IM)
partial_trace_project (sX, trX) = Some (sY, trY)
↔ not_equivocating_equivocator_descriptors IM
    final_descriptors (finite_trace_last sX trX)
  ∧ (∃ initial_descriptors : equivocator_descriptors
                               IM,
       equivocators_trace_project final_descriptors
         trX = Some (trY, initial_descriptors)
       ∧ equivocators_state_project IM
           initial_descriptors sX = sY)
Proof.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
final_descriptors: equivocator_descriptors IM
X:= free_composite_vlsm (equivocator_IM IM): VLSM message
partial_trace_project:= equivocators_partial_trace_project
  final_descriptors: composite_state
  (equivocator_IM IM) *
list
  (composite_transition_item
     (equivocator_IM IM))
→ option
    (composite_state IM *
     list
       (composite_transition_item
          IM))
sX: composite_state (equivocator_IM IM)
trX: list
  (composite_transition_item (equivocator_IM IM))
sY: composite_state IM
trY: list (composite_transition_item IM)
partial_trace_project (sX, trX) = Some (sY, trY)
↔ not_equivocating_equivocator_descriptors IM
    final_descriptors (finite_trace_last sX trX)
  ∧ (∃ initial_descriptors : equivocator_descriptors
                               IM,
       equivocators_trace_project final_descriptors
         trX = Some (trY, initial_descriptors)
       ∧ equivocators_state_project IM
           initial_descriptors sX = sY)
  unfold partial_trace_project, equivocators_partial_trace_project.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
final_descriptors: equivocator_descriptors IM
X:= free_composite_vlsm (equivocator_IM IM): VLSM message
partial_trace_project:= equivocators_partial_trace_project
  final_descriptors: composite_state
  (equivocator_IM IM) *
list
  (composite_transition_item
     (equivocator_IM IM))
→ option
    (composite_state IM *
     list
       (composite_transition_item
          IM))
sX: composite_state (equivocator_IM IM)
trX: list
  (composite_transition_item (equivocator_IM IM))
sY: composite_state IM
trY: list (composite_transition_item IM)
(if
  decide
    (not_equivocating_equivocator_descriptors IM
       final_descriptors (finite_trace_last sX trX))
 then
  match
    equivocators_trace_project final_descriptors trX
  with
  | Some (trX, initial_descriptors) =>
      Some
        (equivocators_state_project IM
           initial_descriptors sX, trX)
  | None => None
  end
 else None) = Some (sY, trY)
↔ not_equivocating_equivocator_descriptors IM
    final_descriptors (finite_trace_last sX trX)
  ∧ (∃ initial_descriptors : equivocator_descriptors
                               IM,
       equivocators_trace_project final_descriptors
         trX = Some (trY, initial_descriptors)
       ∧ equivocators_state_project IM
           initial_descriptors sX = sY)
  split.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
final_descriptors: equivocator_descriptors IM
X:= free_composite_vlsm (equivocator_IM IM): VLSM message
partial_trace_project:= equivocators_partial_trace_project
  final_descriptors: composite_state
  (equivocator_IM IM) *
list
  (composite_transition_item
     (equivocator_IM IM))
→ option
    (composite_state IM *
     list
       (composite_transition_item
          IM))
sX: composite_state (equivocator_IM IM)
trX: list
  (composite_transition_item (equivocator_IM IM))
sY: composite_state IM
trY: list (composite_transition_item IM)
(if
  decide
    (not_equivocating_equivocator_descriptors IM
       final_descriptors (finite_trace_last sX trX))
 then
  match
    equivocators_trace_project final_descriptors trX
  with
  | Some (trX, initial_descriptors) =>
      Some
        (equivocators_state_project IM
           initial_descriptors sX, trX)
  | None => None
  end
 else None) = Some (sY, trY)
→ not_equivocating_equivocator_descriptors IM
    final_descriptors (finite_trace_last sX trX)
  ∧ (∃ initial_descriptors : equivocator_descriptors
                               IM,
       equivocators_trace_project final_descriptors
         trX = Some (trY, initial_descriptors)
       ∧ equivocators_state_project IM
           initial_descriptors sX = sY)
message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
final_descriptors: equivocator_descriptors IM
X:= free_composite_vlsm (equivocator_IM IM): VLSM message
partial_trace_project:= equivocators_partial_trace_project
  final_descriptors: composite_state
  (equivocator_IM IM) *
list
  (composite_transition_item
     (equivocator_IM IM))
→ option
    (composite_state IM *
     list
       (composite_transition_item
          IM))
sX: composite_state (equivocator_IM IM)
trX: list
  (composite_transition_item (equivocator_IM IM))
sY: composite_state IM
trY: list (composite_transition_item IM)
not_equivocating_equivocator_descriptors IM
  final_descriptors (finite_trace_last sX trX)
∧ (∃ initial_descriptors : equivocator_descriptors IM,
     equivocators_trace_project final_descriptors trX =
     Some (trY, initial_descriptors)
     ∧ equivocators_state_project IM
         initial_descriptors sX = sY)
→ (if
    decide
      (not_equivocating_equivocator_descriptors IM
         final_descriptors 
         (finite_trace_last sX trX))
   then
    match
      equivocators_trace_project final_descriptors trX
    with
    | Some (trX, initial_descriptors) =>
        Some
          (equivocators_state_project IM
             initial_descriptors sX, trX)
    | None => None
    end
   else None) = Some (sY, trY)
  -message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
final_descriptors: equivocator_descriptors IM
X:= free_composite_vlsm (equivocator_IM IM): VLSM message
partial_trace_project:= equivocators_partial_trace_project
  final_descriptors: composite_state
  (equivocator_IM IM) *
list
  (composite_transition_item
     (equivocator_IM IM))
→ option
    (composite_state IM *
     list
       (composite_transition_item
          IM))
sX: composite_state (equivocator_IM IM)
trX: list
  (composite_transition_item (equivocator_IM IM))
sY: composite_state IM
trY: list (composite_transition_item IM)
(if
  decide
    (not_equivocating_equivocator_descriptors IM
       final_descriptors (finite_trace_last sX trX))
 then
  match
    equivocators_trace_project final_descriptors trX
  with
  | Some (trX, initial_descriptors) =>
      Some
        (equivocators_state_project IM
           initial_descriptors sX, trX)
  | None => None
  end
 else None) = Some (sY, trY)
→ not_equivocating_equivocator_descriptors IM
    final_descriptors (finite_trace_last sX trX)
  ∧ (∃ initial_descriptors : equivocator_descriptors
                               IM,
       equivocators_trace_project final_descriptors
         trX = Some (trY, initial_descriptors)
       ∧ equivocators_state_project IM
           initial_descriptors sX = sY) intros Hpr_tr.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
final_descriptors: equivocator_descriptors IM
X:= free_composite_vlsm (equivocator_IM IM): VLSM message
partial_trace_project:= equivocators_partial_trace_project
  final_descriptors: composite_state
  (equivocator_IM IM) *
list
  (composite_transition_item
     (equivocator_IM IM))
→ option
    (composite_state IM *
     list
       (composite_transition_item
          IM))
sX: composite_state (equivocator_IM IM)
trX: list
  (composite_transition_item (equivocator_IM IM))
sY: composite_state IM
trY: list (composite_transition_item IM)
Hpr_tr: (if
  decide
    (not_equivocating_equivocator_descriptors
       IM final_descriptors
       (finite_trace_last sX trX))
 then
  match
    equivocators_trace_project
      final_descriptors trX
  with
  | Some (trX, initial_descriptors) =>
      Some
        (equivocators_state_project IM
           initial_descriptors sX, trX)
  | None => None
  end
 else None) = Some (sY, trY)
not_equivocating_equivocator_descriptors IM
  final_descriptors (finite_trace_last sX trX)
∧ (∃ initial_descriptors : equivocator_descriptors IM,
     equivocators_trace_project final_descriptors trX =
     Some (trY, initial_descriptors)
     ∧ equivocators_state_project IM
         initial_descriptors sX = sY)
    case_decide; [| by congruence].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
final_descriptors: equivocator_descriptors IM
X:= free_composite_vlsm (equivocator_IM IM): VLSM message
partial_trace_project:= equivocators_partial_trace_project
  final_descriptors: composite_state
  (equivocator_IM IM) *
list
  (composite_transition_item
     (equivocator_IM IM))
→ option
    (composite_state IM *
     list
       (composite_transition_item
          IM))
sX: composite_state (equivocator_IM IM)
trX: list
  (composite_transition_item (equivocator_IM IM))
sY: composite_state IM
trY: list (composite_transition_item IM)
H1: not_equivocating_equivocator_descriptors IM
  final_descriptors (finite_trace_last sX trX)
Hpr_tr: match
  equivocators_trace_project
    final_descriptors trX
with
| Some (trX, initial_descriptors) =>
    Some
      (equivocators_state_project IM
         initial_descriptors sX, trX)
| None => None
end = Some (sY, trY)
not_equivocating_equivocator_descriptors IM
  final_descriptors (finite_trace_last sX trX)
∧ (∃ initial_descriptors : equivocator_descriptors IM,
     equivocators_trace_project final_descriptors trX =
     Some (trY, initial_descriptors)
     ∧ equivocators_state_project IM
         initial_descriptors sX = sY)
    destruct (equivocators_trace_project final_descriptors trX)
      as [(_trY, initial_descriptors) |] eqn: Htr_project
    ; [| by congruence].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
final_descriptors: equivocator_descriptors IM
X:= free_composite_vlsm (equivocator_IM IM): VLSM message
partial_trace_project:= equivocators_partial_trace_project
  final_descriptors: composite_state
  (equivocator_IM IM) *
list
  (composite_transition_item
     (equivocator_IM IM))
→ option
    (composite_state IM *
     list
       (composite_transition_item
          IM))
sX: composite_state (equivocator_IM IM)
trX: list
  (composite_transition_item (equivocator_IM IM))
sY: composite_state IM
trY: list (composite_transition_item IM)
H1: not_equivocating_equivocator_descriptors IM
  final_descriptors (finite_trace_last sX trX)
_trY: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project
  final_descriptors trX =
Some (_trY, initial_descriptors)
Hpr_tr: Some
  (equivocators_state_project IM
     initial_descriptors sX, _trY) =
Some (sY, trY)
not_equivocating_equivocator_descriptors IM
  final_descriptors (finite_trace_last sX trX)
∧ (∃ initial_descriptors0 : equivocator_descriptors IM,
     Some (_trY, initial_descriptors) =
     Some (trY, initial_descriptors0)
     ∧ equivocators_state_project IM
         initial_descriptors0 sX = sY)
    by inversion Hpr_tr; subst _trY; clear Hpr_tr; eauto.
  -message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
final_descriptors: equivocator_descriptors IM
X:= free_composite_vlsm (equivocator_IM IM): VLSM message
partial_trace_project:= equivocators_partial_trace_project
  final_descriptors: composite_state
  (equivocator_IM IM) *
list
  (composite_transition_item
     (equivocator_IM IM))
→ option
    (composite_state IM *
     list
       (composite_transition_item
          IM))
sX: composite_state (equivocator_IM IM)
trX: list
  (composite_transition_item (equivocator_IM IM))
sY: composite_state IM
trY: list (composite_transition_item IM)
not_equivocating_equivocator_descriptors IM
  final_descriptors (finite_trace_last sX trX)
∧ (∃ initial_descriptors : equivocator_descriptors IM,
     equivocators_trace_project final_descriptors trX =
     Some (trY, initial_descriptors)
     ∧ equivocators_state_project IM
         initial_descriptors sX = sY)
→ (if
    decide
      (not_equivocating_equivocator_descriptors IM
         final_descriptors (finite_trace_last sX trX))
   then
    match
      equivocators_trace_project final_descriptors trX
    with
    | Some (trX, initial_descriptors) =>
        Some
          (equivocators_state_project IM
             initial_descriptors sX, trX)
    | None => None
    end
   else None) = Some (sY, trY) intros [Hnot_equiv [initial_descriptors [Hpr_tr Hpr_s]]].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
final_descriptors: equivocator_descriptors IM
X:= free_composite_vlsm (equivocator_IM IM): VLSM message
partial_trace_project:= equivocators_partial_trace_project
  final_descriptors: composite_state
  (equivocator_IM IM) *
list
  (composite_transition_item
     (equivocator_IM IM))
→ option
    (composite_state IM *
     list
       (composite_transition_item
          IM))
sX: composite_state (equivocator_IM IM)
trX: list
  (composite_transition_item (equivocator_IM IM))
sY: composite_state IM
trY: list (composite_transition_item IM)
Hnot_equiv: not_equivocating_equivocator_descriptors
  IM final_descriptors
  (finite_trace_last sX trX)
initial_descriptors: equivocator_descriptors IM
Hpr_tr: equivocators_trace_project final_descriptors
  trX = Some (trY, initial_descriptors)
Hpr_s: equivocators_state_project IM
  initial_descriptors sX = sY
(if
  decide
    (not_equivocating_equivocator_descriptors IM
       final_descriptors (finite_trace_last sX trX))
 then
  match
    equivocators_trace_project final_descriptors trX
  with
  | Some (trX, initial_descriptors) =>
      Some
        (equivocators_state_project IM
           initial_descriptors sX, trX)
  | None => None
  end
 else None) = Some (sY, trY)
    by rewrite decide_True, Hpr_tr; subst.
Qed.

Lemma destruct_equivocators_partial_trace_project
  {final_descriptors : equivocator_descriptors IM}
  (X := free_composite_vlsm (equivocator_IM IM))
  (partial_trace_project := equivocators_partial_trace_project final_descriptors)
  {sX trX sY trY}
  (Hpr_tr : partial_trace_project (sX, trX) = Some (sY, trY)) :
    not_equivocating_equivocator_descriptors IM final_descriptors (finite_trace_last sX trX) /\
      exists initial_descriptors,
        equivocators_trace_project final_descriptors trX = Some (trY, initial_descriptors) /\
        equivocators_state_project IM initial_descriptors sX = sY.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
final_descriptors: equivocator_descriptors IM
X:= free_composite_vlsm (equivocator_IM IM): VLSM message
partial_trace_project:= equivocators_partial_trace_project
  final_descriptors: composite_state
  (equivocator_IM IM) *
list
  (composite_transition_item
     (equivocator_IM IM))
→ option
    (composite_state IM *
     list
       (composite_transition_item
          IM))
sX: composite_state (equivocator_IM IM)
trX: list
  (composite_transition_item (equivocator_IM IM))
sY: composite_state IM
trY: list (composite_transition_item IM)
Hpr_tr: partial_trace_project (sX, trX) =
Some (sY, trY)
not_equivocating_equivocator_descriptors IM
  final_descriptors (finite_trace_last sX trX)
∧ (∃ initial_descriptors : equivocator_descriptors IM,
     equivocators_trace_project final_descriptors trX =
     Some (trY, initial_descriptors)
     ∧ equivocators_state_project IM
         initial_descriptors sX = sY)
Proof.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
final_descriptors: equivocator_descriptors IM
X:= free_composite_vlsm (equivocator_IM IM): VLSM message
partial_trace_project:= equivocators_partial_trace_project
  final_descriptors: composite_state
  (equivocator_IM IM) *
list
  (composite_transition_item
     (equivocator_IM IM))
→ option
    (composite_state IM *
     list
       (composite_transition_item
          IM))
sX: composite_state (equivocator_IM IM)
trX: list
  (composite_transition_item (equivocator_IM IM))
sY: composite_state IM
trY: list (composite_transition_item IM)
Hpr_tr: partial_trace_project (sX, trX) =
Some (sY, trY)
not_equivocating_equivocator_descriptors IM
  final_descriptors (finite_trace_last sX trX)
∧ (∃ initial_descriptors : equivocator_descriptors IM,
     equivocators_trace_project final_descriptors trX =
     Some (trY, initial_descriptors)
     ∧ equivocators_state_project IM
         initial_descriptors sX = sY)
  exact (proj1 (equivocators_partial_trace_project_characterization
    final_descriptors sX trX sY trY) Hpr_tr).
Qed.

Lemma construct_equivocators_partial_trace_project
  {final_descriptors : equivocator_descriptors IM}
  (X := free_composite_vlsm (equivocator_IM IM))
  (partial_trace_project := equivocators_partial_trace_project final_descriptors)
  {sX trX sY trY}
  (Hed : not_equivocating_equivocator_descriptors IM final_descriptors (finite_trace_last sX trX) /\
    exists initial_descriptors,
      equivocators_trace_project final_descriptors trX = Some (trY, initial_descriptors) /\
      equivocators_state_project IM initial_descriptors sX = sY) :
    partial_trace_project (sX, trX) = Some (sY, trY).message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
final_descriptors: equivocator_descriptors IM
X:= free_composite_vlsm (equivocator_IM IM): VLSM message
partial_trace_project:= equivocators_partial_trace_project
  final_descriptors: composite_state
  (equivocator_IM IM) *
list
  (composite_transition_item
     (equivocator_IM IM))
→ option
    (composite_state IM *
     list
       (composite_transition_item
          IM))
sX: state (free_composite_vlsm (equivocator_IM IM))
trX: list transition_item
sY: state (free_composite_vlsm IM)
trY: list (composite_transition_item IM)
Hed: not_equivocating_equivocator_descriptors IM
  final_descriptors (finite_trace_last sX trX)
∧ (∃ initial_descriptors : 
      equivocator_descriptors IM,
     equivocators_trace_project
       final_descriptors trX =
     Some (trY, initial_descriptors)
     ∧ equivocators_state_project IM
         initial_descriptors sX = sY)
partial_trace_project (sX, trX) = Some (sY, trY)
Proof.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
final_descriptors: equivocator_descriptors IM
X:= free_composite_vlsm (equivocator_IM IM): VLSM message
partial_trace_project:= equivocators_partial_trace_project
  final_descriptors: composite_state
  (equivocator_IM IM) *
list
  (composite_transition_item
     (equivocator_IM IM))
→ option
    (composite_state IM *
     list
       (composite_transition_item
          IM))
sX: state (free_composite_vlsm (equivocator_IM IM))
trX: list transition_item
sY: state (free_composite_vlsm IM)
trY: list (composite_transition_item IM)
Hed: not_equivocating_equivocator_descriptors IM
  final_descriptors (finite_trace_last sX trX)
∧ (∃ initial_descriptors : 
      equivocator_descriptors IM,
     equivocators_trace_project
       final_descriptors trX =
     Some (trY, initial_descriptors)
     ∧ equivocators_state_project IM
         initial_descriptors sX = sY)
partial_trace_project (sX, trX) = Some (sY, trY)
  exact (proj2 (equivocators_partial_trace_project_characterization
    final_descriptors sX trX sY trY) Hed).
Qed.

Lemma equivocators_partial_trace_project_extends_left
  (final_descriptors : equivocator_descriptors IM)
  (X := free_composite_vlsm (equivocator_IM IM))
  (partial_trace_project := equivocators_partial_trace_project final_descriptors)
  : forall sX trX sY trY,
  partial_trace_project (sX, trX) = Some (sY, trY) ->
  forall s'X preX,
    finite_trace_last s'X preX = sX ->
    finite_constrained_trace_from X s'X (preX ++ trX) ->
    exists s'Y preY,
      partial_trace_project (s'X, preX ++ trX) = Some (s'Y, preY ++ trY) /\
      finite_trace_last s'Y preY = sY.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
final_descriptors: equivocator_descriptors IM
X:= free_composite_vlsm (equivocator_IM IM): VLSM message
partial_trace_project:= equivocators_partial_trace_project
  final_descriptors: composite_state
  (equivocator_IM IM) *
list
  (composite_transition_item
     (equivocator_IM IM))
→ option
    (composite_state IM *
     list
       (composite_transition_item
          IM))
∀ (sX : composite_state (equivocator_IM IM)) (trX : 
                                              list
                                                (composite_transition_item
                                                 (equivocator_IM
                                                 IM))) 
  (sY : composite_state IM) (trY : list
                                     (composite_transition_item
                                        IM)),
  partial_trace_project (sX, trX) = Some (sY, trY)
  → ∀ (s'X : state
               (composite_type (equivocator_IM IM))) 
      (preX : list transition_item),
      finite_trace_last s'X preX = sX
      → finite_constrained_trace_from X s'X
          (preX ++ trX)
        → ∃ (s'Y : composite_state IM) (preY : list
                                                 (composite_transition_item
                                                 IM)),
            partial_trace_project (s'X, preX ++ trX) =
            Some (s'Y, preY ++ trY)
            ∧ finite_trace_last s'Y preY = sY
Proof.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
final_descriptors: equivocator_descriptors IM
X:= free_composite_vlsm (equivocator_IM IM): VLSM message
partial_trace_project:= equivocators_partial_trace_project
  final_descriptors: composite_state
  (equivocator_IM IM) *
list
  (composite_transition_item
     (equivocator_IM IM))
→ option
    (composite_state IM *
     list
       (composite_transition_item
          IM))
∀ (sX : composite_state (equivocator_IM IM)) (trX : 
                                              list
                                                (composite_transition_item
                                                 (equivocator_IM
                                                 IM))) 
  (sY : composite_state IM) (trY : list
                                     (composite_transition_item
                                        IM)),
  partial_trace_project (sX, trX) = Some (sY, trY)
  → ∀ (s'X : state
               (composite_type (equivocator_IM IM))) 
      (preX : list transition_item),
      finite_trace_last s'X preX = sX
      → finite_constrained_trace_from X s'X
          (preX ++ trX)
        → ∃ (s'Y : composite_state IM) (preY : list
                                                 (composite_transition_item
                                                 IM)),
            partial_trace_project (s'X, preX ++ trX) =
            Some (s'Y, preY ++ trY)
            ∧ finite_trace_last s'Y preY = sY
  intros s tr sX trX Hpr_tr s_pre pre Hs_lst Hpre_tr.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
final_descriptors: equivocator_descriptors IM
X:= free_composite_vlsm (equivocator_IM IM): VLSM message
partial_trace_project:= equivocators_partial_trace_project
  final_descriptors: composite_state
  (equivocator_IM IM) *
list
  (composite_transition_item
     (equivocator_IM IM))
→ option
    (composite_state IM *
     list
       (composite_transition_item
          IM))
s: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
sX: composite_state IM
trX: list (composite_transition_item IM)
Hpr_tr: partial_trace_project (s, tr) =
Some (sX, trX)
s_pre: state (composite_type (equivocator_IM IM))
pre: list transition_item
Hs_lst: finite_trace_last s_pre pre = s
Hpre_tr: finite_constrained_trace_from X s_pre
  (pre ++ tr)
∃ (s'Y : composite_state IM) (preY : list
                                       (composite_transition_item
                                          IM)),
  partial_trace_project (s_pre, pre ++ tr) =
  Some (s'Y, preY ++ trX)
  ∧ finite_trace_last s'Y preY = sX
  destruct (destruct_equivocators_partial_trace_project Hpr_tr)
    as [Hnot_equiv [initial_descriptors [Htr_project Hs_project]]].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
final_descriptors: equivocator_descriptors IM
X:= free_composite_vlsm (equivocator_IM IM): VLSM message
partial_trace_project:= equivocators_partial_trace_project
  final_descriptors: composite_state
  (equivocator_IM IM) *
list
  (composite_transition_item
     (equivocator_IM IM))
→ option
    (composite_state IM *
     list
       (composite_transition_item
          IM))
s: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
sX: composite_state IM
trX: list (composite_transition_item IM)
Hpr_tr: partial_trace_project (s, tr) =
Some (sX, trX)
s_pre: state (composite_type (equivocator_IM IM))
pre: list transition_item
Hs_lst: finite_trace_last s_pre pre = s
Hpre_tr: finite_constrained_trace_from X s_pre
  (pre ++ tr)
Hnot_equiv: not_equivocating_equivocator_descriptors
  IM final_descriptors
  (finite_trace_last s tr)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project
  final_descriptors tr =
Some (trX, initial_descriptors)
Hs_project: equivocators_state_project IM
  initial_descriptors s = sX
∃ (s'Y : composite_state IM) (preY : list
                                       (composite_transition_item
                                          IM)),
  partial_trace_project (s_pre, pre ++ tr) =
  Some (s'Y, preY ++ trX)
  ∧ finite_trace_last s'Y preY = sX
  apply (finite_valid_trace_from_app_iff PreFreeE) in Hpre_tr.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
final_descriptors: equivocator_descriptors IM
X:= free_composite_vlsm (equivocator_IM IM): VLSM message
partial_trace_project:= equivocators_partial_trace_project
  final_descriptors: composite_state
  (equivocator_IM IM) *
list
  (composite_transition_item
     (equivocator_IM IM))
→ option
    (composite_state IM *
     list
       (composite_transition_item
          IM))
s: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
sX: composite_state IM
trX: list (composite_transition_item IM)
Hpr_tr: partial_trace_project (s, tr) =
Some (sX, trX)
s_pre: state (composite_type (equivocator_IM IM))
pre: list transition_item
Hs_lst: finite_trace_last s_pre pre = s
Hpre_tr: finite_valid_trace_from PreFreeE s_pre pre
∧ finite_valid_trace_from PreFreeE
    (finite_trace_last s_pre pre) tr
Hnot_equiv: not_equivocating_equivocator_descriptors
  IM final_descriptors
  (finite_trace_last s tr)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project
  final_descriptors tr =
Some (trX, initial_descriptors)
Hs_project: equivocators_state_project IM
  initial_descriptors s = sX
∃ (s'Y : composite_state IM) (preY : list
                                       (composite_transition_item
                                          IM)),
  partial_trace_project (s_pre, pre ++ tr) =
  Some (s'Y, preY ++ trX)
  ∧ finite_trace_last s'Y preY = sX
  destruct Hpre_tr as [Hpre Htr].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
final_descriptors: equivocator_descriptors IM
X:= free_composite_vlsm (equivocator_IM IM): VLSM message
partial_trace_project:= equivocators_partial_trace_project
  final_descriptors: composite_state
  (equivocator_IM IM) *
list
  (composite_transition_item
     (equivocator_IM IM))
→ option
    (composite_state IM *
     list
       (composite_transition_item
          IM))
s: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
sX: composite_state IM
trX: list (composite_transition_item IM)
Hpr_tr: partial_trace_project (s, tr) =
Some (sX, trX)
s_pre: state (composite_type (equivocator_IM IM))
pre: list transition_item
Hs_lst: finite_trace_last s_pre pre = s
Hpre: finite_valid_trace_from PreFreeE s_pre pre
Htr: finite_valid_trace_from PreFreeE
  (finite_trace_last s_pre pre) tr
Hnot_equiv: not_equivocating_equivocator_descriptors
  IM final_descriptors
  (finite_trace_last s tr)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project
  final_descriptors tr =
Some (trX, initial_descriptors)
Hs_project: equivocators_state_project IM
  initial_descriptors s = sX
∃ (s'Y : composite_state IM) (preY : list
                                       (composite_transition_item
                                          IM)),
  partial_trace_project (s_pre, pre ++ tr) =
  Some (s'Y, preY ++ trX)
  ∧ finite_trace_last s'Y preY = sX subst s sX.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
final_descriptors: equivocator_descriptors IM
X:= free_composite_vlsm (equivocator_IM IM): VLSM message
partial_trace_project:= equivocators_partial_trace_project
  final_descriptors: composite_state
  (equivocator_IM IM) *
list
  (composite_transition_item
     (equivocator_IM IM))
→ option
    (composite_state IM *
     list
       (composite_transition_item
          IM))
tr: list
  (composite_transition_item (equivocator_IM IM))
trX: list (composite_transition_item IM)
s_pre: state (composite_type (equivocator_IM IM))
pre: list transition_item
initial_descriptors: equivocator_descriptors IM
Hpr_tr: partial_trace_project
  (finite_trace_last s_pre pre, tr) =
Some
  (equivocators_state_project IM
     initial_descriptors
     (finite_trace_last s_pre pre), trX)
Hpre: finite_valid_trace_from PreFreeE s_pre pre
Htr: finite_valid_trace_from PreFreeE
  (finite_trace_last s_pre pre) tr
Hnot_equiv: not_equivocating_equivocator_descriptors
  IM final_descriptors
  (finite_trace_last
     (finite_trace_last s_pre pre) tr)
Htr_project: equivocators_trace_project
  final_descriptors tr =
Some (trX, initial_descriptors)
∃ (s'Y : composite_state IM) (preY : list
                                       (composite_transition_item
                                          IM)),
  partial_trace_project (s_pre, pre ++ tr) =
  Some (s'Y, preY ++ trX)
  ∧ finite_trace_last s'Y preY =
    equivocators_state_project IM initial_descriptors
      (finite_trace_last s_pre pre)
  apply not_equivocating_equivocator_descriptors_proper in Hnot_equiv as Hproper.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
final_descriptors: equivocator_descriptors IM
X:= free_composite_vlsm (equivocator_IM IM): VLSM message
partial_trace_project:= equivocators_partial_trace_project
  final_descriptors: composite_state
  (equivocator_IM IM) *
list
  (composite_transition_item
     (equivocator_IM IM))
→ option
    (composite_state IM *
     list
       (composite_transition_item
          IM))
tr: list
  (composite_transition_item (equivocator_IM IM))
trX: list (composite_transition_item IM)
s_pre: state (composite_type (equivocator_IM IM))
pre: list transition_item
initial_descriptors: equivocator_descriptors IM
Hpr_tr: partial_trace_project
  (finite_trace_last s_pre pre, tr) =
Some
  (equivocators_state_project IM
     initial_descriptors
     (finite_trace_last s_pre pre), trX)
Hpre: finite_valid_trace_from PreFreeE s_pre pre
Htr: finite_valid_trace_from PreFreeE
  (finite_trace_last s_pre pre) tr
Hnot_equiv: not_equivocating_equivocator_descriptors
  IM final_descriptors
  (finite_trace_last
     (finite_trace_last s_pre pre) tr)
Htr_project: equivocators_trace_project
  final_descriptors tr =
Some (trX, initial_descriptors)
Hproper: proper_equivocator_descriptors IM
  final_descriptors
  (finite_trace_last
     (finite_trace_last s_pre pre) tr)
∃ (s'Y : composite_state IM) (preY : list
                                       (composite_transition_item
                                          IM)),
  partial_trace_project (s_pre, pre ++ tr) =
  Some (s'Y, preY ++ trX)
  ∧ finite_trace_last s'Y preY =
    equivocators_state_project IM initial_descriptors
      (finite_trace_last s_pre pre)
  specialize
    (preloaded_equivocators_valid_trace_project_proper_initial _ _ Htr
      _ _ _ Htr_project Hproper)
    as Hinitial_descriptors.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
final_descriptors: equivocator_descriptors IM
X:= free_composite_vlsm (equivocator_IM IM): VLSM message
partial_trace_project:= equivocators_partial_trace_project
  final_descriptors: composite_state
  (equivocator_IM IM) *
list
  (composite_transition_item
     (equivocator_IM IM))
→ option
    (composite_state IM *
     list
       (composite_transition_item
          IM))
tr: list
  (composite_transition_item (equivocator_IM IM))
trX: list (composite_transition_item IM)
s_pre: state (composite_type (equivocator_IM IM))
pre: list transition_item
initial_descriptors: equivocator_descriptors IM
Hpr_tr: partial_trace_project
  (finite_trace_last s_pre pre, tr) =
Some
  (equivocators_state_project IM
     initial_descriptors
     (finite_trace_last s_pre pre), trX)
Hpre: finite_valid_trace_from PreFreeE s_pre pre
Htr: finite_valid_trace_from PreFreeE
  (finite_trace_last s_pre pre) tr
Hnot_equiv: not_equivocating_equivocator_descriptors
  IM final_descriptors
  (finite_trace_last
     (finite_trace_last s_pre pre) tr)
Htr_project: equivocators_trace_project
  final_descriptors tr =
Some (trX, initial_descriptors)
Hproper: proper_equivocator_descriptors IM
  final_descriptors
  (finite_trace_last
     (finite_trace_last s_pre pre) tr)
Hinitial_descriptors: proper_equivocator_descriptors
  IM initial_descriptors
  (finite_trace_last s_pre pre)
∃ (s'Y : composite_state IM) (preY : list
                                       (composite_transition_item
                                          IM)),
  partial_trace_project (s_pre, pre ++ tr) =
  Some (s'Y, preY ++ trX)
  ∧ finite_trace_last s'Y preY =
    equivocators_state_project IM initial_descriptors
      (finite_trace_last s_pre pre)
  destruct
    (preloaded_equivocators_valid_trace_from_project
      _ _ _ Hinitial_descriptors Hpre)
    as [preX [pre_descriptors [Hpre_project [Hpre_desciptors Hs_project]]]].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
final_descriptors: equivocator_descriptors IM
X:= free_composite_vlsm (equivocator_IM IM): VLSM message
partial_trace_project:= equivocators_partial_trace_project
  final_descriptors: composite_state
  (equivocator_IM IM) *
list
  (composite_transition_item
     (equivocator_IM IM))
→ option
    (composite_state IM *
     list
       (composite_transition_item
          IM))
tr: list
  (composite_transition_item (equivocator_IM IM))
trX: list (composite_transition_item IM)
s_pre: state (composite_type (equivocator_IM IM))
pre: list transition_item
initial_descriptors: equivocator_descriptors IM
Hpr_tr: partial_trace_project
  (finite_trace_last s_pre pre, tr) =
Some
  (equivocators_state_project IM
     initial_descriptors
     (finite_trace_last s_pre pre), trX)
Hpre: finite_valid_trace_from PreFreeE s_pre pre
Htr: finite_valid_trace_from PreFreeE
  (finite_trace_last s_pre pre) tr
Hnot_equiv: not_equivocating_equivocator_descriptors
  IM final_descriptors
  (finite_trace_last
     (finite_trace_last s_pre pre) tr)
Htr_project: equivocators_trace_project
  final_descriptors tr =
Some (trX, initial_descriptors)
Hproper: proper_equivocator_descriptors IM
  final_descriptors
  (finite_trace_last
     (finite_trace_last s_pre pre) tr)
Hinitial_descriptors: proper_equivocator_descriptors
  IM initial_descriptors
  (finite_trace_last s_pre pre)
preX: list (composite_transition_item IM)
pre_descriptors: equivocator_descriptors IM
Hpre_project: equivocators_trace_project
  initial_descriptors pre =
Some (preX, pre_descriptors)
Hpre_desciptors: proper_equivocator_descriptors IM
  pre_descriptors s_pre
Hs_project: equivocators_state_project IM
  initial_descriptors
  (finite_trace_last s_pre pre) =
finite_trace_last
  (equivocators_state_project IM
     pre_descriptors s_pre) preX
∃ (s'Y : composite_state IM) (preY : list
                                       (composite_transition_item
                                          IM)),
  partial_trace_project (s_pre, pre ++ tr) =
  Some (s'Y, preY ++ trX)
  ∧ finite_trace_last s'Y preY =
    equivocators_state_project IM initial_descriptors
      (finite_trace_last s_pre pre)
  exists (equivocators_state_project IM pre_descriptors s_pre), preX.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
final_descriptors: equivocator_descriptors IM
X:= free_composite_vlsm (equivocator_IM IM): VLSM message
partial_trace_project:= equivocators_partial_trace_project
  final_descriptors: composite_state
  (equivocator_IM IM) *
list
  (composite_transition_item
     (equivocator_IM IM))
→ option
    (composite_state IM *
     list
       (composite_transition_item
          IM))
tr: list
  (composite_transition_item (equivocator_IM IM))
trX: list (composite_transition_item IM)
s_pre: state (composite_type (equivocator_IM IM))
pre: list transition_item
initial_descriptors: equivocator_descriptors IM
Hpr_tr: partial_trace_project
  (finite_trace_last s_pre pre, tr) =
Some
  (equivocators_state_project IM
     initial_descriptors
     (finite_trace_last s_pre pre), trX)
Hpre: finite_valid_trace_from PreFreeE s_pre pre
Htr: finite_valid_trace_from PreFreeE
  (finite_trace_last s_pre pre) tr
Hnot_equiv: not_equivocating_equivocator_descriptors
  IM final_descriptors
  (finite_trace_last
     (finite_trace_last s_pre pre) tr)
Htr_project: equivocators_trace_project
  final_descriptors tr =
Some (trX, initial_descriptors)
Hproper: proper_equivocator_descriptors IM
  final_descriptors
  (finite_trace_last
     (finite_trace_last s_pre pre) tr)
Hinitial_descriptors: proper_equivocator_descriptors
  IM initial_descriptors
  (finite_trace_last s_pre pre)
preX: list (composite_transition_item IM)
pre_descriptors: equivocator_descriptors IM
Hpre_project: equivocators_trace_project
  initial_descriptors pre =
Some (preX, pre_descriptors)
Hpre_desciptors: proper_equivocator_descriptors IM
  pre_descriptors s_pre
Hs_project: equivocators_state_project IM
  initial_descriptors
  (finite_trace_last s_pre pre) =
finite_trace_last
  (equivocators_state_project IM
     pre_descriptors s_pre) preX
partial_trace_project (s_pre, pre ++ tr) =
Some
  (equivocators_state_project IM pre_descriptors s_pre,
   preX ++ trX)
∧ finite_trace_last
    (equivocators_state_project IM pre_descriptors
       s_pre) preX =
  equivocators_state_project IM initial_descriptors
    (finite_trace_last s_pre pre)
  split; [| done].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
final_descriptors: equivocator_descriptors IM
X:= free_composite_vlsm (equivocator_IM IM): VLSM message
partial_trace_project:= equivocators_partial_trace_project
  final_descriptors: composite_state
  (equivocator_IM IM) *
list
  (composite_transition_item
     (equivocator_IM IM))
→ option
    (composite_state IM *
     list
       (composite_transition_item
          IM))
tr: list
  (composite_transition_item (equivocator_IM IM))
trX: list (composite_transition_item IM)
s_pre: state (composite_type (equivocator_IM IM))
pre: list transition_item
initial_descriptors: equivocator_descriptors IM
Hpr_tr: partial_trace_project
  (finite_trace_last s_pre pre, tr) =
Some
  (equivocators_state_project IM
     initial_descriptors
     (finite_trace_last s_pre pre), trX)
Hpre: finite_valid_trace_from PreFreeE s_pre pre
Htr: finite_valid_trace_from PreFreeE
  (finite_trace_last s_pre pre) tr
Hnot_equiv: not_equivocating_equivocator_descriptors
  IM final_descriptors
  (finite_trace_last
     (finite_trace_last s_pre pre) tr)
Htr_project: equivocators_trace_project
  final_descriptors tr =
Some (trX, initial_descriptors)
Hproper: proper_equivocator_descriptors IM
  final_descriptors
  (finite_trace_last
     (finite_trace_last s_pre pre) tr)
Hinitial_descriptors: proper_equivocator_descriptors
  IM initial_descriptors
  (finite_trace_last s_pre pre)
preX: list (composite_transition_item IM)
pre_descriptors: equivocator_descriptors IM
Hpre_project: equivocators_trace_project
  initial_descriptors pre =
Some (preX, pre_descriptors)
Hpre_desciptors: proper_equivocator_descriptors IM
  pre_descriptors s_pre
Hs_project: equivocators_state_project IM
  initial_descriptors
  (finite_trace_last s_pre pre) =
finite_trace_last
  (equivocators_state_project IM
     pre_descriptors s_pre) preX
partial_trace_project (s_pre, pre ++ tr) =
Some
  (equivocators_state_project IM pre_descriptors s_pre,
   preX ++ trX)
  apply construct_equivocators_partial_trace_project.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
final_descriptors: equivocator_descriptors IM
X:= free_composite_vlsm (equivocator_IM IM): VLSM message
partial_trace_project:= equivocators_partial_trace_project
  final_descriptors: composite_state
  (equivocator_IM IM) *
list
  (composite_transition_item
     (equivocator_IM IM))
→ option
    (composite_state IM *
     list
       (composite_transition_item
          IM))
tr: list
  (composite_transition_item (equivocator_IM IM))
trX: list (composite_transition_item IM)
s_pre: state (composite_type (equivocator_IM IM))
pre: list transition_item
initial_descriptors: equivocator_descriptors IM
Hpr_tr: partial_trace_project
  (finite_trace_last s_pre pre, tr) =
Some
  (equivocators_state_project IM
     initial_descriptors
     (finite_trace_last s_pre pre), trX)
Hpre: finite_valid_trace_from PreFreeE s_pre pre
Htr: finite_valid_trace_from PreFreeE
  (finite_trace_last s_pre pre) tr
Hnot_equiv: not_equivocating_equivocator_descriptors
  IM final_descriptors
  (finite_trace_last
     (finite_trace_last s_pre pre) tr)
Htr_project: equivocators_trace_project
  final_descriptors tr =
Some (trX, initial_descriptors)
Hproper: proper_equivocator_descriptors IM
  final_descriptors
  (finite_trace_last
     (finite_trace_last s_pre pre) tr)
Hinitial_descriptors: proper_equivocator_descriptors
  IM initial_descriptors
  (finite_trace_last s_pre pre)
preX: list (composite_transition_item IM)
pre_descriptors: equivocator_descriptors IM
Hpre_project: equivocators_trace_project
  initial_descriptors pre =
Some (preX, pre_descriptors)
Hpre_desciptors: proper_equivocator_descriptors IM
  pre_descriptors s_pre
Hs_project: equivocators_state_project IM
  initial_descriptors
  (finite_trace_last s_pre pre) =
finite_trace_last
  (equivocators_state_project IM
     pre_descriptors s_pre) preX
not_equivocating_equivocator_descriptors IM
  final_descriptors
  (finite_trace_last s_pre (pre ++ tr))
∧ (∃ initial_descriptors : equivocator_descriptors IM,
     equivocators_trace_project final_descriptors
       (pre ++ tr) =
     Some (preX ++ trX, initial_descriptors)
     ∧ equivocators_state_project IM
         initial_descriptors s_pre =
       equivocators_state_project IM pre_descriptors
         s_pre)
  split; [by rewrite finite_trace_last_app |].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
final_descriptors: equivocator_descriptors IM
X:= free_composite_vlsm (equivocator_IM IM): VLSM message
partial_trace_project:= equivocators_partial_trace_project
  final_descriptors: composite_state
  (equivocator_IM IM) *
list
  (composite_transition_item
     (equivocator_IM IM))
→ option
    (composite_state IM *
     list
       (composite_transition_item
          IM))
tr: list
  (composite_transition_item (equivocator_IM IM))
trX: list (composite_transition_item IM)
s_pre: state (composite_type (equivocator_IM IM))
pre: list transition_item
initial_descriptors: equivocator_descriptors IM
Hpr_tr: partial_trace_project
  (finite_trace_last s_pre pre, tr) =
Some
  (equivocators_state_project IM
     initial_descriptors
     (finite_trace_last s_pre pre), trX)
Hpre: finite_valid_trace_from PreFreeE s_pre pre
Htr: finite_valid_trace_from PreFreeE
  (finite_trace_last s_pre pre) tr
Hnot_equiv: not_equivocating_equivocator_descriptors
  IM final_descriptors
  (finite_trace_last
     (finite_trace_last s_pre pre) tr)
Htr_project: equivocators_trace_project
  final_descriptors tr =
Some (trX, initial_descriptors)
Hproper: proper_equivocator_descriptors IM
  final_descriptors
  (finite_trace_last
     (finite_trace_last s_pre pre) tr)
Hinitial_descriptors: proper_equivocator_descriptors
  IM initial_descriptors
  (finite_trace_last s_pre pre)
preX: list (composite_transition_item IM)
pre_descriptors: equivocator_descriptors IM
Hpre_project: equivocators_trace_project
  initial_descriptors pre =
Some (preX, pre_descriptors)
Hpre_desciptors: proper_equivocator_descriptors IM
  pre_descriptors s_pre
Hs_project: equivocators_state_project IM
  initial_descriptors
  (finite_trace_last s_pre pre) =
finite_trace_last
  (equivocators_state_project IM
     pre_descriptors s_pre) preX
∃ initial_descriptors : equivocator_descriptors IM,
  equivocators_trace_project final_descriptors
    (pre ++ tr) =
  Some (preX ++ trX, initial_descriptors)
  ∧ equivocators_state_project IM initial_descriptors
      s_pre =
    equivocators_state_project IM pre_descriptors
      s_pre
  exists pre_descriptors.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
final_descriptors: equivocator_descriptors IM
X:= free_composite_vlsm (equivocator_IM IM): VLSM message
partial_trace_project:= equivocators_partial_trace_project
  final_descriptors: composite_state
  (equivocator_IM IM) *
list
  (composite_transition_item
     (equivocator_IM IM))
→ option
    (composite_state IM *
     list
       (composite_transition_item
          IM))
tr: list
  (composite_transition_item (equivocator_IM IM))
trX: list (composite_transition_item IM)
s_pre: state (composite_type (equivocator_IM IM))
pre: list transition_item
initial_descriptors: equivocator_descriptors IM
Hpr_tr: partial_trace_project
  (finite_trace_last s_pre pre, tr) =
Some
  (equivocators_state_project IM
     initial_descriptors
     (finite_trace_last s_pre pre), trX)
Hpre: finite_valid_trace_from PreFreeE s_pre pre
Htr: finite_valid_trace_from PreFreeE
  (finite_trace_last s_pre pre) tr
Hnot_equiv: not_equivocating_equivocator_descriptors
  IM final_descriptors
  (finite_trace_last
     (finite_trace_last s_pre pre) tr)
Htr_project: equivocators_trace_project
  final_descriptors tr =
Some (trX, initial_descriptors)
Hproper: proper_equivocator_descriptors IM
  final_descriptors
  (finite_trace_last
     (finite_trace_last s_pre pre) tr)
Hinitial_descriptors: proper_equivocator_descriptors
  IM initial_descriptors
  (finite_trace_last s_pre pre)
preX: list (composite_transition_item IM)
pre_descriptors: equivocator_descriptors IM
Hpre_project: equivocators_trace_project
  initial_descriptors pre =
Some (preX, pre_descriptors)
Hpre_desciptors: proper_equivocator_descriptors IM
  pre_descriptors s_pre
Hs_project: equivocators_state_project IM
  initial_descriptors
  (finite_trace_last s_pre pre) =
finite_trace_last
  (equivocators_state_project IM
     pre_descriptors s_pre) preX
equivocators_trace_project final_descriptors
  (pre ++ tr) = Some (preX ++ trX, pre_descriptors)
∧ equivocators_state_project IM pre_descriptors s_pre =
  equivocators_state_project IM pre_descriptors s_pre split; [| done].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
final_descriptors: equivocator_descriptors IM
X:= free_composite_vlsm (equivocator_IM IM): VLSM message
partial_trace_project:= equivocators_partial_trace_project
  final_descriptors: composite_state
  (equivocator_IM IM) *
list
  (composite_transition_item
     (equivocator_IM IM))
→ option
    (composite_state IM *
     list
       (composite_transition_item
          IM))
tr: list
  (composite_transition_item (equivocator_IM IM))
trX: list (composite_transition_item IM)
s_pre: state (composite_type (equivocator_IM IM))
pre: list transition_item
initial_descriptors: equivocator_descriptors IM
Hpr_tr: partial_trace_project
  (finite_trace_last s_pre pre, tr) =
Some
  (equivocators_state_project IM
     initial_descriptors
     (finite_trace_last s_pre pre), trX)
Hpre: finite_valid_trace_from PreFreeE s_pre pre
Htr: finite_valid_trace_from PreFreeE
  (finite_trace_last s_pre pre) tr
Hnot_equiv: not_equivocating_equivocator_descriptors
  IM final_descriptors
  (finite_trace_last
     (finite_trace_last s_pre pre) tr)
Htr_project: equivocators_trace_project
  final_descriptors tr =
Some (trX, initial_descriptors)
Hproper: proper_equivocator_descriptors IM
  final_descriptors
  (finite_trace_last
     (finite_trace_last s_pre pre) tr)
Hinitial_descriptors: proper_equivocator_descriptors
  IM initial_descriptors
  (finite_trace_last s_pre pre)
preX: list (composite_transition_item IM)
pre_descriptors: equivocator_descriptors IM
Hpre_project: equivocators_trace_project
  initial_descriptors pre =
Some (preX, pre_descriptors)
Hpre_desciptors: proper_equivocator_descriptors IM
  pre_descriptors s_pre
Hs_project: equivocators_state_project IM
  initial_descriptors
  (finite_trace_last s_pre pre) =
finite_trace_last
  (equivocators_state_project IM
     pre_descriptors s_pre) preX
equivocators_trace_project final_descriptors
  (pre ++ tr) = Some (preX ++ trX, pre_descriptors)
  apply equivocators_trace_project_app_iff.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
final_descriptors: equivocator_descriptors IM
X:= free_composite_vlsm (equivocator_IM IM): VLSM message
partial_trace_project:= equivocators_partial_trace_project
  final_descriptors: composite_state
  (equivocator_IM IM) *
list
  (composite_transition_item
     (equivocator_IM IM))
→ option
    (composite_state IM *
     list
       (composite_transition_item
          IM))
tr: list
  (composite_transition_item (equivocator_IM IM))
trX: list (composite_transition_item IM)
s_pre: state (composite_type (equivocator_IM IM))
pre: list transition_item
initial_descriptors: equivocator_descriptors IM
Hpr_tr: partial_trace_project
  (finite_trace_last s_pre pre, tr) =
Some
  (equivocators_state_project IM
     initial_descriptors
     (finite_trace_last s_pre pre), trX)
Hpre: finite_valid_trace_from PreFreeE s_pre pre
Htr: finite_valid_trace_from PreFreeE
  (finite_trace_last s_pre pre) tr
Hnot_equiv: not_equivocating_equivocator_descriptors
  IM final_descriptors
  (finite_trace_last
     (finite_trace_last s_pre pre) tr)
Htr_project: equivocators_trace_project
  final_descriptors tr =
Some (trX, initial_descriptors)
Hproper: proper_equivocator_descriptors IM
  final_descriptors
  (finite_trace_last
     (finite_trace_last s_pre pre) tr)
Hinitial_descriptors: proper_equivocator_descriptors
  IM initial_descriptors
  (finite_trace_last s_pre pre)
preX: list (composite_transition_item IM)
pre_descriptors: equivocator_descriptors IM
Hpre_project: equivocators_trace_project
  initial_descriptors pre =
Some (preX, pre_descriptors)
Hpre_desciptors: proper_equivocator_descriptors IM
  pre_descriptors s_pre
Hs_project: equivocators_state_project IM
  initial_descriptors
  (finite_trace_last s_pre pre) =
finite_trace_last
  (equivocators_state_project IM
     pre_descriptors s_pre) preX
∃ (preX0 sufX : list (composite_transition_item IM)) 
  (eqv_descriptors' : equivocator_descriptors IM),
  equivocators_trace_project final_descriptors tr =
  Some (sufX, eqv_descriptors')
  ∧ equivocators_trace_project eqv_descriptors' pre =
    Some (preX0, pre_descriptors)
    ∧ preX ++ trX = preX0 ++ sufX
  by exists preX, trX, initial_descriptors.
Qed.

The projection of an composite equivocator state using zero_descriptors which is guaranteed to always succeed.

Definition equivocators_total_state_project := equivocators_state_project IM (zero_descriptor IM).

Definition equivocators_total_label_project
  (l : composite_label (equivocator_IM IM)) : option (composite_label IM) :=
  let (i, li) := l in
  option_map (existT i) (equivocator_label_zero_project _ li).

Definition equivocators_total_trace_project
  (tr : list (composite_transition_item (equivocator_IM IM)))
  : list (composite_transition_item IM)
  :=
  from_option fst [] (equivocators_trace_project (zero_descriptor IM) tr).

The projection of an composite equivocator trace using zero_descriptors which is guaranteed to always succeed.

Lemma equivocators_total_trace_project_characterization
  {s tr}
  (Hpre_tr : finite_constrained_trace_from FreeE s tr)
  : equivocators_trace_project (zero_descriptor IM) tr =
    Some (equivocators_total_trace_project tr, zero_descriptor IM).message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
s: state (preloaded_with_all_messages_vlsm FreeE)
tr: list transition_item
Hpre_tr: finite_constrained_trace_from FreeE s tr
equivocators_trace_project (zero_descriptor IM) tr =
Some
  (equivocators_total_trace_project tr,
   zero_descriptor IM)
Proof.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
s: state (preloaded_with_all_messages_vlsm FreeE)
tr: list transition_item
Hpre_tr: finite_constrained_trace_from FreeE s tr
equivocators_trace_project (zero_descriptor IM) tr =
Some
  (equivocators_total_trace_project tr,
   zero_descriptor IM)
  unfold equivocators_total_trace_project.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
s: state (preloaded_with_all_messages_vlsm FreeE)
tr: list transition_item
Hpre_tr: finite_constrained_trace_from FreeE s tr
equivocators_trace_project (zero_descriptor IM) tr =
Some
  (from_option fst []
     (equivocators_trace_project (zero_descriptor IM)
        tr), zero_descriptor IM)
  by destruct (equivocators_trace_project_zero_descriptors _ _ Hpre_tr) as [_trX ->].
Qed.

Lemma equivocators_total_trace_project_app
  (X := FreeE)
  (trace_project := equivocators_total_trace_project)
  : forall tr1X tr2X,
      (exists sX, finite_constrained_trace_from X sX (tr1X ++ tr2X)) ->
      trace_project (tr1X ++ tr2X) = trace_project tr1X ++ trace_project tr2X.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
X:= FreeE: VLSM message
trace_project:= equivocators_total_trace_project: list
  (composite_transition_item
     (equivocator_IM IM))
→ list (composite_transition_item IM)
∀ tr1X tr2X : list transition_item,
  (∃ sX : state (preloaded_with_all_messages_vlsm X),
     finite_constrained_trace_from X sX (tr1X ++ tr2X))
  → trace_project (tr1X ++ tr2X) =
    trace_project tr1X ++ trace_project tr2X
Proof.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
X:= FreeE: VLSM message
trace_project:= equivocators_total_trace_project: list
  (composite_transition_item
     (equivocator_IM IM))
→ list (composite_transition_item IM)
∀ tr1X tr2X : list transition_item,
  (∃ sX : state (preloaded_with_all_messages_vlsm X),
     finite_constrained_trace_from X sX (tr1X ++ tr2X))
  → trace_project (tr1X ++ tr2X) =
    trace_project tr1X ++ trace_project tr2X
  intros tr1X tr2X [sX Hpre_tr].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
X:= FreeE: VLSM message
trace_project:= equivocators_total_trace_project: list
  (composite_transition_item
     (equivocator_IM IM))
→ list (composite_transition_item IM)
tr1X, tr2X: list transition_item
sX: state (preloaded_with_all_messages_vlsm X)
Hpre_tr: finite_constrained_trace_from X sX
  (tr1X ++ tr2X)
trace_project (tr1X ++ tr2X) =
trace_project tr1X ++ trace_project tr2X
  specialize (equivocators_total_trace_project_characterization Hpre_tr) as Htr12_pr.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
X:= FreeE: VLSM message
trace_project:= equivocators_total_trace_project: list
  (composite_transition_item
     (equivocator_IM IM))
→ list (composite_transition_item IM)
tr1X, tr2X: list transition_item
sX: state (preloaded_with_all_messages_vlsm X)
Hpre_tr: finite_constrained_trace_from X sX
  (tr1X ++ tr2X)
Htr12_pr: equivocators_trace_project
  (zero_descriptor IM) 
  (tr1X ++ tr2X) =
Some
  (equivocators_total_trace_project
     (tr1X ++ tr2X), zero_descriptor IM)
trace_project (tr1X ++ tr2X) =
trace_project tr1X ++ trace_project tr2X
  apply equivocators_trace_project_app_iff in Htr12_pr.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
X:= FreeE: VLSM message
trace_project:= equivocators_total_trace_project: list
  (composite_transition_item
     (equivocator_IM IM))
→ list (composite_transition_item IM)
tr1X, tr2X: list transition_item
sX: state (preloaded_with_all_messages_vlsm X)
Hpre_tr: finite_constrained_trace_from X sX
  (tr1X ++ tr2X)
Htr12_pr: ∃ (preX
   sufX : list
            (composite_transition_item IM)) 
  (eqv_descriptors' : equivocator_descriptors
                      IM),
  equivocators_trace_project
    (zero_descriptor IM) tr2X =
  Some (sufX, eqv_descriptors')
  ∧ equivocators_trace_project
      eqv_descriptors' tr1X =
    Some (preX, zero_descriptor IM)
    ∧ equivocators_total_trace_project
        (tr1X ++ tr2X) = 
      preX ++ sufX
trace_project (tr1X ++ tr2X) =
trace_project tr1X ++ trace_project tr2X
  destruct Htr12_pr as [tr1Y [tr2Y [descriptors [Htr2_pr [Htr1_pr Htr12_eq]]]]].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
X:= FreeE: VLSM message
trace_project:= equivocators_total_trace_project: list
  (composite_transition_item
     (equivocator_IM IM))
→ list (composite_transition_item IM)
tr1X, tr2X: list transition_item
sX: state (preloaded_with_all_messages_vlsm X)
Hpre_tr: finite_constrained_trace_from X sX
  (tr1X ++ tr2X)
tr1Y, tr2Y: list (composite_transition_item IM)
descriptors: equivocator_descriptors IM
Htr2_pr: equivocators_trace_project
  (zero_descriptor IM) tr2X =
Some (tr2Y, descriptors)
Htr1_pr: equivocators_trace_project descriptors tr1X =
Some (tr1Y, zero_descriptor IM)
Htr12_eq: equivocators_total_trace_project
  (tr1X ++ tr2X) = tr1Y ++ tr2Y
trace_project (tr1X ++ tr2X) =
trace_project tr1X ++ trace_project tr2X
  apply (finite_valid_trace_from_app_iff PreFreeE) in Hpre_tr.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
X:= FreeE: VLSM message
trace_project:= equivocators_total_trace_project: list
  (composite_transition_item
     (equivocator_IM IM))
→ list (composite_transition_item IM)
tr1X, tr2X: list transition_item
sX: state (preloaded_with_all_messages_vlsm X)
Hpre_tr: finite_valid_trace_from PreFreeE sX tr1X
∧ finite_valid_trace_from PreFreeE
    (finite_trace_last sX tr1X) tr2X
tr1Y, tr2Y: list (composite_transition_item IM)
descriptors: equivocator_descriptors IM
Htr2_pr: equivocators_trace_project
  (zero_descriptor IM) tr2X =
Some (tr2Y, descriptors)
Htr1_pr: equivocators_trace_project descriptors tr1X =
Some (tr1Y, zero_descriptor IM)
Htr12_eq: equivocators_total_trace_project
  (tr1X ++ tr2X) = tr1Y ++ tr2Y
trace_project (tr1X ++ tr2X) =
trace_project tr1X ++ trace_project tr2X
  destruct Hpre_tr as [Hpre_tr1 Hpre_tr2].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
X:= FreeE: VLSM message
trace_project:= equivocators_total_trace_project: list
  (composite_transition_item
     (equivocator_IM IM))
→ list (composite_transition_item IM)
tr1X, tr2X: list transition_item
sX: state (preloaded_with_all_messages_vlsm X)
Hpre_tr1: finite_valid_trace_from PreFreeE sX tr1X
Hpre_tr2: finite_valid_trace_from PreFreeE
  (finite_trace_last sX tr1X) tr2X
tr1Y, tr2Y: list (composite_transition_item IM)
descriptors: equivocator_descriptors IM
Htr2_pr: equivocators_trace_project
  (zero_descriptor IM) tr2X =
Some (tr2Y, descriptors)
Htr1_pr: equivocators_trace_project descriptors tr1X =
Some (tr1Y, zero_descriptor IM)
Htr12_eq: equivocators_total_trace_project
  (tr1X ++ tr2X) = tr1Y ++ tr2Y
trace_project (tr1X ++ tr2X) =
trace_project tr1X ++ trace_project tr2X
  rewrite (equivocators_total_trace_project_characterization Hpre_tr2) in Htr2_pr.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
X:= FreeE: VLSM message
trace_project:= equivocators_total_trace_project: list
  (composite_transition_item
     (equivocator_IM IM))
→ list (composite_transition_item IM)
tr1X, tr2X: list transition_item
sX: state (preloaded_with_all_messages_vlsm X)
Hpre_tr1: finite_valid_trace_from PreFreeE sX tr1X
Hpre_tr2: finite_valid_trace_from PreFreeE
  (finite_trace_last sX tr1X) tr2X
tr1Y, tr2Y: list (composite_transition_item IM)
descriptors: equivocator_descriptors IM
Htr2_pr: Some
  (equivocators_total_trace_project tr2X,
   zero_descriptor IM) =
Some (tr2Y, descriptors)
Htr1_pr: equivocators_trace_project descriptors tr1X =
Some (tr1Y, zero_descriptor IM)
Htr12_eq: equivocators_total_trace_project
  (tr1X ++ tr2X) = tr1Y ++ tr2Y
trace_project (tr1X ++ tr2X) =
trace_project tr1X ++ trace_project tr2X
  inversion Htr2_pr.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
X:= FreeE: VLSM message
trace_project:= equivocators_total_trace_project: list
  (composite_transition_item
     (equivocator_IM IM))
→ list (composite_transition_item IM)
tr1X, tr2X: list transition_item
sX: state (preloaded_with_all_messages_vlsm X)
Hpre_tr1: finite_valid_trace_from PreFreeE sX tr1X
Hpre_tr2: finite_valid_trace_from PreFreeE
  (finite_trace_last sX tr1X) tr2X
tr1Y, tr2Y: list (composite_transition_item IM)
descriptors: equivocator_descriptors IM
Htr2_pr: Some
  (equivocators_total_trace_project tr2X,
   zero_descriptor IM) =
Some (tr2Y, descriptors)
Htr1_pr: equivocators_trace_project descriptors tr1X =
Some (tr1Y, zero_descriptor IM)
Htr12_eq: equivocators_total_trace_project
  (tr1X ++ tr2X) = tr1Y ++ tr2Y
H2: equivocators_total_trace_project tr2X = tr2Y
H3: zero_descriptor IM = descriptors
trace_project (tr1X ++ tr2X) =
trace_project tr1X ++ trace_project tr2X subst.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
X:= FreeE: VLSM message
trace_project:= equivocators_total_trace_project: list
  (composite_transition_item
     (equivocator_IM IM))
→ list (composite_transition_item IM)
tr1X, tr2X: list transition_item
sX: state (preloaded_with_all_messages_vlsm X)
Hpre_tr1: finite_valid_trace_from PreFreeE sX tr1X
Hpre_tr2: finite_valid_trace_from PreFreeE
  (finite_trace_last sX tr1X) tr2X
tr1Y: list (composite_transition_item IM)
Htr1_pr: equivocators_trace_project
  (zero_descriptor IM) tr1X =
Some (tr1Y, zero_descriptor IM)
Htr2_pr: Some
  (equivocators_total_trace_project tr2X,
   zero_descriptor IM) =
Some
  (equivocators_total_trace_project tr2X,
   zero_descriptor IM)
Htr12_eq: equivocators_total_trace_project
  (tr1X ++ tr2X) =
tr1Y ++
equivocators_total_trace_project tr2X
trace_project (tr1X ++ tr2X) =
trace_project tr1X ++ trace_project tr2X clear Htr2_pr.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
X:= FreeE: VLSM message
trace_project:= equivocators_total_trace_project: list
  (composite_transition_item
     (equivocator_IM IM))
→ list (composite_transition_item IM)
tr1X, tr2X: list transition_item
sX: state (preloaded_with_all_messages_vlsm X)
Hpre_tr1: finite_valid_trace_from PreFreeE sX tr1X
Hpre_tr2: finite_valid_trace_from PreFreeE
  (finite_trace_last sX tr1X) tr2X
tr1Y: list (composite_transition_item IM)
Htr1_pr: equivocators_trace_project
  (zero_descriptor IM) tr1X =
Some (tr1Y, zero_descriptor IM)
Htr12_eq: equivocators_total_trace_project
  (tr1X ++ tr2X) =
tr1Y ++
equivocators_total_trace_project tr2X
trace_project (tr1X ++ tr2X) =
trace_project tr1X ++ trace_project tr2X
  rewrite (equivocators_total_trace_project_characterization Hpre_tr1) in Htr1_pr.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
X:= FreeE: VLSM message
trace_project:= equivocators_total_trace_project: list
  (composite_transition_item
     (equivocator_IM IM))
→ list (composite_transition_item IM)
tr1X, tr2X: list transition_item
sX: state (preloaded_with_all_messages_vlsm X)
Hpre_tr1: finite_valid_trace_from PreFreeE sX tr1X
Hpre_tr2: finite_valid_trace_from PreFreeE
  (finite_trace_last sX tr1X) tr2X
tr1Y: list (composite_transition_item IM)
Htr1_pr: Some
  (equivocators_total_trace_project tr1X,
   zero_descriptor IM) =
Some (tr1Y, zero_descriptor IM)
Htr12_eq: equivocators_total_trace_project
  (tr1X ++ tr2X) =
tr1Y ++
equivocators_total_trace_project tr2X
trace_project (tr1X ++ tr2X) =
trace_project tr1X ++ trace_project tr2X
  by inversion Htr1_pr; subst.
Qed.

Lemma equivocators_total_VLSM_projection_finite_trace_project
  {s tr}
  (Hpre_tr : finite_constrained_trace_from FreeE s tr)
  : pre_VLSM_projection_finite_trace_project PreFreeE _ equivocators_total_label_project
      equivocators_total_state_project tr = equivocators_total_trace_project tr.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
s: state (preloaded_with_all_messages_vlsm FreeE)
tr: list transition_item
Hpre_tr: finite_constrained_trace_from FreeE s tr
pre_VLSM_projection_finite_trace_project PreFreeE
  (composite_type IM) equivocators_total_label_project
  equivocators_total_state_project tr =
equivocators_total_trace_project tr
Proof.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
s: state (preloaded_with_all_messages_vlsm FreeE)
tr: list transition_item
Hpre_tr: finite_constrained_trace_from FreeE s tr
pre_VLSM_projection_finite_trace_project PreFreeE
  (composite_type IM) equivocators_total_label_project
  equivocators_total_state_project tr =
equivocators_total_trace_project tr
  induction tr using rev_ind; [done |].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
s: state (preloaded_with_all_messages_vlsm FreeE)
x: transition_item
tr: list transition_item
Hpre_tr: finite_constrained_trace_from FreeE s
  (tr ++ [x])
IHtr: finite_constrained_trace_from FreeE s tr
→ pre_VLSM_projection_finite_trace_project
    PreFreeE (composite_type IM)
    equivocators_total_label_project
    equivocators_total_state_project tr =
  equivocators_total_trace_project tr
pre_VLSM_projection_finite_trace_project PreFreeE
  (composite_type IM) equivocators_total_label_project
  equivocators_total_state_project (tr ++ [x]) =
equivocators_total_trace_project (tr ++ [x])
  rewrite equivocators_total_trace_project_app by (eexists; done).message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
s: state (preloaded_with_all_messages_vlsm FreeE)
x: transition_item
tr: list transition_item
Hpre_tr: finite_constrained_trace_from FreeE s
  (tr ++ [x])
IHtr: finite_constrained_trace_from FreeE s tr
→ pre_VLSM_projection_finite_trace_project
    PreFreeE (composite_type IM)
    equivocators_total_label_project
    equivocators_total_state_project tr =
  equivocators_total_trace_project tr
pre_VLSM_projection_finite_trace_project PreFreeE
  (composite_type IM) equivocators_total_label_project
  equivocators_total_state_project (tr ++ [x]) =
equivocators_total_trace_project tr ++
equivocators_total_trace_project [x]
  rewrite pre_VLSM_projection_finite_trace_project_app.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
s: state (preloaded_with_all_messages_vlsm FreeE)
x: transition_item
tr: list transition_item
Hpre_tr: finite_constrained_trace_from FreeE s
  (tr ++ [x])
IHtr: finite_constrained_trace_from FreeE s tr
→ pre_VLSM_projection_finite_trace_project
    PreFreeE (composite_type IM)
    equivocators_total_label_project
    equivocators_total_state_project tr =
  equivocators_total_trace_project tr
pre_VLSM_projection_finite_trace_project
  (preloaded_with_all_messages_vlsm FreeE)
  (composite_type IM) equivocators_total_label_project
  equivocators_total_state_project tr ++
pre_VLSM_projection_finite_trace_project
  (preloaded_with_all_messages_vlsm FreeE)
  (composite_type IM) equivocators_total_label_project
  equivocators_total_state_project [x] =
equivocators_total_trace_project tr ++
equivocators_total_trace_project [x]
  apply finite_valid_trace_from_app_iff in Hpre_tr as [Hpre_tr Hpre_x].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
s: state (preloaded_with_all_messages_vlsm FreeE)
x: transition_item
tr: list transition_item
Hpre_tr: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) s
  tr
Hpre_x: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE)
  (finite_trace_last s tr) [x]
IHtr: finite_constrained_trace_from FreeE s tr
→ pre_VLSM_projection_finite_trace_project
    PreFreeE (composite_type IM)
    equivocators_total_label_project
    equivocators_total_state_project tr =
  equivocators_total_trace_project tr
pre_VLSM_projection_finite_trace_project
  (preloaded_with_all_messages_vlsm FreeE)
  (composite_type IM) equivocators_total_label_project
  equivocators_total_state_project tr ++
pre_VLSM_projection_finite_trace_project
  (preloaded_with_all_messages_vlsm FreeE)
  (composite_type IM) equivocators_total_label_project
  equivocators_total_state_project [x] =
equivocators_total_trace_project tr ++
equivocators_total_trace_project [x]
  specialize (IHtr Hpre_tr).message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
s: state (preloaded_with_all_messages_vlsm FreeE)
x: transition_item
tr: list transition_item
Hpre_tr: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) s
  tr
Hpre_x: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE)
  (finite_trace_last s tr) [x]
IHtr: pre_VLSM_projection_finite_trace_project
  PreFreeE (composite_type IM)
  equivocators_total_label_project
  equivocators_total_state_project tr =
equivocators_total_trace_project tr
pre_VLSM_projection_finite_trace_project
  (preloaded_with_all_messages_vlsm FreeE)
  (composite_type IM) equivocators_total_label_project
  equivocators_total_state_project tr ++
pre_VLSM_projection_finite_trace_project
  (preloaded_with_all_messages_vlsm FreeE)
  (composite_type IM) equivocators_total_label_project
  equivocators_total_state_project [x] =
equivocators_total_trace_project tr ++
equivocators_total_trace_project [x]
  unfold PreFreeE in IHtr; rewrite IHtr.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
s: state (preloaded_with_all_messages_vlsm FreeE)
x: transition_item
tr: list transition_item
Hpre_tr: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) s
  tr
Hpre_x: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE)
  (finite_trace_last s tr) [x]
IHtr: pre_VLSM_projection_finite_trace_project
  (preloaded_with_all_messages_vlsm FreeE)
  (composite_type IM)
  equivocators_total_label_project
  equivocators_total_state_project tr =
equivocators_total_trace_project tr
equivocators_total_trace_project tr ++
pre_VLSM_projection_finite_trace_project
  (preloaded_with_all_messages_vlsm FreeE)
  (composite_type IM) equivocators_total_label_project
  equivocators_total_state_project [x] =
equivocators_total_trace_project tr ++
equivocators_total_trace_project [x]
  f_equal.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
s: state (preloaded_with_all_messages_vlsm FreeE)
x: transition_item
tr: list transition_item
Hpre_tr: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) s
  tr
Hpre_x: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE)
  (finite_trace_last s tr) [x]
IHtr: pre_VLSM_projection_finite_trace_project
  (preloaded_with_all_messages_vlsm FreeE)
  (composite_type IM)
  equivocators_total_label_project
  equivocators_total_state_project tr =
equivocators_total_trace_project tr
pre_VLSM_projection_finite_trace_project
  (preloaded_with_all_messages_vlsm FreeE)
  (composite_type IM) equivocators_total_label_project
  equivocators_total_state_project [x] =
equivocators_total_trace_project [x]
  inversion Hpre_x.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
s: state (preloaded_with_all_messages_vlsm FreeE)
x: transition_item
tr: list transition_item
Hpre_tr: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) s
  tr
Hpre_x: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE)
  (finite_trace_last s tr) [x]
IHtr: pre_VLSM_projection_finite_trace_project
  (preloaded_with_all_messages_vlsm FreeE)
  (composite_type IM)
  equivocators_total_label_project
  equivocators_total_state_project tr =
equivocators_total_trace_project tr
s0: state (preloaded_with_all_messages_vlsm FreeE)
tl: list transition_item
s': state (preloaded_with_all_messages_vlsm FreeE)
iom, oom: option message
l: label (preloaded_with_all_messages_vlsm FreeE)
Htl: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) s0 []
Ht: input_valid_transition
  (preloaded_with_all_messages_vlsm FreeE) l
  (finite_trace_last s tr, iom) (
  s0, oom)
H1: s' = finite_trace_last s tr
H3: {|
  l := l;
  input := iom;
  destination := s0;
  output := oom
|} = x
H4: tl = []
pre_VLSM_projection_finite_trace_project
  (preloaded_with_all_messages_vlsm FreeE)
  (composite_type IM) equivocators_total_label_project
  equivocators_total_state_project
  [{|
     l := l;
     input := iom;
     destination := s0;
     output := oom
   |}] =
equivocators_total_trace_project
  [{|
     l := l;
     input := iom;
     destination := s0;
     output := oom
   |}] subst.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
s: state (preloaded_with_all_messages_vlsm FreeE)
tr: list transition_item
Hpre_tr: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) s
  tr
s0: state (preloaded_with_all_messages_vlsm FreeE)
iom, oom: option message
l: label (preloaded_with_all_messages_vlsm FreeE)
Hpre_x: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE)
  (finite_trace_last s tr)
  [{|
     l := l;
     input := iom;
     destination := s0;
     output := oom
   |}]
IHtr: pre_VLSM_projection_finite_trace_project
  (preloaded_with_all_messages_vlsm FreeE)
  (composite_type IM)
  equivocators_total_label_project
  equivocators_total_state_project tr =
equivocators_total_trace_project tr
Htl: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) s0 []
Ht: input_valid_transition
  (preloaded_with_all_messages_vlsm FreeE) l
  (finite_trace_last s tr, iom) (
  s0, oom)
pre_VLSM_projection_finite_trace_project
  (preloaded_with_all_messages_vlsm FreeE)
  (composite_type IM) equivocators_total_label_project
  equivocators_total_state_project
  [{|
     l := l;
     input := iom;
     destination := s0;
     output := oom
   |}] =
equivocators_total_trace_project
  [{|
     l := l;
     input := iom;
     destination := s0;
     output := oom
   |}]
  destruct Ht as [[_ [_ Hv]] Ht].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
s: state (preloaded_with_all_messages_vlsm FreeE)
tr: list transition_item
Hpre_tr: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) s
  tr
s0: state (preloaded_with_all_messages_vlsm FreeE)
iom, oom: option message
l: label (preloaded_with_all_messages_vlsm FreeE)
Hpre_x: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE)
  (finite_trace_last s tr)
  [{|
     l := l;
     input := iom;
     destination := s0;
     output := oom
   |}]
IHtr: pre_VLSM_projection_finite_trace_project
  (preloaded_with_all_messages_vlsm FreeE)
  (composite_type IM)
  equivocators_total_label_project
  equivocators_total_state_project tr =
equivocators_total_trace_project tr
Htl: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) s0 []
Hv: valid l (finite_trace_last s tr, iom)
Ht: transition l (finite_trace_last s tr, iom) =
(s0, oom)
pre_VLSM_projection_finite_trace_project
  (preloaded_with_all_messages_vlsm FreeE)
  (composite_type IM) equivocators_total_label_project
  equivocators_total_state_project
  [{|
     l := l;
     input := iom;
     destination := s0;
     output := oom
   |}] =
equivocators_total_trace_project
  [{|
     l := l;
     input := iom;
     destination := s0;
     output := oom
   |}] destruct l as (i, [sn | ji li | ji li])
  ; unfold equivocators_total_trace_project; cbn in *
  ; unfold equivocators_transition_item_project; cbn in *
  ; rewrite !equivocator_state_project_zero.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
s: composite_state (equivocator_IM IM)
tr: list transition_item
Hpre_tr: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) s
  tr
s0: composite_state (equivocator_IM IM)
iom, oom: option message
i: index
sn: state (IM i)
Hpre_x: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE)
  (finite_trace_last s tr)
  [{|
     l := existT i (Spawn sn);
     input := iom;
     destination := s0;
     output := oom
   |}]
IHtr: pre_VLSM_projection_finite_trace_project
  (composite_type (equivocator_IM IM))
  (composite_type IM)
  equivocators_total_label_project
  equivocators_total_state_project tr =
equivocators_total_trace_project tr
Htl: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) s0 []
Hv: initial_state_prop sn ∧ iom = None
Ht: (state_update (equivocator_IM IM)
   (finite_trace_last s tr) i
   (equivocator_state_extend
      (finite_trace_last s tr i) sn), None) =
(s0, oom)
[] =
from_option fst []
  match
    match
      (if decide (0 = equivocator_state_last (s0 i))
       then Some (None, NewMachine sn)
       else Some (None, Existing 0))
    with
    | Some (Some item', deqv') =>
        Some
          (Some
             {|
               l := existT i (l item');
               input := iom;
               destination :=
                 equivocators_state_project IM
                   (zero_descriptor IM) s0;
               output := oom
             |},
           equivocator_descriptors_update IM
             (zero_descriptor IM) i deqv')
    | Some (None, deqv') =>
        Some
          (None,
           equivocator_descriptors_update IM
             (zero_descriptor IM) i deqv')
    | None => None
    end
  with
  | Some (Some item', odescriptor) =>
      Some ([item'], odescriptor)
  | Some (None, odescriptor) => Some ([], odescriptor)
  | None => None
  end
message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
s: composite_state (equivocator_IM IM)
tr: list transition_item
Hpre_tr: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) s
  tr
s0: composite_state (equivocator_IM IM)
iom, oom: option message
i: index
ji: nat
li: label (IM i)
Hpre_x: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE)
  (finite_trace_last s tr)
  [{|
     l := existT i (ContinueWith ji li);
     input := iom;
     destination := s0;
     output := oom
   |}]
IHtr: pre_VLSM_projection_finite_trace_project
  (composite_type (equivocator_IM IM))
  (composite_type IM)
  equivocators_total_label_project
  equivocators_total_state_project tr =
equivocators_total_trace_project tr
Htl: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) s0 []
Hv: match
  equivocator_state_project
    (finite_trace_last s tr i) ji
with
| Some si => valid li (si, iom)
| None => False
end
Ht: (let (si', om') :=
   match
     equivocator_state_project
       (finite_trace_last s tr i) ji
   with
   | Some si =>
       let (si', om') :=
         transition li (si, iom) in
       (equivocator_state_update
          (finite_trace_last s tr i) ji si', om')
   | None => (finite_trace_last s tr i, iom)
   end in
 (state_update (equivocator_IM IM)
    (finite_trace_last s tr) i si', om')) =
(s0, oom)
match
  pre_VLSM_projection_transition_item_project
    (composite_type (equivocator_IM IM))
    (composite_type IM)
    equivocators_total_label_project
    equivocators_total_state_project
    {|
      l := existT i (ContinueWith ji li);
      input := iom;
      destination := s0;
      output := oom
    |}
with
| Some y => [y]
| None => []
end =
from_option fst []
  match
    match
      (if decide (ji = 0)
       then
        Some
          (Some
             {|
               l := li;
               input := iom;
               destination :=
                 equivocator_state_zero (s0 i);
               output := oom
             |}, Existing ji)
       else Some (None, Existing 0))
    with
    | Some (Some item', deqv') =>
        Some
          (Some
             {|
               l := existT i (l item');
               input := iom;
               destination :=
                 equivocators_state_project IM
                   (zero_descriptor IM) s0;
               output := oom
             |},
           equivocator_descriptors_update IM
             (zero_descriptor IM) i deqv')
    | Some (None, deqv') =>
        Some
          (None,
           equivocator_descriptors_update IM
             (zero_descriptor IM) i deqv')
    | None => None
    end
  with
  | Some (Some item', odescriptor) =>
      Some ([item'], odescriptor)
  | Some (None, odescriptor) => Some ([], odescriptor)
  | None => None
  end
message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
s: composite_state (equivocator_IM IM)
tr: list transition_item
Hpre_tr: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) s
  tr
s0: composite_state (equivocator_IM IM)
iom, oom: option message
i: index
ji: nat
li: label (IM i)
Hpre_x: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE)
  (finite_trace_last s tr)
  [{|
     l := existT i (ForkWith ji li);
     input := iom;
     destination := s0;
     output := oom
   |}]
IHtr: pre_VLSM_projection_finite_trace_project
  (composite_type (equivocator_IM IM))
  (composite_type IM)
  equivocators_total_label_project
  equivocators_total_state_project tr =
equivocators_total_trace_project tr
Htl: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) s0 []
Hv: match
  equivocator_state_project
    (finite_trace_last s tr i) ji
with
| Some si => valid li (si, iom)
| None => False
end
Ht: (let (si', om') :=
   match
     equivocator_state_project
       (finite_trace_last s tr i) ji
   with
   | Some si =>
       let (si', om') :=
         transition li (si, iom) in
       (equivocator_state_extend
          (finite_trace_last s tr i) si', om')
   | None => (finite_trace_last s tr i, iom)
   end in
 (state_update (equivocator_IM IM)
    (finite_trace_last s tr) i si', om')) =
(s0, oom)
[] =
from_option fst []
  match
    match
      (if decide (0 = equivocator_state_last (s0 i))
       then
        Some
          (Some
             {|
               l := li;
               input := iom;
               destination :=
                 equivocator_state_zero (s0 i);
               output := oom
             |}, Existing ji)
       else Some (None, Existing 0))
    with
    | Some (Some item', deqv') =>
        Some
          (Some
             {|
               l := existT i (l item');
               input := iom;
               destination :=
                 equivocators_state_project IM
                   (zero_descriptor IM) s0;
               output := oom
             |},
           equivocator_descriptors_update IM
             (zero_descriptor IM) i deqv')
    | Some (None, deqv') =>
        Some
          (None,
           equivocator_descriptors_update IM
             (zero_descriptor IM) i deqv')
    | None => None
    end
  with
  | Some (Some item', odescriptor) =>
      Some ([item'], odescriptor)
  | Some (None, odescriptor) => Some ([], odescriptor)
  | None => None
  end
  -message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
s: composite_state (equivocator_IM IM)
tr: list transition_item
Hpre_tr: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) s
  tr
s0: composite_state (equivocator_IM IM)
iom, oom: option message
i: index
sn: state (IM i)
Hpre_x: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE)
  (finite_trace_last s tr)
  [{|
     l := existT i (Spawn sn);
     input := iom;
     destination := s0;
     output := oom
   |}]
IHtr: pre_VLSM_projection_finite_trace_project
  (composite_type (equivocator_IM IM))
  (composite_type IM)
  equivocators_total_label_project
  equivocators_total_state_project tr =
equivocators_total_trace_project tr
Htl: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) s0 []
Hv: initial_state_prop sn ∧ iom = None
Ht: (state_update (equivocator_IM IM)
   (finite_trace_last s tr) i
   (equivocator_state_extend
      (finite_trace_last s tr i) sn), None) =
(s0, oom)
[] =
from_option fst []
  match
    match
      (if decide (0 = equivocator_state_last (s0 i))
       then Some (None, NewMachine sn)
       else Some (None, Existing 0))
    with
    | Some (Some item', deqv') =>
        Some
          (Some
             {|
               l := existT i (l item');
               input := iom;
               destination :=
                 equivocators_state_project IM
                   (zero_descriptor IM) s0;
               output := oom
             |},
           equivocator_descriptors_update IM
             (zero_descriptor IM) i deqv')
    | Some (None, deqv') =>
        Some
          (None,
           equivocator_descriptors_update IM
             (zero_descriptor IM) i deqv')
    | None => None
    end
  with
  | Some (Some item', odescriptor) =>
      Some ([item'], odescriptor)
  | Some (None, odescriptor) => Some ([], odescriptor)
  | None => None
  end inversion_clear Ht.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
s: composite_state (equivocator_IM IM)
tr: list transition_item
Hpre_tr: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) s
  tr
s0: composite_state (equivocator_IM IM)
iom, oom: option message
i: index
sn: state (IM i)
Hpre_x: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE)
  (finite_trace_last s tr)
  [{|
     l := existT i (Spawn sn);
     input := iom;
     destination := s0;
     output := oom
   |}]
IHtr: pre_VLSM_projection_finite_trace_project
  (composite_type (equivocator_IM IM))
  (composite_type IM)
  equivocators_total_label_project
  equivocators_total_state_project tr =
equivocators_total_trace_project tr
Htl: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) s0 []
Hv: initial_state_prop sn ∧ iom = None
[] =
from_option fst []
  match
    match
      (if
        decide
          (0 =
           equivocator_state_last
             (state_update (equivocator_IM IM)
                (finite_trace_last s tr) i
                (equivocator_state_extend
                   (finite_trace_last s tr i) sn) i))
       then Some (None, NewMachine sn)
       else Some (None, Existing 0))
    with
    | Some (Some item', deqv') =>
        Some
          (Some
             {|
               l := existT i (l item');
               input := iom;
               destination :=
                 equivocators_state_project IM
                   (zero_descriptor IM)
                   (state_update (equivocator_IM IM)
                      (finite_trace_last s tr) i
                      (equivocator_state_extend
                         (finite_trace_last s tr i) sn));
               output := None
             |},
           equivocator_descriptors_update IM
             (zero_descriptor IM) i deqv')
    | Some (None, deqv') =>
        Some
          (None,
           equivocator_descriptors_update IM
             (zero_descriptor IM) i deqv')
    | None => None
    end
  with
  | Some (Some item', odescriptor) =>
      Some ([item'], odescriptor)
  | Some (None, odescriptor) => Some ([], odescriptor)
  | None => None
  end
    rewrite decide_False; [done |].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
s: composite_state (equivocator_IM IM)
tr: list transition_item
Hpre_tr: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) s
  tr
s0: composite_state (equivocator_IM IM)
iom, oom: option message
i: index
sn: state (IM i)
Hpre_x: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE)
  (finite_trace_last s tr)
  [{|
     l := existT i (Spawn sn);
     input := iom;
     destination := s0;
     output := oom
   |}]
IHtr: pre_VLSM_projection_finite_trace_project
  (composite_type (equivocator_IM IM))
  (composite_type IM)
  equivocators_total_label_project
  equivocators_total_state_project tr =
equivocators_total_trace_project tr
Htl: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) s0 []
Hv: initial_state_prop sn ∧ iom = None
0
≠ equivocator_state_last
    (state_update (equivocator_IM IM)
       (finite_trace_last s tr) i
       (equivocator_state_extend
          (finite_trace_last s tr i) sn) i)
    by state_update_simpl; cbn.
  -message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
s: composite_state (equivocator_IM IM)
tr: list transition_item
Hpre_tr: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) s
  tr
s0: composite_state (equivocator_IM IM)
iom, oom: option message
i: index
ji: nat
li: label (IM i)
Hpre_x: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE)
  (finite_trace_last s tr)
  [{|
     l := existT i (ContinueWith ji li);
     input := iom;
     destination := s0;
     output := oom
   |}]
IHtr: pre_VLSM_projection_finite_trace_project
  (composite_type (equivocator_IM IM))
  (composite_type IM)
  equivocators_total_label_project
  equivocators_total_state_project tr =
equivocators_total_trace_project tr
Htl: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) s0 []
Hv: match
  equivocator_state_project
    (finite_trace_last s tr i) ji
with
| Some si => valid li (si, iom)
| None => False
end
Ht: (let (si', om') :=
   match
     equivocator_state_project
       (finite_trace_last s tr i) ji
   with
   | Some si =>
       let (si', om') :=
         transition li (si, iom) in
       (equivocator_state_update
          (finite_trace_last s tr i) ji si', om')
   | None => (finite_trace_last s tr i, iom)
   end in
 (state_update (equivocator_IM IM)
    (finite_trace_last s tr) i si', om')) =
(s0, oom)
match
  pre_VLSM_projection_transition_item_project
    (composite_type (equivocator_IM IM))
    (composite_type IM)
    equivocators_total_label_project
    equivocators_total_state_project
    {|
      l := existT i (ContinueWith ji li);
      input := iom;
      destination := s0;
      output := oom
    |}
with
| Some y => [y]
| None => []
end =
from_option fst []
  match
    match
      (if decide (ji = 0)
       then
        Some
          (Some
             {|
               l := li;
               input := iom;
               destination :=
                 equivocator_state_zero (s0 i);
               output := oom
             |}, Existing ji)
       else Some (None, Existing 0))
    with
    | Some (Some item', deqv') =>
        Some
          (Some
             {|
               l := existT i (l item');
               input := iom;
               destination :=
                 equivocators_state_project IM
                   (zero_descriptor IM) s0;
               output := oom
             |},
           equivocator_descriptors_update IM
             (zero_descriptor IM) i deqv')
    | Some (None, deqv') =>
        Some
          (None,
           equivocator_descriptors_update IM
             (zero_descriptor IM) i deqv')
    | None => None
    end
  with
  | Some (Some item', odescriptor) =>
      Some ([item'], odescriptor)
  | Some (None, odescriptor) => Some ([], odescriptor)
  | None => None
  end destruct (equivocator_state_project _ _) as [s_i |]; [| done].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
s: composite_state (equivocator_IM IM)
tr: list transition_item
Hpre_tr: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) s
  tr
s0: composite_state (equivocator_IM IM)
iom, oom: option message
i: index
ji: nat
li: label (IM i)
Hpre_x: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE)
  (finite_trace_last s tr)
  [{|
     l := existT i (ContinueWith ji li);
     input := iom;
     destination := s0;
     output := oom
   |}]
IHtr: pre_VLSM_projection_finite_trace_project
  (composite_type (equivocator_IM IM))
  (composite_type IM)
  equivocators_total_label_project
  equivocators_total_state_project tr =
equivocators_total_trace_project tr
Htl: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) s0 []
s_i: state (IM i)
Hv: valid li (s_i, iom)
Ht: (let (si', om') :=
   let (si', om') := transition li (s_i, iom) in
   (equivocator_state_update
      (finite_trace_last s tr i) ji si', om') in
 (state_update (equivocator_IM IM)
    (finite_trace_last s tr) i si', om')) =
(s0, oom)
match
  pre_VLSM_projection_transition_item_project
    (composite_type (equivocator_IM IM))
    (composite_type IM)
    equivocators_total_label_project
    equivocators_total_state_project
    {|
      l := existT i (ContinueWith ji li);
      input := iom;
      destination := s0;
      output := oom
    |}
with
| Some y => [y]
| None => []
end =
from_option fst []
  match
    match
      (if decide (ji = 0)
       then
        Some
          (Some
             {|
               l := li;
               input := iom;
               destination :=
                 equivocator_state_zero (s0 i);
               output := oom
             |}, Existing ji)
       else Some (None, Existing 0))
    with
    | Some (Some item', deqv') =>
        Some
          (Some
             {|
               l := existT i (l item');
               input := iom;
               destination :=
                 equivocators_state_project IM
                   (zero_descriptor IM) s0;
               output := oom
             |},
           equivocator_descriptors_update IM
             (zero_descriptor IM) i deqv')
    | Some (None, deqv') =>
        Some
          (None,
           equivocator_descriptors_update IM
             (zero_descriptor IM) i deqv')
    | None => None
    end
  with
  | Some (Some item', odescriptor) =>
      Some ([item'], odescriptor)
  | Some (None, odescriptor) => Some ([], odescriptor)
  | None => None
  end
    destruct (transition _ _ _) as (si', _om').message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
s: composite_state (equivocator_IM IM)
tr: list transition_item
Hpre_tr: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) s
  tr
s0: composite_state (equivocator_IM IM)
iom, oom: option message
i: index
ji: nat
li: label (IM i)
Hpre_x: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE)
  (finite_trace_last s tr)
  [{|
     l := existT i (ContinueWith ji li);
     input := iom;
     destination := s0;
     output := oom
   |}]
IHtr: pre_VLSM_projection_finite_trace_project
  (composite_type (equivocator_IM IM))
  (composite_type IM)
  equivocators_total_label_project
  equivocators_total_state_project tr =
equivocators_total_trace_project tr
Htl: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) s0 []
s_i: state (IM i)
Hv: valid li (s_i, iom)
si': state (IM i)
_om': option message
Ht: (state_update (equivocator_IM IM)
   (finite_trace_last s tr) i
   (equivocator_state_update
      (finite_trace_last s tr i) ji si'), _om') =
(s0, oom)
match
  pre_VLSM_projection_transition_item_project
    (composite_type (equivocator_IM IM))
    (composite_type IM)
    equivocators_total_label_project
    equivocators_total_state_project
    {|
      l := existT i (ContinueWith ji li);
      input := iom;
      destination := s0;
      output := oom
    |}
with
| Some y => [y]
| None => []
end =
from_option fst []
  match
    match
      (if decide (ji = 0)
       then
        Some
          (Some
             {|
               l := li;
               input := iom;
               destination :=
                 equivocator_state_zero (s0 i);
               output := oom
             |}, Existing ji)
       else Some (None, Existing 0))
    with
    | Some (Some item', deqv') =>
        Some
          (Some
             {|
               l := existT i (l item');
               input := iom;
               destination :=
                 equivocators_state_project IM
                   (zero_descriptor IM) s0;
               output := oom
             |},
           equivocator_descriptors_update IM
             (zero_descriptor IM) i deqv')
    | Some (None, deqv') =>
        Some
          (None,
           equivocator_descriptors_update IM
             (zero_descriptor IM) i deqv')
    | None => None
    end
  with
  | Some (Some item', odescriptor) =>
      Some ([item'], odescriptor)
  | Some (None, odescriptor) => Some ([], odescriptor)
  | None => None
  end
    inversion_clear Ht.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
s: composite_state (equivocator_IM IM)
tr: list transition_item
Hpre_tr: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) s
  tr
s0: composite_state (equivocator_IM IM)
iom, oom: option message
i: index
ji: nat
li: label (IM i)
Hpre_x: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE)
  (finite_trace_last s tr)
  [{|
     l := existT i (ContinueWith ji li);
     input := iom;
     destination := s0;
     output := oom
   |}]
IHtr: pre_VLSM_projection_finite_trace_project
  (composite_type (equivocator_IM IM))
  (composite_type IM)
  equivocators_total_label_project
  equivocators_total_state_project tr =
equivocators_total_trace_project tr
Htl: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) s0 []
s_i: state (IM i)
Hv: valid li (s_i, iom)
si': state (IM i)
_om': option message
match
  pre_VLSM_projection_transition_item_project
    (composite_type (equivocator_IM IM))
    (composite_type IM)
    equivocators_total_label_project
    equivocators_total_state_project
    {|
      l := existT i (ContinueWith ji li);
      input := iom;
      destination :=
        state_update (equivocator_IM IM)
          (finite_trace_last s tr) i
          (equivocator_state_update
             (finite_trace_last s tr i) ji si');
      output := oom
    |}
with
| Some y => [y]
| None => []
end =
from_option fst []
  match
    match
      (if decide (ji = 0)
       then
        Some
          (Some
             {|
               l := li;
               input := iom;
               destination :=
                 equivocator_state_zero
                   (state_update (equivocator_IM IM)
                      (finite_trace_last s tr) i
                      (equivocator_state_update
                         (finite_trace_last s tr i) ji
                         si') i);
               output := oom
             |}, Existing ji)
       else Some (None, Existing 0))
    with
    | Some (Some item', deqv') =>
        Some
          (Some
             {|
               l := existT i (l item');
               input := iom;
               destination :=
                 equivocators_state_project IM
                   (zero_descriptor IM)
                   (state_update (equivocator_IM IM)
                      (finite_trace_last s tr) i
                      (equivocator_state_update
                         (finite_trace_last s tr i) ji
                         si'));
               output := oom
             |},
           equivocator_descriptors_update IM
             (zero_descriptor IM) i deqv')
    | Some (None, deqv') =>
        Some
          (None,
           equivocator_descriptors_update IM
             (zero_descriptor IM) i deqv')
    | None => None
    end
  with
  | Some (Some item', odescriptor) =>
      Some ([item'], odescriptor)
  | Some (None, odescriptor) => Some ([], odescriptor)
  | None => None
  end
    destruct ji as [| ji].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
s: composite_state (equivocator_IM IM)
tr: list transition_item
Hpre_tr: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) s
  tr
s0: composite_state (equivocator_IM IM)
iom, oom: option message
i: index
li: label (IM i)
Hpre_x: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE)
  (finite_trace_last s tr)
  [{|
     l := existT i (ContinueWith 0 li);
     input := iom;
     destination := s0;
     output := oom
   |}]
IHtr: pre_VLSM_projection_finite_trace_project
  (composite_type (equivocator_IM IM))
  (composite_type IM)
  equivocators_total_label_project
  equivocators_total_state_project tr =
equivocators_total_trace_project tr
Htl: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) s0 []
s_i: state (IM i)
Hv: valid li (s_i, iom)
si': state (IM i)
_om': option message
match
  pre_VLSM_projection_transition_item_project
    (composite_type (equivocator_IM IM))
    (composite_type IM)
    equivocators_total_label_project
    equivocators_total_state_project
    {|
      l := existT i (ContinueWith 0 li);
      input := iom;
      destination :=
        state_update (equivocator_IM IM)
          (finite_trace_last s tr) i
          (equivocator_state_update
             (finite_trace_last s tr i) 0 si');
      output := oom
    |}
with
| Some y => [y]
| None => []
end =
from_option fst []
  match
    match
      (if decide (0 = 0)
       then
        Some
          (Some
             {|
               l := li;
               input := iom;
               destination :=
                 equivocator_state_zero
                   (state_update (equivocator_IM IM)
                      (finite_trace_last s tr) i
                      (equivocator_state_update
                         (finite_trace_last s tr i) 0
                         si') i);
               output := oom
             |}, Existing 0)
       else Some (None, Existing 0))
    with
    | Some (Some item', deqv') =>
        Some
          (Some
             {|
               l := existT i (l item');
               input := iom;
               destination :=
                 equivocators_state_project IM
                   (zero_descriptor IM)
                   (state_update (equivocator_IM IM)
                      (finite_trace_last s tr) i
                      (equivocator_state_update
                         (finite_trace_last s tr i) 0
                         si'));
               output := oom
             |},
           equivocator_descriptors_update IM
             (zero_descriptor IM) i deqv')
    | Some (None, deqv') =>
        Some
          (None,
           equivocator_descriptors_update IM
             (zero_descriptor IM) i deqv')
    | None => None
    end
  with
  | Some (Some item', odescriptor) =>
      Some ([item'], odescriptor)
  | Some (None, odescriptor) => Some ([], odescriptor)
  | None => None
  end
message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
s: composite_state (equivocator_IM IM)
tr: list transition_item
Hpre_tr: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) s
  tr
s0: composite_state (equivocator_IM IM)
iom, oom: option message
i: index
ji: nat
li: label (IM i)
Hpre_x: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE)
  (finite_trace_last s tr)
  [{|
     l := existT i (ContinueWith (S ji) li);
     input := iom;
     destination := s0;
     output := oom
   |}]
IHtr: pre_VLSM_projection_finite_trace_project
  (composite_type (equivocator_IM IM))
  (composite_type IM)
  equivocators_total_label_project
  equivocators_total_state_project tr =
equivocators_total_trace_project tr
Htl: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) s0 []
s_i: state (IM i)
Hv: valid li (s_i, iom)
si': state (IM i)
_om': option message
match
  pre_VLSM_projection_transition_item_project
    (composite_type (equivocator_IM IM))
    (composite_type IM)
    equivocators_total_label_project
    equivocators_total_state_project
    {|
      l := existT i (ContinueWith (S ji) li);
      input := iom;
      destination :=
        state_update (equivocator_IM IM)
          (finite_trace_last s tr) i
          (equivocator_state_update
             (finite_trace_last s tr i) 
             (S ji) si');
      output := oom
    |}
with
| Some y => [y]
| None => []
end =
from_option fst []
  match
    match
      (if decide (S ji = 0)
       then
        Some
          (Some
             {|
               l := li;
               input := iom;
               destination :=
                 equivocator_state_zero
                   (state_update 
                      (equivocator_IM IM)
                      (finite_trace_last s tr) i
                      (equivocator_state_update
                      (finite_trace_last s tr i)
                      (S ji) si') i);
               output := oom
             |}, Existing (S ji))
       else Some (None, Existing 0))
    with
    | Some (Some item', deqv') =>
        Some
          (Some
             {|
               l := existT i (l item');
               input := iom;
               destination :=
                 equivocators_state_project IM
                   (zero_descriptor IM)
                   (state_update 
                      (equivocator_IM IM)
                      (finite_trace_last s tr) i
                      (equivocator_state_update
                      (finite_trace_last s tr i)
                      (S ji) si'));
               output := oom
             |},
           equivocator_descriptors_update IM
             (zero_descriptor IM) i deqv')
    | Some (None, deqv') =>
        Some
          (None,
           equivocator_descriptors_update IM
             (zero_descriptor IM) i deqv')
    | None => None
    end
  with
  | Some (Some item', odescriptor) =>
      Some ([item'], odescriptor)
  | Some (None, odescriptor) => Some ([], odescriptor)
  | None => None
  end
    +message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
s: composite_state (equivocator_IM IM)
tr: list transition_item
Hpre_tr: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) s
  tr
s0: composite_state (equivocator_IM IM)
iom, oom: option message
i: index
li: label (IM i)
Hpre_x: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE)
  (finite_trace_last s tr)
  [{|
     l := existT i (ContinueWith 0 li);
     input := iom;
     destination := s0;
     output := oom
   |}]
IHtr: pre_VLSM_projection_finite_trace_project
  (composite_type (equivocator_IM IM))
  (composite_type IM)
  equivocators_total_label_project
  equivocators_total_state_project tr =
equivocators_total_trace_project tr
Htl: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) s0 []
s_i: state (IM i)
Hv: valid li (s_i, iom)
si': state (IM i)
_om': option message
match
  pre_VLSM_projection_transition_item_project
    (composite_type (equivocator_IM IM))
    (composite_type IM)
    equivocators_total_label_project
    equivocators_total_state_project
    {|
      l := existT i (ContinueWith 0 li);
      input := iom;
      destination :=
        state_update (equivocator_IM IM)
          (finite_trace_last s tr) i
          (equivocator_state_update
             (finite_trace_last s tr i) 0 si');
      output := oom
    |}
with
| Some y => [y]
| None => []
end =
from_option fst []
  match
    match
      (if decide (0 = 0)
       then
        Some
          (Some
             {|
               l := li;
               input := iom;
               destination :=
                 equivocator_state_zero
                   (state_update (equivocator_IM IM)
                      (finite_trace_last s tr) i
                      (equivocator_state_update
                         (finite_trace_last s tr i) 0
                         si') i);
               output := oom
             |}, Existing 0)
       else Some (None, Existing 0))
    with
    | Some (Some item', deqv') =>
        Some
          (Some
             {|
               l := existT i (l item');
               input := iom;
               destination :=
                 equivocators_state_project IM
                   (zero_descriptor IM)
                   (state_update (equivocator_IM IM)
                      (finite_trace_last s tr) i
                      (equivocator_state_update
                         (finite_trace_last s tr i) 0
                         si'));
               output := oom
             |},
           equivocator_descriptors_update IM
             (zero_descriptor IM) i deqv')
    | Some (None, deqv') =>
        Some
          (None,
           equivocator_descriptors_update IM
             (zero_descriptor IM) i deqv')
    | None => None
    end
  with
  | Some (Some item', odescriptor) =>
      Some ([item'], odescriptor)
  | Some (None, odescriptor) => Some ([], odescriptor)
  | None => None
  end by rewrite decide_True.
    +message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
s: composite_state (equivocator_IM IM)
tr: list transition_item
Hpre_tr: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) s
  tr
s0: composite_state (equivocator_IM IM)
iom, oom: option message
i: index
ji: nat
li: label (IM i)
Hpre_x: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE)
  (finite_trace_last s tr)
  [{|
     l := existT i (ContinueWith (S ji) li);
     input := iom;
     destination := s0;
     output := oom
   |}]
IHtr: pre_VLSM_projection_finite_trace_project
  (composite_type (equivocator_IM IM))
  (composite_type IM)
  equivocators_total_label_project
  equivocators_total_state_project tr =
equivocators_total_trace_project tr
Htl: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) s0 []
s_i: state (IM i)
Hv: valid li (s_i, iom)
si': state (IM i)
_om': option message
match
  pre_VLSM_projection_transition_item_project
    (composite_type (equivocator_IM IM))
    (composite_type IM)
    equivocators_total_label_project
    equivocators_total_state_project
    {|
      l := existT i (ContinueWith (S ji) li);
      input := iom;
      destination :=
        state_update (equivocator_IM IM)
          (finite_trace_last s tr) i
          (equivocator_state_update
             (finite_trace_last s tr i) (S ji) si');
      output := oom
    |}
with
| Some y => [y]
| None => []
end =
from_option fst []
  match
    match
      (if decide (S ji = 0)
       then
        Some
          (Some
             {|
               l := li;
               input := iom;
               destination :=
                 equivocator_state_zero
                   (state_update (equivocator_IM IM)
                      (finite_trace_last s tr) i
                      (equivocator_state_update
                         (finite_trace_last s tr i)
                         (S ji) si') i);
               output := oom
             |}, Existing (S ji))
       else Some (None, Existing 0))
    with
    | Some (Some item', deqv') =>
        Some
          (Some
             {|
               l := existT i (l item');
               input := iom;
               destination :=
                 equivocators_state_project IM
                   (zero_descriptor IM)
                   (state_update (equivocator_IM IM)
                      (finite_trace_last s tr) i
                      (equivocator_state_update
                         (finite_trace_last s tr i)
                         (S ji) si'));
               output := oom
             |},
           equivocator_descriptors_update IM
             (zero_descriptor IM) i deqv')
    | Some (None, deqv') =>
        Some
          (None,
           equivocator_descriptors_update IM
             (zero_descriptor IM) i deqv')
    | None => None
    end
  with
  | Some (Some item', odescriptor) =>
      Some ([item'], odescriptor)
  | Some (None, odescriptor) => Some ([], odescriptor)
  | None => None
  end by rewrite decide_False.
  -message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
s: composite_state (equivocator_IM IM)
tr: list transition_item
Hpre_tr: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) s
  tr
s0: composite_state (equivocator_IM IM)
iom, oom: option message
i: index
ji: nat
li: label (IM i)
Hpre_x: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE)
  (finite_trace_last s tr)
  [{|
     l := existT i (ForkWith ji li);
     input := iom;
     destination := s0;
     output := oom
   |}]
IHtr: pre_VLSM_projection_finite_trace_project
  (composite_type (equivocator_IM IM))
  (composite_type IM)
  equivocators_total_label_project
  equivocators_total_state_project tr =
equivocators_total_trace_project tr
Htl: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) s0 []
Hv: match
  equivocator_state_project
    (finite_trace_last s tr i) ji
with
| Some si => valid li (si, iom)
| None => False
end
Ht: (let (si', om') :=
   match
     equivocator_state_project
       (finite_trace_last s tr i) ji
   with
   | Some si =>
       let (si', om') :=
         transition li (si, iom) in
       (equivocator_state_extend
          (finite_trace_last s tr i) si', om')
   | None => (finite_trace_last s tr i, iom)
   end in
 (state_update (equivocator_IM IM)
    (finite_trace_last s tr) i si', om')) =
(s0, oom)
[] =
from_option fst []
  match
    match
      (if decide (0 = equivocator_state_last (s0 i))
       then
        Some
          (Some
             {|
               l := li;
               input := iom;
               destination :=
                 equivocator_state_zero (s0 i);
               output := oom
             |}, Existing ji)
       else Some (None, Existing 0))
    with
    | Some (Some item', deqv') =>
        Some
          (Some
             {|
               l := existT i (l item');
               input := iom;
               destination :=
                 equivocators_state_project IM
                   (zero_descriptor IM) s0;
               output := oom
             |},
           equivocator_descriptors_update IM
             (zero_descriptor IM) i deqv')
    | Some (None, deqv') =>
        Some
          (None,
           equivocator_descriptors_update IM
             (zero_descriptor IM) i deqv')
    | None => None
    end
  with
  | Some (Some item', odescriptor) =>
      Some ([item'], odescriptor)
  | Some (None, odescriptor) => Some ([], odescriptor)
  | None => None
  end destruct (equivocator_state_project _ _) as [s_i |]; [| done].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
s: composite_state (equivocator_IM IM)
tr: list transition_item
Hpre_tr: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) s
  tr
s0: composite_state (equivocator_IM IM)
iom, oom: option message
i: index
ji: nat
li: label (IM i)
Hpre_x: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE)
  (finite_trace_last s tr)
  [{|
     l := existT i (ForkWith ji li);
     input := iom;
     destination := s0;
     output := oom
   |}]
IHtr: pre_VLSM_projection_finite_trace_project
  (composite_type (equivocator_IM IM))
  (composite_type IM)
  equivocators_total_label_project
  equivocators_total_state_project tr =
equivocators_total_trace_project tr
Htl: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) s0 []
s_i: state (IM i)
Hv: valid li (s_i, iom)
Ht: (let (si', om') :=
   let (si', om') := transition li (s_i, iom) in
   (equivocator_state_extend
      (finite_trace_last s tr i) si', om') in
 (state_update (equivocator_IM IM)
    (finite_trace_last s tr) i si', om')) =
(s0, oom)
[] =
from_option fst []
  match
    match
      (if decide (0 = equivocator_state_last (s0 i))
       then
        Some
          (Some
             {|
               l := li;
               input := iom;
               destination :=
                 equivocator_state_zero (s0 i);
               output := oom
             |}, Existing ji)
       else Some (None, Existing 0))
    with
    | Some (Some item', deqv') =>
        Some
          (Some
             {|
               l := existT i (l item');
               input := iom;
               destination :=
                 equivocators_state_project IM
                   (zero_descriptor IM) s0;
               output := oom
             |},
           equivocator_descriptors_update IM
             (zero_descriptor IM) i deqv')
    | Some (None, deqv') =>
        Some
          (None,
           equivocator_descriptors_update IM
             (zero_descriptor IM) i deqv')
    | None => None
    end
  with
  | Some (Some item', odescriptor) =>
      Some ([item'], odescriptor)
  | Some (None, odescriptor) => Some ([], odescriptor)
  | None => None
  end
    destruct (transition _ _ _) as (si', _om').message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
s: composite_state (equivocator_IM IM)
tr: list transition_item
Hpre_tr: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) s
  tr
s0: composite_state (equivocator_IM IM)
iom, oom: option message
i: index
ji: nat
li: label (IM i)
Hpre_x: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE)
  (finite_trace_last s tr)
  [{|
     l := existT i (ForkWith ji li);
     input := iom;
     destination := s0;
     output := oom
   |}]
IHtr: pre_VLSM_projection_finite_trace_project
  (composite_type (equivocator_IM IM))
  (composite_type IM)
  equivocators_total_label_project
  equivocators_total_state_project tr =
equivocators_total_trace_project tr
Htl: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) s0 []
s_i: state (IM i)
Hv: valid li (s_i, iom)
si': state (IM i)
_om': option message
Ht: (state_update (equivocator_IM IM)
   (finite_trace_last s tr) i
   (equivocator_state_extend
      (finite_trace_last s tr i) si'), _om') =
(s0, oom)
[] =
from_option fst []
  match
    match
      (if decide (0 = equivocator_state_last (s0 i))
       then
        Some
          (Some
             {|
               l := li;
               input := iom;
               destination :=
                 equivocator_state_zero (s0 i);
               output := oom
             |}, Existing ji)
       else Some (None, Existing 0))
    with
    | Some (Some item', deqv') =>
        Some
          (Some
             {|
               l := existT i (l item');
               input := iom;
               destination :=
                 equivocators_state_project IM
                   (zero_descriptor IM) s0;
               output := oom
             |},
           equivocator_descriptors_update IM
             (zero_descriptor IM) i deqv')
    | Some (None, deqv') =>
        Some
          (None,
           equivocator_descriptors_update IM
             (zero_descriptor IM) i deqv')
    | None => None
    end
  with
  | Some (Some item', odescriptor) =>
      Some ([item'], odescriptor)
  | Some (None, odescriptor) => Some ([], odescriptor)
  | None => None
  end
    inversion_clear Ht.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
s: composite_state (equivocator_IM IM)
tr: list transition_item
Hpre_tr: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) s
  tr
s0: composite_state (equivocator_IM IM)
iom, oom: option message
i: index
ji: nat
li: label (IM i)
Hpre_x: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE)
  (finite_trace_last s tr)
  [{|
     l := existT i (ForkWith ji li);
     input := iom;
     destination := s0;
     output := oom
   |}]
IHtr: pre_VLSM_projection_finite_trace_project
  (composite_type (equivocator_IM IM))
  (composite_type IM)
  equivocators_total_label_project
  equivocators_total_state_project tr =
equivocators_total_trace_project tr
Htl: finite_valid_trace_from
  (preloaded_with_all_messages_vlsm FreeE) s0 []
s_i: state (IM i)
Hv: valid li (s_i, iom)
si': state (IM i)
_om': option message
[] =
from_option fst []
  match
    match
      (if
        decide
          (0 =
           equivocator_state_last
             (state_update (equivocator_IM IM)
                (finite_trace_last s tr) i
                (equivocator_state_extend
                   (finite_trace_last s tr i) si') i))
       then
        Some
          (Some
             {|
               l := li;
               input := iom;
               destination :=
                 equivocator_state_zero
                   (state_update (equivocator_IM IM)
                      (finite_trace_last s tr) i
                      (equivocator_state_extend
                         (finite_trace_last s tr i)
                         si') i);
               output := oom
             |}, Existing ji)
       else Some (None, Existing 0))
    with
    | Some (Some item', deqv') =>
        Some
          (Some
             {|
               l := existT i (l item');
               input := iom;
               destination :=
                 equivocators_state_project IM
                   (zero_descriptor IM)
                   (state_update (equivocator_IM IM)
                      (finite_trace_last s tr) i
                      (equivocator_state_extend
                         (finite_trace_last s tr i)
                         si'));
               output := oom
             |},
           equivocator_descriptors_update IM
             (zero_descriptor IM) i deqv')
    | Some (None, deqv') =>
        Some
          (None,
           equivocator_descriptors_update IM
             (zero_descriptor IM) i deqv')
    | None => None
    end
  with
  | Some (Some item', odescriptor) =>
      Some ([item'], odescriptor)
  | Some (None, odescriptor) => Some ([], odescriptor)
  | None => None
  end
    by state_update_simpl; cbn; rewrite decide_False.
Qed.

Lemma equivocators_total_trace_project_final_state
  (X := FreeE)
  (state_project := equivocators_total_state_project)
  (trace_project := equivocators_total_trace_project)
  : forall sX trX,
      finite_constrained_trace_from X sX trX ->
      state_project (finite_trace_last sX trX) =
      finite_trace_last (state_project sX) (trace_project trX).message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
X:= FreeE: VLSM message
state_project:= equivocators_total_state_project: state
  (free_composite_vlsm
     (equivocator_IM IM))
→ state (free_composite_vlsm IM)
trace_project:= equivocators_total_trace_project: list
  (composite_transition_item
     (equivocator_IM IM))
→ list (composite_transition_item IM)
∀ (sX : state (preloaded_with_all_messages_vlsm X)) 
  (trX : list transition_item),
  finite_constrained_trace_from X sX trX
  → state_project (finite_trace_last sX trX) =
    finite_trace_last (state_project sX)
      (trace_project trX)
Proof.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
X:= FreeE: VLSM message
state_project:= equivocators_total_state_project: state
  (free_composite_vlsm
     (equivocator_IM IM))
→ state (free_composite_vlsm IM)
trace_project:= equivocators_total_trace_project: list
  (composite_transition_item
     (equivocator_IM IM))
→ list (composite_transition_item IM)
∀ (sX : state (preloaded_with_all_messages_vlsm X)) 
  (trX : list transition_item),
  finite_constrained_trace_from X sX trX
  → state_project (finite_trace_last sX trX) =
    finite_trace_last (state_project sX)
      (trace_project trX)
  intros sX trX Hpre_tr.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
X:= FreeE: VLSM message
state_project:= equivocators_total_state_project: state
  (free_composite_vlsm
     (equivocator_IM IM))
→ state (free_composite_vlsm IM)
trace_project:= equivocators_total_trace_project: list
  (composite_transition_item
     (equivocator_IM IM))
→ list (composite_transition_item IM)
sX: state (preloaded_with_all_messages_vlsm X)
trX: list transition_item
Hpre_tr: finite_constrained_trace_from X sX trX
state_project (finite_trace_last sX trX) =
finite_trace_last (state_project sX)
  (trace_project trX)
  specialize (equivocators_total_trace_project_characterization Hpre_tr) as Htr_pr.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
X:= FreeE: VLSM message
state_project:= equivocators_total_state_project: state
  (free_composite_vlsm
     (equivocator_IM IM))
→ state (free_composite_vlsm IM)
trace_project:= equivocators_total_trace_project: list
  (composite_transition_item
     (equivocator_IM IM))
→ list (composite_transition_item IM)
sX: state (preloaded_with_all_messages_vlsm X)
trX: list transition_item
Hpre_tr: finite_constrained_trace_from X sX trX
Htr_pr: equivocators_trace_project
  (zero_descriptor IM) trX =
Some
  (equivocators_total_trace_project trX,
   zero_descriptor IM)
state_project (finite_trace_last sX trX) =
finite_trace_last (state_project sX)
  (trace_project trX)
  specialize
    (preloaded_equivocators_valid_trace_from_project (zero_descriptor IM) sX trX)
    as Hproject.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
X:= FreeE: VLSM message
state_project:= equivocators_total_state_project: state
  (free_composite_vlsm
     (equivocator_IM IM))
→ state (free_composite_vlsm IM)
trace_project:= equivocators_total_trace_project: list
  (composite_transition_item
     (equivocator_IM IM))
→ list (composite_transition_item IM)
sX: state (preloaded_with_all_messages_vlsm X)
trX: list transition_item
Hpre_tr: finite_constrained_trace_from X sX trX
Htr_pr: equivocators_trace_project
  (zero_descriptor IM) trX =
Some
  (equivocators_total_trace_project trX,
   zero_descriptor IM)
Hproject: let final_state := finite_trace_last sX trX
  in
proper_equivocator_descriptors IM
  (zero_descriptor IM) final_state
→ finite_constrained_trace_from FreeE sX
    trX
  → ∃ (trX0 : list
                (composite_transition_item
                   IM)) 
      (initial_descriptors : 
       equivocator_descriptors IM),
      equivocators_trace_project
        (zero_descriptor IM) trX =
      Some (trX0, initial_descriptors)
      ∧ proper_equivocator_descriptors IM
          initial_descriptors sX
        ∧ equivocators_state_project IM
            (zero_descriptor IM)
            (finite_trace_last sX trX) =
          finite_trace_last
            (equivocators_state_project IM
               initial_descriptors sX) trX0
state_project (finite_trace_last sX trX) =
finite_trace_last (state_project sX)
  (trace_project trX)
  simpl in Hproject; spec Hproject; [by apply zero_descriptor_proper |].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
X:= FreeE: VLSM message
state_project:= equivocators_total_state_project: state
  (free_composite_vlsm
     (equivocator_IM IM))
→ state (free_composite_vlsm IM)
trace_project:= equivocators_total_trace_project: list
  (composite_transition_item
     (equivocator_IM IM))
→ list (composite_transition_item IM)
sX: state (preloaded_with_all_messages_vlsm X)
trX: list transition_item
Hpre_tr: finite_constrained_trace_from X sX trX
Htr_pr: equivocators_trace_project
  (zero_descriptor IM) trX =
Some
  (equivocators_total_trace_project trX,
   zero_descriptor IM)
Hproject: finite_constrained_trace_from FreeE sX trX
→ ∃ (trX0 : list
              (composite_transition_item IM)) 
    (initial_descriptors : 
     equivocator_descriptors IM),
    equivocators_trace_project
      (zero_descriptor IM) trX =
    Some (trX0, initial_descriptors)
    ∧ proper_equivocator_descriptors IM
        initial_descriptors sX
      ∧ equivocators_state_project IM
          (zero_descriptor IM)
          (finite_trace_last sX trX) =
        finite_trace_last
          (equivocators_state_project IM
             initial_descriptors sX) trX0
state_project (finite_trace_last sX trX) =
finite_trace_last (state_project sX)
  (trace_project trX)
  specialize (Hproject Hpre_tr).message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
X:= FreeE: VLSM message
state_project:= equivocators_total_state_project: state
  (free_composite_vlsm
     (equivocator_IM IM))
→ state (free_composite_vlsm IM)
trace_project:= equivocators_total_trace_project: list
  (composite_transition_item
     (equivocator_IM IM))
→ list (composite_transition_item IM)
sX: state (preloaded_with_all_messages_vlsm X)
trX: list transition_item
Hpre_tr: finite_constrained_trace_from X sX trX
Htr_pr: equivocators_trace_project
  (zero_descriptor IM) trX =
Some
  (equivocators_total_trace_project trX,
   zero_descriptor IM)
Hproject: ∃ (trX0 : list
            (composite_transition_item IM)) 
  (initial_descriptors : 
   equivocator_descriptors IM),
  equivocators_trace_project
    (zero_descriptor IM) trX =
  Some (trX0, initial_descriptors)
  ∧ proper_equivocator_descriptors IM
      initial_descriptors sX
    ∧ equivocators_state_project IM
        (zero_descriptor IM)
        (finite_trace_last sX trX) =
      finite_trace_last
        (equivocators_state_project IM
           initial_descriptors sX) trX0
state_project (finite_trace_last sX trX) =
finite_trace_last (state_project sX)
  (trace_project trX)
  destruct Hproject as [_trX [initial_descriptors [_Htr_pr [_ Hlst]]]].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
X:= FreeE: VLSM message
state_project:= equivocators_total_state_project: state
  (free_composite_vlsm
     (equivocator_IM IM))
→ state (free_composite_vlsm IM)
trace_project:= equivocators_total_trace_project: list
  (composite_transition_item
     (equivocator_IM IM))
→ list (composite_transition_item IM)
sX: state (preloaded_with_all_messages_vlsm X)
trX: list transition_item
Hpre_tr: finite_constrained_trace_from X sX trX
Htr_pr: equivocators_trace_project
  (zero_descriptor IM) trX =
Some
  (equivocators_total_trace_project trX,
   zero_descriptor IM)
_trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
_Htr_pr: equivocators_trace_project
  (zero_descriptor IM) trX =
Some (_trX, initial_descriptors)
Hlst: equivocators_state_project IM
  (zero_descriptor IM)
  (finite_trace_last sX trX) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors sX) _trX
state_project (finite_trace_last sX trX) =
finite_trace_last (state_project sX)
  (trace_project trX)
  rewrite Htr_pr in _Htr_pr.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
X:= FreeE: VLSM message
state_project:= equivocators_total_state_project: state
  (free_composite_vlsm
     (equivocator_IM IM))
→ state (free_composite_vlsm IM)
trace_project:= equivocators_total_trace_project: list
  (composite_transition_item
     (equivocator_IM IM))
→ list (composite_transition_item IM)
sX: state (preloaded_with_all_messages_vlsm X)
trX: list transition_item
Hpre_tr: finite_constrained_trace_from X sX trX
Htr_pr: equivocators_trace_project
  (zero_descriptor IM) trX =
Some
  (equivocators_total_trace_project trX,
   zero_descriptor IM)
_trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
_Htr_pr: Some
  (equivocators_total_trace_project trX,
   zero_descriptor IM) =
Some (_trX, initial_descriptors)
Hlst: equivocators_state_project IM
  (zero_descriptor IM)
  (finite_trace_last sX trX) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors sX) _trX
state_project (finite_trace_last sX trX) =
finite_trace_last (state_project sX)
  (trace_project trX)
  by inversion _Htr_pr; subst.
Qed.

Lemma PreFreeE_PreFree_vlsm_partial_projection
  (final_descriptors : equivocator_descriptors IM)
  : VLSM_partial_projection PreFreeE PreFree (equivocators_partial_trace_project final_descriptors).message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
final_descriptors: equivocator_descriptors IM
VLSM_partial_projection PreFreeE PreFree
  (equivocators_partial_trace_project
     final_descriptors)
Proof.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
final_descriptors: equivocator_descriptors IM
VLSM_partial_projection PreFreeE PreFree
  (equivocators_partial_trace_project
     final_descriptors)
  split; [by split; apply equivocators_partial_trace_project_extends_left |].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
final_descriptors: equivocator_descriptors IM
∀ (sX : state PreFreeE) (trX : list transition_item) 
  (sY : state PreFree) (trY : list transition_item),
  equivocators_partial_trace_project final_descriptors
    (sX, trX) = Some (sY, trY)
  → finite_valid_trace PreFreeE sX trX
    → finite_valid_trace PreFree sY trY
  intros s tr sX trX Hpr_tr Htr.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
final_descriptors: equivocator_descriptors IM
s: state PreFreeE
tr: list transition_item
sX: state PreFree
trX: list transition_item
Hpr_tr: equivocators_partial_trace_project
  final_descriptors (s, tr) = 
Some (sX, trX)
Htr: finite_valid_trace PreFreeE s tr
finite_valid_trace PreFree sX trX
  destruct (destruct_equivocators_partial_trace_project Hpr_tr)
    as [Hnot_equiv [initial_descriptors [Htr_project Hs_project]]].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
final_descriptors: equivocator_descriptors IM
s: state PreFreeE
tr: list transition_item
sX: state PreFree
trX: list transition_item
Hpr_tr: equivocators_partial_trace_project
  final_descriptors (s, tr) = 
Some (sX, trX)
Htr: finite_valid_trace PreFreeE s tr
Hnot_equiv: not_equivocating_equivocator_descriptors
  IM final_descriptors
  (finite_trace_last s tr)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project
  final_descriptors tr =
Some (trX, initial_descriptors)
Hs_project: equivocators_state_project IM
  initial_descriptors s = sX
finite_valid_trace PreFree sX trX
  apply valid_trace_add_default_last in Htr.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
final_descriptors: equivocator_descriptors IM
s: state PreFreeE
tr: list transition_item
sX: state PreFree
trX: list transition_item
Hpr_tr: equivocators_partial_trace_project
  final_descriptors (s, tr) = 
Some (sX, trX)
Htr: finite_valid_trace_init_to PreFreeE s
  (finite_trace_last s tr) tr
Hnot_equiv: not_equivocating_equivocator_descriptors
  IM final_descriptors
  (finite_trace_last s tr)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project
  final_descriptors tr =
Some (trX, initial_descriptors)
Hs_project: equivocators_state_project IM
  initial_descriptors s = sX
finite_valid_trace PreFree sX trX
  apply not_equivocating_equivocator_descriptors_proper in Hnot_equiv as Hproper.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
final_descriptors: equivocator_descriptors IM
s: state PreFreeE
tr: list transition_item
sX: state PreFree
trX: list transition_item
Hpr_tr: equivocators_partial_trace_project
  final_descriptors (s, tr) = 
Some (sX, trX)
Htr: finite_valid_trace_init_to PreFreeE s
  (finite_trace_last s tr) tr
Hnot_equiv: not_equivocating_equivocator_descriptors
  IM final_descriptors
  (finite_trace_last s tr)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project
  final_descriptors tr =
Some (trX, initial_descriptors)
Hs_project: equivocators_state_project IM
  initial_descriptors s = sX
Hproper: proper_equivocator_descriptors IM
  final_descriptors (finite_trace_last s tr)
finite_valid_trace PreFree sX trX
  destruct (pre_equivocators_valid_trace_project _ _ _ Htr _ Hproper)
    as [_initial_descriptors [_ [_trX [_Htr_project HtrX]]]].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
final_descriptors: equivocator_descriptors IM
s: state PreFreeE
tr: list transition_item
sX: state PreFree
trX: list transition_item
Hpr_tr: equivocators_partial_trace_project
  final_descriptors (s, tr) = 
Some (sX, trX)
Htr: finite_valid_trace_init_to PreFreeE s
  (finite_trace_last s tr) tr
Hnot_equiv: not_equivocating_equivocator_descriptors
  IM final_descriptors
  (finite_trace_last s tr)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project
  final_descriptors tr =
Some (trX, initial_descriptors)
Hs_project: equivocators_state_project IM
  initial_descriptors s = sX
Hproper: proper_equivocator_descriptors IM
  final_descriptors (finite_trace_last s tr)
_initial_descriptors: equivocator_descriptors IM
_trX: list (composite_transition_item IM)
_Htr_project: equivocators_trace_project
  final_descriptors tr =
Some (_trX, _initial_descriptors)
HtrX: finite_constrained_trace_init_to Free
  (equivocators_state_project IM
     _initial_descriptors s)
  (equivocators_state_project IM
     final_descriptors 
     (finite_trace_last s tr)) _trX
finite_valid_trace PreFree sX trX
  rewrite Htr_project in _Htr_project.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
final_descriptors: equivocator_descriptors IM
s: state PreFreeE
tr: list transition_item
sX: state PreFree
trX: list transition_item
Hpr_tr: equivocators_partial_trace_project
  final_descriptors (s, tr) = 
Some (sX, trX)
Htr: finite_valid_trace_init_to PreFreeE s
  (finite_trace_last s tr) tr
Hnot_equiv: not_equivocating_equivocator_descriptors
  IM final_descriptors
  (finite_trace_last s tr)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project
  final_descriptors tr =
Some (trX, initial_descriptors)
Hs_project: equivocators_state_project IM
  initial_descriptors s = sX
Hproper: proper_equivocator_descriptors IM
  final_descriptors (finite_trace_last s tr)
_initial_descriptors: equivocator_descriptors IM
_trX: list (composite_transition_item IM)
_Htr_project: Some (trX, initial_descriptors) =
Some (_trX, _initial_descriptors)
HtrX: finite_constrained_trace_init_to Free
  (equivocators_state_project IM
     _initial_descriptors s)
  (equivocators_state_project IM
     final_descriptors 
     (finite_trace_last s tr)) _trX
finite_valid_trace PreFree sX trX
  inversion _Htr_project; subst.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
final_descriptors: equivocator_descriptors IM
s: state PreFreeE
tr: list transition_item
_trX: list (composite_transition_item IM)
_initial_descriptors: equivocator_descriptors IM
Hpr_tr: equivocators_partial_trace_project
  final_descriptors (s, tr) =
Some
  (equivocators_state_project IM
     _initial_descriptors s, _trX)
_Htr_project: Some (_trX, _initial_descriptors) =
Some (_trX, _initial_descriptors)
Htr: finite_valid_trace_init_to PreFreeE s
  (finite_trace_last s tr) tr
Hnot_equiv: not_equivocating_equivocator_descriptors
  IM final_descriptors
  (finite_trace_last s tr)
Htr_project: equivocators_trace_project
  final_descriptors tr =
Some (_trX, _initial_descriptors)
Hproper: proper_equivocator_descriptors IM
  final_descriptors (finite_trace_last s tr)
HtrX: finite_constrained_trace_init_to Free
  (equivocators_state_project IM
     _initial_descriptors s)
  (equivocators_state_project IM
     final_descriptors 
     (finite_trace_last s tr)) _trX
finite_valid_trace PreFree
  (equivocators_state_project IM _initial_descriptors
     s) _trX
  by red in HtrX; apply valid_trace_forget_last in HtrX.
Qed.

End sec_equivocators_composition_projections.

Section sec_equivocators_composition_sub_projections.

Context
  {message : Type}
  `{finite.Finite index}
  (IM : index -> VLSM message)
  `{forall i : index, HasBeenSentCapability (IM i)}
  (selection : list index)
  .

A generalization of equivocators_trace_project_finite_trace_projection_list_commute to projections over a set of indices.

We can project a trace over the composition of equivocators in two ways:

first project to a subset of equivocator components, then project that to the corresponding subset of the composition of the original components
first project to the composition of original components, then project to a subset of them

The results below (fist for a single item, then for the full trace say that the two ways lead to the same result.

Lemma equivocators_trace_project_finite_trace_sub_projection_item_commute
  (item : composite_transition_item (equivocator_IM IM))
  (final_descriptors' final_descriptors : equivocator_descriptors IM)
  (final_sub_descriptors := fun i : sub_index selection => final_descriptors (` i))
  (pr_item : list (composite_transition_item IM))
  (Hpr_item : equivocators_trace_project IM final_descriptors [item] =
                Some (pr_item, final_descriptors'))
  (pr_sub_item : list (composite_transition_item (sub_IM IM selection)))
  (final_sub_descriptors' : equivocator_descriptors (sub_IM IM selection))
  (Hpr_sub_item :
    equivocators_trace_project (sub_IM IM selection) final_sub_descriptors
      (finite_trace_sub_projection (equivocator_IM IM) selection [item]) =
    Some (pr_sub_item, final_sub_descriptors'))
  : final_sub_descriptors' = (fun i : sub_index selection => final_descriptors' (` i))
  /\ finite_trace_sub_projection IM selection pr_item = pr_sub_item.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
item: composite_transition_item (equivocator_IM IM)
final_descriptors', final_descriptors: equivocator_descriptors IM
final_sub_descriptors:= λ i : sub_index selection,
  final_descriptors (`i): ∀ i : sub_index selection,
  MachineDescriptor (IM (`i))
pr_item: list (composite_transition_item IM)
Hpr_item: equivocators_trace_project IM
  final_descriptors [item] =
Some (pr_item, final_descriptors')
pr_sub_item: list
  (composite_transition_item
     (sub_IM IM selection))
final_sub_descriptors': equivocator_descriptors
  (sub_IM IM selection)
Hpr_sub_item: equivocators_trace_project
  (sub_IM IM selection)
  final_sub_descriptors
  (finite_trace_sub_projection
     (equivocator_IM IM) selection
     [item]) =
Some
  (pr_sub_item, final_sub_descriptors')
final_sub_descriptors' =
(λ i : sub_index selection, final_descriptors' (`i))
∧ finite_trace_sub_projection IM selection pr_item =
  pr_sub_item
Proof.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
item: composite_transition_item (equivocator_IM IM)
final_descriptors', final_descriptors: equivocator_descriptors IM
final_sub_descriptors:= λ i : sub_index selection,
  final_descriptors (`i): ∀ i : sub_index selection,
  MachineDescriptor (IM (`i))
pr_item: list (composite_transition_item IM)
Hpr_item: equivocators_trace_project IM
  final_descriptors [item] =
Some (pr_item, final_descriptors')
pr_sub_item: list
  (composite_transition_item
     (sub_IM IM selection))
final_sub_descriptors': equivocator_descriptors
  (sub_IM IM selection)
Hpr_sub_item: equivocators_trace_project
  (sub_IM IM selection)
  final_sub_descriptors
  (finite_trace_sub_projection
     (equivocator_IM IM) selection
     [item]) =
Some
  (pr_sub_item, final_sub_descriptors')
final_sub_descriptors' =
(λ i : sub_index selection, final_descriptors' (`i))
∧ finite_trace_sub_projection IM selection pr_item =
  pr_sub_item
  unfold equivocators_trace_project in Hpr_item.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
item: composite_transition_item (equivocator_IM IM)
final_descriptors', final_descriptors: equivocator_descriptors IM
final_sub_descriptors:= λ i : sub_index selection,
  final_descriptors (`i): ∀ i : sub_index selection,
  MachineDescriptor (IM (`i))
pr_item: list (composite_transition_item IM)
Hpr_item: foldr
  (equivocators_trace_project_folder IM)
  (Some ([], final_descriptors)) [item] =
Some (pr_item, final_descriptors')
pr_sub_item: list
  (composite_transition_item
     (sub_IM IM selection))
final_sub_descriptors': equivocator_descriptors
  (sub_IM IM selection)
Hpr_sub_item: equivocators_trace_project
  (sub_IM IM selection)
  final_sub_descriptors
  (finite_trace_sub_projection
     (equivocator_IM IM) selection
     [item]) =
Some
  (pr_sub_item, final_sub_descriptors')
final_sub_descriptors' =
(λ i : sub_index selection, final_descriptors' (`i))
∧ finite_trace_sub_projection IM selection pr_item =
  pr_sub_item unfold sub_IM in *.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
item: composite_transition_item (equivocator_IM IM)
final_descriptors', final_descriptors: equivocator_descriptors IM
final_sub_descriptors:= λ i : sub_index selection,
  final_descriptors (`i): ∀ i : sub_index selection,
  MachineDescriptor (IM (`i))
pr_item: list (composite_transition_item IM)
Hpr_item: foldr
  (equivocators_trace_project_folder IM)
  (Some ([], final_descriptors)) [item] =
Some (pr_item, final_descriptors')
pr_sub_item: list
  (composite_transition_item
     (λ ei : sub_index selection,
        IM (`ei)))
final_sub_descriptors': equivocator_descriptors
  (λ ei : sub_index selection,
     IM (`ei))
Hpr_sub_item: equivocators_trace_project
  (λ ei : sub_index selection, IM (`ei))
  final_sub_descriptors
  (finite_trace_sub_projection
     (equivocator_IM IM) selection
     [item]) =
Some
  (pr_sub_item, final_sub_descriptors')
final_sub_descriptors' =
(λ i : sub_index selection, final_descriptors' (`i))
∧ finite_trace_sub_projection IM selection pr_item =
  pr_sub_item
  simpl in *.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
item: composite_transition_item (equivocator_IM IM)
final_descriptors', final_descriptors: equivocator_descriptors IM
final_sub_descriptors:= λ i : sub_index selection,
  final_descriptors (`i): ∀ i : sub_index selection,
  MachineDescriptor (IM (`i))
pr_item: list (composite_transition_item IM)
Hpr_item: match
  equivocators_transition_item_project IM
    final_descriptors item
with
| Some (Some item', odescriptor) =>
    Some ([item'], odescriptor)
| Some (None, odescriptor) =>
    Some ([], odescriptor)
| None => None
end = Some (pr_item, final_descriptors')
pr_sub_item: list
  (composite_transition_item
     (λ ei : sub_index selection,
        IM (`ei)))
final_sub_descriptors': equivocator_descriptors
  (λ ei : sub_index selection,
     IM (`ei))
Hpr_sub_item: equivocators_trace_project
  (λ ei : sub_index selection, IM (`ei))
  final_sub_descriptors
  match
    pre_VLSM_projection_transition_item_project
      (composite_type
         (equivocator_IM IM))
      (composite_type
         (sub_IM (equivocator_IM IM)
            selection))
      (composite_label_sub_projection_option
         (equivocator_IM IM) selection)
      (composite_state_sub_projection
         (equivocator_IM IM) selection)
      item
  with
  | Some y => [y]
  | None => []
  end =
Some
  (pr_sub_item, final_sub_descriptors')
final_sub_descriptors' =
(λ i : sub_index selection, final_descriptors' (`i))
∧ finite_trace_sub_projection IM selection pr_item =
  pr_sub_item
  destruct (equivocators_transition_item_project IM final_descriptors item)
    as [(ox, final') |] eqn: Hpr_item_x
  ; [| by congruence].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
item: composite_transition_item (equivocator_IM IM)
final_descriptors', final_descriptors: equivocator_descriptors IM
final_sub_descriptors:= λ i : sub_index selection,
  final_descriptors (`i): ∀ i : sub_index selection,
  MachineDescriptor (IM (`i))
pr_item: list (composite_transition_item IM)
ox: option (composite_transition_item IM)
final': equivocator_descriptors IM
Hpr_item_x: equivocators_transition_item_project IM
  final_descriptors item =
Some (ox, final')
Hpr_item: match ox with
| Some item' => Some ([item'], final')
| None => Some ([], final')
end = Some (pr_item, final_descriptors')
pr_sub_item: list
  (composite_transition_item
     (λ ei : sub_index selection,
        IM (`ei)))
final_sub_descriptors': equivocator_descriptors
  (λ ei : sub_index selection,
     IM (`ei))
Hpr_sub_item: equivocators_trace_project
  (λ ei : sub_index selection, IM (`ei))
  final_sub_descriptors
  match
    pre_VLSM_projection_transition_item_project
      (composite_type
         (equivocator_IM IM))
      (composite_type
         (sub_IM (equivocator_IM IM)
            selection))
      (composite_label_sub_projection_option
         (equivocator_IM IM) selection)
      (composite_state_sub_projection
         (equivocator_IM IM) selection)
      item
  with
  | Some y => [y]
  | None => []
  end =
Some
  (pr_sub_item, final_sub_descriptors')
final_sub_descriptors' =
(λ i : sub_index selection, final_descriptors' (`i))
∧ finite_trace_sub_projection IM selection pr_item =
  pr_sub_item
  unfold equivocators_transition_item_project in Hpr_item_x.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
item: composite_transition_item (equivocator_IM IM)
final_descriptors', final_descriptors: equivocator_descriptors IM
final_sub_descriptors:= λ i : sub_index selection,
  final_descriptors (`i): ∀ i : sub_index selection,
  MachineDescriptor (IM (`i))
pr_item: list (composite_transition_item IM)
ox: option (composite_transition_item IM)
final': equivocator_descriptors IM
Hpr_item_x: match
  equivocator_vlsm_transition_item_project
    (IM (projT1 (l item)))
    (composite_transition_item_projection
       (equivocator_IM IM) item)
    (final_descriptors (projT1 (l item)))
with
| Some (Some item', deqv') =>
    Some
      (Some
         {|
           l :=
             existT (projT1 (l item))
               (l item');
           input := input item;
           destination :=
             equivocators_state_project
               IM final_descriptors
               (destination item);
           output := output item
         |},
       equivocator_descriptors_update IM
         final_descriptors
         (projT1 (l item)) deqv')
| Some (None, deqv') =>
    Some
      (None,
       equivocator_descriptors_update IM
         final_descriptors
         (projT1 (l item)) deqv')
| None => None
end = Some (ox, final')
Hpr_item: match ox with
| Some item' => Some ([item'], final')
| None => Some ([], final')
end = Some (pr_item, final_descriptors')
pr_sub_item: list
  (composite_transition_item
     (λ ei : sub_index selection,
        IM (`ei)))
final_sub_descriptors': equivocator_descriptors
  (λ ei : sub_index selection,
     IM (`ei))
Hpr_sub_item: equivocators_trace_project
  (λ ei : sub_index selection, IM (`ei))
  final_sub_descriptors
  match
    pre_VLSM_projection_transition_item_project
      (composite_type
         (equivocator_IM IM))
      (composite_type
         (sub_IM (equivocator_IM IM)
            selection))
      (composite_label_sub_projection_option
         (equivocator_IM IM) selection)
      (composite_state_sub_projection
         (equivocator_IM IM) selection)
      item
  with
  | Some y => [y]
  | None => []
  end =
Some
  (pr_sub_item, final_sub_descriptors')
final_sub_descriptors' =
(λ i : sub_index selection, final_descriptors' (`i))
∧ finite_trace_sub_projection IM selection pr_item =
  pr_sub_item
  unfold composite_transition_item_projection in Hpr_item_x.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
item: composite_transition_item (equivocator_IM IM)
final_descriptors', final_descriptors: equivocator_descriptors IM
final_sub_descriptors:= λ i : sub_index selection,
  final_descriptors (`i): ∀ i : sub_index selection,
  MachineDescriptor (IM (`i))
pr_item: list (composite_transition_item IM)
ox: option (composite_transition_item IM)
final': equivocator_descriptors IM
Hpr_item_x: match
  equivocator_vlsm_transition_item_project
    (IM (projT1 (l item)))
    (composite_transition_item_projection_from_eq
       (equivocator_IM IM) item
       (projT1 (l item)) eq_refl)
    (final_descriptors (projT1 (l item)))
with
| Some (Some item', deqv') =>
    Some
      (Some
         {|
           l :=
             existT (projT1 (l item))
               (l item');
           input := input item;
           destination :=
             equivocators_state_project
               IM final_descriptors
               (destination item);
           output := output item
         |},
       equivocator_descriptors_update IM
         final_descriptors
         (projT1 (l item)) deqv')
| Some (None, deqv') =>
    Some
      (None,
       equivocator_descriptors_update IM
         final_descriptors
         (projT1 (l item)) deqv')
| None => None
end = Some (ox, final')
Hpr_item: match ox with
| Some item' => Some ([item'], final')
| None => Some ([], final')
end = Some (pr_item, final_descriptors')
pr_sub_item: list
  (composite_transition_item
     (λ ei : sub_index selection,
        IM (`ei)))
final_sub_descriptors': equivocator_descriptors
  (λ ei : sub_index selection,
     IM (`ei))
Hpr_sub_item: equivocators_trace_project
  (λ ei : sub_index selection, IM (`ei))
  final_sub_descriptors
  match
    pre_VLSM_projection_transition_item_project
      (composite_type
         (equivocator_IM IM))
      (composite_type
         (sub_IM (equivocator_IM IM)
            selection))
      (composite_label_sub_projection_option
         (equivocator_IM IM) selection)
      (composite_state_sub_projection
         (equivocator_IM IM) selection)
      item
  with
  | Some y => [y]
  | None => []
  end =
Some
  (pr_sub_item, final_sub_descriptors')
final_sub_descriptors' =
(λ i : sub_index selection, final_descriptors' (`i))
∧ finite_trace_sub_projection IM selection pr_item =
  pr_sub_item
  remember (equivocator_vlsm_transition_item_project (IM (projT1 (l item)))
    (composite_transition_item_projection_from_eq (equivocator_IM IM) item
      (projT1 (l item)) eq_refl) (final_descriptors (projT1 (l item))))
    as pr_item_x.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
item: composite_transition_item (equivocator_IM IM)
final_descriptors', final_descriptors: equivocator_descriptors IM
final_sub_descriptors:= λ i : sub_index selection,
  final_descriptors (`i): ∀ i : sub_index selection,
  MachineDescriptor (IM (`i))
pr_item: list (composite_transition_item IM)
ox: option (composite_transition_item IM)
final': equivocator_descriptors IM
pr_item_x: option
  (option transition_item *
   MachineDescriptor
     (IM (projT1 (l item))))
Heqpr_item_x: pr_item_x =
equivocator_vlsm_transition_item_project
  (IM (projT1 (l item)))
  (composite_transition_item_projection_from_eq
     (equivocator_IM IM) item
     (projT1 (l item)) eq_refl)
  (final_descriptors (projT1 (l item)))
Hpr_item_x: match pr_item_x with
| Some (Some item', deqv') =>
    Some
      (Some
         {|
           l :=
             existT (projT1 (l item))
               (l item');
           input := input item;
           destination :=
             equivocators_state_project
               IM final_descriptors
               (destination item);
           output := output item
         |},
       equivocator_descriptors_update IM
         final_descriptors
         (projT1 (l item)) deqv')
| Some (None, deqv') =>
    Some
      (None,
       equivocator_descriptors_update IM
         final_descriptors
         (projT1 (l item)) deqv')
| None => None
end = Some (ox, final')
Hpr_item: match ox with
| Some item' => Some ([item'], final')
| None => Some ([], final')
end = Some (pr_item, final_descriptors')
pr_sub_item: list
  (composite_transition_item
     (λ ei : sub_index selection,
        IM (`ei)))
final_sub_descriptors': equivocator_descriptors
  (λ ei : sub_index selection,
     IM (`ei))
Hpr_sub_item: equivocators_trace_project
  (λ ei : sub_index selection, IM (`ei))
  final_sub_descriptors
  match
    pre_VLSM_projection_transition_item_project
      (composite_type
         (equivocator_IM IM))
      (composite_type
         (sub_IM (equivocator_IM IM)
            selection))
      (composite_label_sub_projection_option
         (equivocator_IM IM) selection)
      (composite_state_sub_projection
         (equivocator_IM IM) selection)
      item
  with
  | Some y => [y]
  | None => []
  end =
Some
  (pr_sub_item, final_sub_descriptors')
final_sub_descriptors' =
(λ i : sub_index selection, final_descriptors' (`i))
∧ finite_trace_sub_projection IM selection pr_item =
  pr_sub_item
  destruct pr_item_x as [(oitem', descriptor') |]; [| by congruence].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
item: composite_transition_item (equivocator_IM IM)
final_descriptors', final_descriptors: equivocator_descriptors IM
final_sub_descriptors:= λ i : sub_index selection,
  final_descriptors (`i): ∀ i : sub_index selection,
  MachineDescriptor (IM (`i))
pr_item: list (composite_transition_item IM)
ox: option (composite_transition_item IM)
final': equivocator_descriptors IM
oitem': option transition_item
descriptor': MachineDescriptor (IM (projT1 (l item)))
Heqpr_item_x: Some (oitem', descriptor') =
equivocator_vlsm_transition_item_project
  (IM (projT1 (l item)))
  (composite_transition_item_projection_from_eq
     (equivocator_IM IM) item
     (projT1 (l item)) eq_refl)
  (final_descriptors (projT1 (l item)))
Hpr_item_x: match oitem' with
| Some item' =>
    Some
      (Some
         {|
           l :=
             existT (projT1 (l item))
               (l item');
           input := input item;
           destination :=
             equivocators_state_project
               IM final_descriptors
               (destination item);
           output := output item
         |},
       equivocator_descriptors_update IM
         final_descriptors
         (projT1 (l item)) descriptor')
| None =>
    Some
      (None,
       equivocator_descriptors_update IM
         final_descriptors
         (projT1 (l item)) descriptor')
end = Some (ox, final')
Hpr_item: match ox with
| Some item' => Some ([item'], final')
| None => Some ([], final')
end = Some (pr_item, final_descriptors')
pr_sub_item: list
  (composite_transition_item
     (λ ei : sub_index selection,
        IM (`ei)))
final_sub_descriptors': equivocator_descriptors
  (λ ei : sub_index selection,
     IM (`ei))
Hpr_sub_item: equivocators_trace_project
  (λ ei : sub_index selection, IM (`ei))
  final_sub_descriptors
  match
    pre_VLSM_projection_transition_item_project
      (composite_type
         (equivocator_IM IM))
      (composite_type
         (sub_IM (equivocator_IM IM)
            selection))
      (composite_label_sub_projection_option
         (equivocator_IM IM) selection)
      (composite_state_sub_projection
         (equivocator_IM IM) selection)
      item
  with
  | Some y => [y]
  | None => []
  end =
Some
  (pr_sub_item, final_sub_descriptors')
final_sub_descriptors' =
(λ i : sub_index selection, final_descriptors' (`i))
∧ finite_trace_sub_projection IM selection pr_item =
  pr_sub_item

  unfold composite_transition_item_projection_from_eq in Heqpr_item_x.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
item: composite_transition_item (equivocator_IM IM)
final_descriptors', final_descriptors: equivocator_descriptors IM
final_sub_descriptors:= λ i : sub_index selection,
  final_descriptors (`i): ∀ i : sub_index selection,
  MachineDescriptor (IM (`i))
pr_item: list (composite_transition_item IM)
ox: option (composite_transition_item IM)
final': equivocator_descriptors IM
oitem': option transition_item
descriptor': MachineDescriptor (IM (projT1 (l item)))
Heqpr_item_x: Some (oitem', descriptor') =
equivocator_vlsm_transition_item_project
  (IM (projT1 (l item)))
  {|
    l :=
      eq_rect_r
        (λ n : index,
           label (equivocator_IM IM n))
        (projT2 (l item)) eq_refl;
    input := input item;
    destination :=
      destination item
        (projT1 (l item));
    output := output item
  |}
  (final_descriptors (projT1 (l item)))
Hpr_item_x: match oitem' with
| Some item' =>
    Some
      (Some
         {|
           l :=
             existT (projT1 (l item))
               (l item');
           input := input item;
           destination :=
             equivocators_state_project
               IM final_descriptors
               (destination item);
           output := output item
         |},
       equivocator_descriptors_update IM
         final_descriptors
         (projT1 (l item)) descriptor')
| None =>
    Some
      (None,
       equivocator_descriptors_update IM
         final_descriptors
         (projT1 (l item)) descriptor')
end = Some (ox, final')
Hpr_item: match ox with
| Some item' => Some ([item'], final')
| None => Some ([], final')
end = Some (pr_item, final_descriptors')
pr_sub_item: list
  (composite_transition_item
     (λ ei : sub_index selection,
        IM (`ei)))
final_sub_descriptors': equivocator_descriptors
  (λ ei : sub_index selection,
     IM (`ei))
Hpr_sub_item: equivocators_trace_project
  (λ ei : sub_index selection, IM (`ei))
  final_sub_descriptors
  match
    pre_VLSM_projection_transition_item_project
      (composite_type
         (equivocator_IM IM))
      (composite_type
         (sub_IM (equivocator_IM IM)
            selection))
      (composite_label_sub_projection_option
         (equivocator_IM IM) selection)
      (composite_state_sub_projection
         (equivocator_IM IM) selection)
      item
  with
  | Some y => [y]
  | None => []
  end =
Some
  (pr_sub_item, final_sub_descriptors')
final_sub_descriptors' =
(λ i : sub_index selection, final_descriptors' (`i))
∧ finite_trace_sub_projection IM selection pr_item =
  pr_sub_item
  unfold eq_rect_r in Heqpr_item_x.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
item: composite_transition_item (equivocator_IM IM)
final_descriptors', final_descriptors: equivocator_descriptors IM
final_sub_descriptors:= λ i : sub_index selection,
  final_descriptors (`i): ∀ i : sub_index selection,
  MachineDescriptor (IM (`i))
pr_item: list (composite_transition_item IM)
ox: option (composite_transition_item IM)
final': equivocator_descriptors IM
oitem': option transition_item
descriptor': MachineDescriptor (IM (projT1 (l item)))
Heqpr_item_x: Some (oitem', descriptor') =
equivocator_vlsm_transition_item_project
  (IM (projT1 (l item)))
  {|
    l :=
      eq_rect (projT1 (l item))
        (λ y : index,
           label (equivocator_IM IM y))
        (projT2 (l item))
        (projT1 (l item))
        (eq_sym eq_refl);
    input := input item;
    destination :=
      destination item
        (projT1 (l item));
    output := output item
  |}
  (final_descriptors (projT1 (l item)))
Hpr_item_x: match oitem' with
| Some item' =>
    Some
      (Some
         {|
           l :=
             existT (projT1 (l item))
               (l item');
           input := input item;
           destination :=
             equivocators_state_project
               IM final_descriptors
               (destination item);
           output := output item
         |},
       equivocator_descriptors_update IM
         final_descriptors
         (projT1 (l item)) descriptor')
| None =>
    Some
      (None,
       equivocator_descriptors_update IM
         final_descriptors
         (projT1 (l item)) descriptor')
end = Some (ox, final')
Hpr_item: match ox with
| Some item' => Some ([item'], final')
| None => Some ([], final')
end = Some (pr_item, final_descriptors')
pr_sub_item: list
  (composite_transition_item
     (λ ei : sub_index selection,
        IM (`ei)))
final_sub_descriptors': equivocator_descriptors
  (λ ei : sub_index selection,
     IM (`ei))
Hpr_sub_item: equivocators_trace_project
  (λ ei : sub_index selection, IM (`ei))
  final_sub_descriptors
  match
    pre_VLSM_projection_transition_item_project
      (composite_type
         (equivocator_IM IM))
      (composite_type
         (sub_IM (equivocator_IM IM)
            selection))
      (composite_label_sub_projection_option
         (equivocator_IM IM) selection)
      (composite_state_sub_projection
         (equivocator_IM IM) selection)
      item
  with
  | Some y => [y]
  | None => []
  end =
Some
  (pr_sub_item, final_sub_descriptors')
final_sub_descriptors' =
(λ i : sub_index selection, final_descriptors' (`i))
∧ finite_trace_sub_projection IM selection pr_item =
  pr_sub_item
  simpl in Heqpr_item_x.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
item: composite_transition_item (equivocator_IM IM)
final_descriptors', final_descriptors: equivocator_descriptors IM
final_sub_descriptors:= λ i : sub_index selection,
  final_descriptors (`i): ∀ i : sub_index selection,
  MachineDescriptor (IM (`i))
pr_item: list (composite_transition_item IM)
ox: option (composite_transition_item IM)
final': equivocator_descriptors IM
oitem': option transition_item
descriptor': MachineDescriptor (IM (projT1 (l item)))
Heqpr_item_x: Some (oitem', descriptor') =
equivocator_vlsm_transition_item_project
  (IM (projT1 (l item)))
  {|
    l := projT2 (l item);
    input := input item;
    destination :=
      destination item
        (projT1 (l item));
    output := output item
  |}
  (final_descriptors (projT1 (l item)))
Hpr_item_x: match oitem' with
| Some item' =>
    Some
      (Some
         {|
           l :=
             existT (projT1 (l item))
               (l item');
           input := input item;
           destination :=
             equivocators_state_project
               IM final_descriptors
               (destination item);
           output := output item
         |},
       equivocator_descriptors_update IM
         final_descriptors
         (projT1 (l item)) descriptor')
| None =>
    Some
      (None,
       equivocator_descriptors_update IM
         final_descriptors
         (projT1 (l item)) descriptor')
end = Some (ox, final')
Hpr_item: match ox with
| Some item' => Some ([item'], final')
| None => Some ([], final')
end = Some (pr_item, final_descriptors')
pr_sub_item: list
  (composite_transition_item
     (λ ei : sub_index selection,
        IM (`ei)))
final_sub_descriptors': equivocator_descriptors
  (λ ei : sub_index selection,
     IM (`ei))
Hpr_sub_item: equivocators_trace_project
  (λ ei : sub_index selection, IM (`ei))
  final_sub_descriptors
  match
    pre_VLSM_projection_transition_item_project
      (composite_type
         (equivocator_IM IM))
      (composite_type
         (sub_IM (equivocator_IM IM)
            selection))
      (composite_label_sub_projection_option
         (equivocator_IM IM) selection)
      (composite_state_sub_projection
         (equivocator_IM IM) selection)
      item
  with
  | Some y => [y]
  | None => []
  end =
Some
  (pr_sub_item, final_sub_descriptors')
final_sub_descriptors' =
(λ i : sub_index selection, final_descriptors' (`i))
∧ finite_trace_sub_projection IM selection pr_item =
  pr_sub_item
  unfold pre_VLSM_projection_transition_item_project
    , composite_label_sub_projection_option in Hpr_sub_item.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
item: composite_transition_item (equivocator_IM IM)
final_descriptors', final_descriptors: equivocator_descriptors IM
final_sub_descriptors:= λ i : sub_index selection,
  final_descriptors (`i): ∀ i : sub_index selection,
  MachineDescriptor (IM (`i))
pr_item: list (composite_transition_item IM)
ox: option (composite_transition_item IM)
final': equivocator_descriptors IM
oitem': option transition_item
descriptor': MachineDescriptor (IM (projT1 (l item)))
Heqpr_item_x: Some (oitem', descriptor') =
equivocator_vlsm_transition_item_project
  (IM (projT1 (l item)))
  {|
    l := projT2 (l item);
    input := input item;
    destination :=
      destination item
        (projT1 (l item));
    output := output item
  |}
  (final_descriptors (projT1 (l item)))
Hpr_item_x: match oitem' with
| Some item' =>
    Some
      (Some
         {|
           l :=
             existT (projT1 (l item))
               (l item');
           input := input item;
           destination :=
             equivocators_state_project
               IM final_descriptors
               (destination item);
           output := output item
         |},
       equivocator_descriptors_update IM
         final_descriptors
         (projT1 (l item)) descriptor')
| None =>
    Some
      (None,
       equivocator_descriptors_update IM
         final_descriptors
         (projT1 (l item)) descriptor')
end = Some (ox, final')
Hpr_item: match ox with
| Some item' => Some ([item'], final')
| None => Some ([], final')
end = Some (pr_item, final_descriptors')
pr_sub_item: list
  (composite_transition_item
     (λ ei : sub_index selection,
        IM (`ei)))
final_sub_descriptors': equivocator_descriptors
  (λ ei : sub_index selection,
     IM (`ei))
Hpr_sub_item: equivocators_trace_project
  (λ ei : sub_index selection, IM (`ei))
  final_sub_descriptors
  match
    match
      match
        decide
          (projT1 (l item) ∈ selection)
      with
      | left i_in =>
          Some
            (composite_label_sub_projection
               (equivocator_IM IM)
               selection 
               (l item) i_in)
      | right _ => None
      end
    with
    | Some lY =>
        Some
          {|
            l := lY;
            input := input item;
            destination :=
              composite_state_sub_projection
                (equivocator_IM IM)
                selection
                (destination item);
            output := output item
          |}
    | None => None
    end
  with
  | Some y => [y]
  | None => []
  end =
Some
  (pr_sub_item, final_sub_descriptors')
final_sub_descriptors' =
(λ i : sub_index selection, final_descriptors' (`i))
∧ finite_trace_sub_projection IM selection pr_item =
  pr_sub_item
  case_decide as Hl.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
item: composite_transition_item (equivocator_IM IM)
final_descriptors', final_descriptors: equivocator_descriptors IM
final_sub_descriptors:= λ i : sub_index selection,
  final_descriptors (`i): ∀ i : sub_index selection,
  MachineDescriptor (IM (`i))
pr_item: list (composite_transition_item IM)
ox: option (composite_transition_item IM)
final': equivocator_descriptors IM
oitem': option transition_item
descriptor': MachineDescriptor (IM (projT1 (l item)))
Heqpr_item_x: Some (oitem', descriptor') =
equivocator_vlsm_transition_item_project
  (IM (projT1 (l item)))
  {|
    l := projT2 (l item);
    input := input item;
    destination :=
      destination item
        (projT1 (l item));
    output := output item
  |}
  (final_descriptors (projT1 (l item)))
Hpr_item_x: match oitem' with
| Some item' =>
    Some
      (Some
         {|
           l :=
             existT (projT1 (l item))
               (l item');
           input := input item;
           destination :=
             equivocators_state_project
               IM final_descriptors
               (destination item);
           output := output item
         |},
       equivocator_descriptors_update IM
         final_descriptors
         (projT1 (l item)) descriptor')
| None =>
    Some
      (None,
       equivocator_descriptors_update IM
         final_descriptors
         (projT1 (l item)) descriptor')
end = Some (ox, final')
Hpr_item: match ox with
| Some item' => Some ([item'], final')
| None => Some ([], final')
end = Some (pr_item, final_descriptors')
pr_sub_item: list
  (composite_transition_item
     (λ ei : sub_index selection,
        IM (`ei)))
final_sub_descriptors': equivocator_descriptors
  (λ ei : sub_index selection,
     IM (`ei))
Hl: projT1 (l item) ∈ selection
Hpr_sub_item: equivocators_trace_project
  (λ ei : sub_index selection, IM (`ei))
  final_sub_descriptors
  [{|
     l :=
       composite_label_sub_projection
         (equivocator_IM IM) selection
         (l item) Hl;
     input := input item;
     destination :=
       composite_state_sub_projection
         (equivocator_IM IM) selection
         (destination item);
     output := output item
   |}] =
Some
  (pr_sub_item, final_sub_descriptors')
final_sub_descriptors' =
(λ i : sub_index selection, final_descriptors' (`i))
∧ finite_trace_sub_projection IM selection pr_item =
  pr_sub_item
message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
item: composite_transition_item (equivocator_IM IM)
final_descriptors', final_descriptors: equivocator_descriptors IM
final_sub_descriptors:= λ i : sub_index selection,
  final_descriptors (`i): ∀ i : sub_index selection,
  MachineDescriptor (IM (`i))
pr_item: list (composite_transition_item IM)
ox: option (composite_transition_item IM)
final': equivocator_descriptors IM
oitem': option transition_item
descriptor': MachineDescriptor (IM (projT1 (l item)))
Heqpr_item_x: Some (oitem', descriptor') =
equivocator_vlsm_transition_item_project
  (IM (projT1 (l item)))
  {|
    l := projT2 (l item);
    input := input item;
    destination :=
      destination item
        (projT1 (l item));
    output := output item
  |}
  (final_descriptors (projT1 (l item)))
Hpr_item_x: match oitem' with
| Some item' =>
    Some
      (Some
         {|
           l :=
             existT (projT1 (l item))
               (l item');
           input := input item;
           destination :=
             equivocators_state_project
               IM final_descriptors
               (destination item);
           output := output item
         |},
       equivocator_descriptors_update IM
         final_descriptors
         (projT1 (l item)) descriptor')
| None =>
    Some
      (None,
       equivocator_descriptors_update IM
         final_descriptors
         (projT1 (l item)) descriptor')
end = Some (ox, final')
Hpr_item: match ox with
| Some item' => Some ([item'], final')
| None => Some ([], final')
end = Some (pr_item, final_descriptors')
pr_sub_item: list
  (composite_transition_item
     (λ ei : sub_index selection,
        IM (`ei)))
final_sub_descriptors': equivocator_descriptors
  (λ ei : sub_index selection,
     IM (`ei))
Hl: projT1 (l item) ∉ selection
Hpr_sub_item: equivocators_trace_project
  (λ ei : sub_index selection, IM (`ei))
  final_sub_descriptors [] =
Some
  (pr_sub_item, final_sub_descriptors')
final_sub_descriptors' =
(λ i : sub_index selection, final_descriptors' (`i))
∧ finite_trace_sub_projection IM selection pr_item =
  pr_sub_item
  -message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
item: composite_transition_item (equivocator_IM IM)
final_descriptors', final_descriptors: equivocator_descriptors IM
final_sub_descriptors:= λ i : sub_index selection,
  final_descriptors (`i): ∀ i : sub_index selection,
  MachineDescriptor (IM (`i))
pr_item: list (composite_transition_item IM)
ox: option (composite_transition_item IM)
final': equivocator_descriptors IM
oitem': option transition_item
descriptor': MachineDescriptor (IM (projT1 (l item)))
Heqpr_item_x: Some (oitem', descriptor') =
equivocator_vlsm_transition_item_project
  (IM (projT1 (l item)))
  {|
    l := projT2 (l item);
    input := input item;
    destination :=
      destination item
        (projT1 (l item));
    output := output item
  |}
  (final_descriptors (projT1 (l item)))
Hpr_item_x: match oitem' with
| Some item' =>
    Some
      (Some
         {|
           l :=
             existT (projT1 (l item))
               (l item');
           input := input item;
           destination :=
             equivocators_state_project
               IM final_descriptors
               (destination item);
           output := output item
         |},
       equivocator_descriptors_update IM
         final_descriptors
         (projT1 (l item)) descriptor')
| None =>
    Some
      (None,
       equivocator_descriptors_update IM
         final_descriptors
         (projT1 (l item)) descriptor')
end = Some (ox, final')
Hpr_item: match ox with
| Some item' => Some ([item'], final')
| None => Some ([], final')
end = Some (pr_item, final_descriptors')
pr_sub_item: list
  (composite_transition_item
     (λ ei : sub_index selection,
        IM (`ei)))
final_sub_descriptors': equivocator_descriptors
  (λ ei : sub_index selection,
     IM (`ei))
Hl: projT1 (l item) ∈ selection
Hpr_sub_item: equivocators_trace_project
  (λ ei : sub_index selection, IM (`ei))
  final_sub_descriptors
  [{|
     l :=
       composite_label_sub_projection
         (equivocator_IM IM) selection
         (l item) Hl;
     input := input item;
     destination :=
       composite_state_sub_projection
         (equivocator_IM IM) selection
         (destination item);
     output := output item
   |}] =
Some
  (pr_sub_item, final_sub_descriptors')
final_sub_descriptors' =
(λ i : sub_index selection, final_descriptors' (`i))
∧ finite_trace_sub_projection IM selection pr_item =
  pr_sub_item simpl in Hpr_sub_item.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
item: composite_transition_item (equivocator_IM IM)
final_descriptors', final_descriptors: equivocator_descriptors IM
final_sub_descriptors:= λ i : sub_index selection,
  final_descriptors (`i): ∀ i : sub_index selection,
  MachineDescriptor (IM (`i))
pr_item: list (composite_transition_item IM)
ox: option (composite_transition_item IM)
final': equivocator_descriptors IM
oitem': option transition_item
descriptor': MachineDescriptor (IM (projT1 (l item)))
Heqpr_item_x: Some (oitem', descriptor') =
equivocator_vlsm_transition_item_project
  (IM (projT1 (l item)))
  {|
    l := projT2 (l item);
    input := input item;
    destination :=
      destination item
        (projT1 (l item));
    output := output item
  |}
  (final_descriptors (projT1 (l item)))
Hpr_item_x: match oitem' with
| Some item' =>
    Some
      (Some
         {|
           l :=
             existT (projT1 (l item))
               (l item');
           input := input item;
           destination :=
             equivocators_state_project
               IM final_descriptors
               (destination item);
           output := output item
         |},
       equivocator_descriptors_update IM
         final_descriptors
         (projT1 (l item)) descriptor')
| None =>
    Some
      (None,
       equivocator_descriptors_update IM
         final_descriptors
         (projT1 (l item)) descriptor')
end = Some (ox, final')
Hpr_item: match ox with
| Some item' => Some ([item'], final')
| None => Some ([], final')
end = Some (pr_item, final_descriptors')
pr_sub_item: list
  (composite_transition_item
     (λ ei : sub_index selection,
        IM (`ei)))
final_sub_descriptors': equivocator_descriptors
  (λ ei : sub_index selection,
     IM (`ei))
Hl: projT1 (l item) ∈ selection
Hpr_sub_item: match
  equivocators_transition_item_project
    (λ ei : sub_index selection,
       IM (`ei)) final_sub_descriptors
    {|
      l :=
        composite_label_sub_projection
          (equivocator_IM IM) selection
          (l item) Hl;
      input := input item;
      destination :=
        composite_state_sub_projection
          (equivocator_IM IM) selection
          (destination item);
      output := output item
    |}
with
| Some (Some item', odescriptor) =>
    Some ([item'], odescriptor)
| Some (None, odescriptor) =>
    Some ([], odescriptor)
| None => None
end =
Some
  (pr_sub_item, final_sub_descriptors')
final_sub_descriptors' =
(λ i : sub_index selection, final_descriptors' (`i))
∧ finite_trace_sub_projection IM selection pr_item =
  pr_sub_item
    unfold final_sub_descriptors in *.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
item: composite_transition_item (equivocator_IM IM)
final_descriptors', final_descriptors: equivocator_descriptors IM
final_sub_descriptors:= λ i : sub_index selection,
  final_descriptors (`i): ∀ i : sub_index selection,
  MachineDescriptor (IM (`i))
pr_item: list (composite_transition_item IM)
ox: option (composite_transition_item IM)
final': equivocator_descriptors IM
oitem': option transition_item
descriptor': MachineDescriptor (IM (projT1 (l item)))
Heqpr_item_x: Some (oitem', descriptor') =
equivocator_vlsm_transition_item_project
  (IM (projT1 (l item)))
  {|
    l := projT2 (l item);
    input := input item;
    destination :=
      destination item
        (projT1 (l item));
    output := output item
  |}
  (final_descriptors (projT1 (l item)))
Hpr_item_x: match oitem' with
| Some item' =>
    Some
      (Some
         {|
           l :=
             existT (projT1 (l item))
               (l item');
           input := input item;
           destination :=
             equivocators_state_project
               IM final_descriptors
               (destination item);
           output := output item
         |},
       equivocator_descriptors_update IM
         final_descriptors
         (projT1 (l item)) descriptor')
| None =>
    Some
      (None,
       equivocator_descriptors_update IM
         final_descriptors
         (projT1 (l item)) descriptor')
end = Some (ox, final')
Hpr_item: match ox with
| Some item' => Some ([item'], final')
| None => Some ([], final')
end = Some (pr_item, final_descriptors')
pr_sub_item: list
  (composite_transition_item
     (λ ei : sub_index selection,
        IM (`ei)))
final_sub_descriptors': equivocator_descriptors
  (λ ei : sub_index selection,
     IM (`ei))
Hl: projT1 (l item) ∈ selection
Hpr_sub_item: match
  equivocators_transition_item_project
    (λ ei : sub_index selection,
       IM (`ei))
    (λ i : sub_index selection,
       final_descriptors (`i))
    {|
      l :=
        composite_label_sub_projection
          (equivocator_IM IM) selection
          (l item) Hl;
      input := input item;
      destination :=
        composite_state_sub_projection
          (equivocator_IM IM) selection
          (destination item);
      output := output item
    |}
with
| Some (Some item', odescriptor) =>
    Some ([item'], odescriptor)
| Some (None, odescriptor) =>
    Some ([], odescriptor)
| None => None
end =
Some
  (pr_sub_item, final_sub_descriptors')
final_sub_descriptors' =
(λ i : sub_index selection, final_descriptors' (`i))
∧ finite_trace_sub_projection IM selection pr_item =
  pr_sub_item
    unfold equivocators_transition_item_project in Hpr_sub_item.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
item: composite_transition_item (equivocator_IM IM)
final_descriptors', final_descriptors: equivocator_descriptors IM
final_sub_descriptors:= λ i : sub_index selection,
  final_descriptors (`i): ∀ i : sub_index selection,
  MachineDescriptor (IM (`i))
pr_item: list (composite_transition_item IM)
ox: option (composite_transition_item IM)
final': equivocator_descriptors IM
oitem': option transition_item
descriptor': MachineDescriptor (IM (projT1 (l item)))
Heqpr_item_x: Some (oitem', descriptor') =
equivocator_vlsm_transition_item_project
  (IM (projT1 (l item)))
  {|
    l := projT2 (l item);
    input := input item;
    destination :=
      destination item
        (projT1 (l item));
    output := output item
  |}
  (final_descriptors (projT1 (l item)))
Hpr_item_x: match oitem' with
| Some item' =>
    Some
      (Some
         {|
           l :=
             existT (projT1 (l item))
               (l item');
           input := input item;
           destination :=
             equivocators_state_project
               IM final_descriptors
               (destination item);
           output := output item
         |},
       equivocator_descriptors_update IM
         final_descriptors
         (projT1 (l item)) descriptor')
| None =>
    Some
      (None,
       equivocator_descriptors_update IM
         final_descriptors
         (projT1 (l item)) descriptor')
end = Some (ox, final')
Hpr_item: match ox with
| Some item' => Some ([item'], final')
| None => Some ([], final')
end = Some (pr_item, final_descriptors')
pr_sub_item: list
  (composite_transition_item
     (λ ei : sub_index selection,
        IM (`ei)))
final_sub_descriptors': equivocator_descriptors
  (λ ei : sub_index selection,
     IM (`ei))
Hl: projT1 (l item) ∈ selection
Hpr_sub_item: match
  match
    equivocator_vlsm_transition_item_project
      (IM
         (`(projT1
              (l
                 {|
                   l :=
                     composite_label_sub_projection
                      (equivocator_IM
                      IM) selection
                      (l item) Hl;
                   input := input item;
                   destination :=
                     composite_state_sub_projection
                      (equivocator_IM
                      IM) selection
                      (destination item);
                   output :=
                     output item
                 |}))))
      (composite_transition_item_projection
         (equivocator_IM
            (λ ei : sub_index selection,
               IM (`ei)))
         {|
           l :=
             composite_label_sub_projection
               (equivocator_IM IM)
               selection 
               (l item) Hl;
           input := input item;
           destination :=
             composite_state_sub_projection
               (equivocator_IM IM)
               selection
               (destination item);
           output := output item
         |})
      (final_descriptors
         (`(projT1
              (l
                 {|
                   l :=
                     composite_label_sub_projection
                      (equivocator_IM
                      IM) selection
                      (l item) Hl;
                   input := input item;
                   destination :=
                     composite_state_sub_projection
                      (equivocator_IM
                      IM) selection
                      (destination item);
                   output :=
                     output item
                 |}))))
  with
  | Some (Some item', deqv') =>
      Some
        (Some
           {|
             l :=
               existT
                 (projT1
                    (l
                      {|
                      l :=
                      composite_label_sub_projection
                      (equivocator_IM
                      IM) selection
                      (l item) Hl;
                      input :=
                      input item;
                      destination :=
                      composite_state_sub_projection
                      (equivocator_IM
                      IM) selection
                      (destination item);
                      output :=
                      output item
                      |})) 
                 (l item');
             input :=
               input
                 {|
                   l :=
                     composite_label_sub_projection
                      (equivocator_IM
                      IM) selection
                      (l item) Hl;
                   input := input item;
                   destination :=
                     composite_state_sub_projection
                      (equivocator_IM
                      IM) selection
                      (destination item);
                   output :=
                     output item
                 |};
             destination :=
               equivocators_state_project
                 (λ ei : 
                     sub_index
                      selection,
                    IM (`ei))
                 (λ i : 
                     sub_index
                      selection,
                    final_descriptors
                      (`i))
                 (destination
                    {|
                      l :=
                      composite_label_sub_projection
                      (equivocator_IM
                      IM) selection
                      (l item) Hl;
                      input :=
                      input item;
                      destination :=
                      composite_state_sub_projection
                      (equivocator_IM
                      IM) selection
                      (destination item);
                      output :=
                      output item
                    |});
             output :=
               output
                 {|
                   l :=
                     composite_label_sub_projection
                      (equivocator_IM
                      IM) selection
                      (l item) Hl;
                   input := input item;
                   destination :=
                     composite_state_sub_projection
                      (equivocator_IM
                      IM) selection
                      (destination item);
                   output :=
                     output item
                 |}
           |},
         equivocator_descriptors_update
           (λ ei : sub_index selection,
              IM (`ei))
           (λ i : sub_index selection,
              final_descriptors (`i))
           (projT1
              (l
                 {|
                   l :=
                     composite_label_sub_projection
                      (equivocator_IM
                      IM) selection
                      (l item) Hl;
                   input := input item;
                   destination :=
                     composite_state_sub_projection
                      (equivocator_IM
                      IM) selection
                      (destination item);
                   output :=
                     output item
                 |})) deqv')
  | Some (None, deqv') =>
      Some
        (None,
         equivocator_descriptors_update
           (λ ei : sub_index selection,
              IM (`ei))
           (λ i : sub_index selection,
              final_descriptors (`i))
           (projT1
              (l
                 {|
                   l :=
                     composite_label_sub_projection
                      (equivocator_IM
                      IM) selection
                      (l item) Hl;
                   input := input item;
                   destination :=
                     composite_state_sub_projection
                      (equivocator_IM
                      IM) selection
                      (destination item);
                   output :=
                     output item
                 |})) deqv')
  | None => None
  end
with
| Some (Some item', odescriptor) =>
    Some ([item'], odescriptor)
| Some (None, odescriptor) =>
    Some ([], odescriptor)
| None => None
end =
Some
  (pr_sub_item, final_sub_descriptors')
final_sub_descriptors' =
(λ i : sub_index selection, final_descriptors' (`i))
∧ finite_trace_sub_projection IM selection pr_item =
  pr_sub_item
    cbn in Hpr_sub_item.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
item: composite_transition_item (equivocator_IM IM)
final_descriptors', final_descriptors: equivocator_descriptors IM
final_sub_descriptors:= λ i : sub_index selection,
  final_descriptors (`i): ∀ i : sub_index selection,
  MachineDescriptor (IM (`i))
pr_item: list (composite_transition_item IM)
ox: option (composite_transition_item IM)
final': equivocator_descriptors IM
oitem': option transition_item
descriptor': MachineDescriptor (IM (projT1 (l item)))
Heqpr_item_x: Some (oitem', descriptor') =
equivocator_vlsm_transition_item_project
  (IM (projT1 (l item)))
  {|
    l := projT2 (l item);
    input := input item;
    destination :=
      destination item
        (projT1 (l item));
    output := output item
  |}
  (final_descriptors (projT1 (l item)))
Hpr_item_x: match oitem' with
| Some item' =>
    Some
      (Some
         {|
           l :=
             existT (projT1 (l item))
               (l item');
           input := input item;
           destination :=
             equivocators_state_project
               IM final_descriptors
               (destination item);
           output := output item
         |},
       equivocator_descriptors_update IM
         final_descriptors
         (projT1 (l item)) descriptor')
| None =>
    Some
      (None,
       equivocator_descriptors_update IM
         final_descriptors
         (projT1 (l item)) descriptor')
end = Some (ox, final')
Hpr_item: match ox with
| Some item' => Some ([item'], final')
| None => Some ([], final')
end = Some (pr_item, final_descriptors')
pr_sub_item: list
  (composite_transition_item
     (λ ei : sub_index selection,
        IM (`ei)))
final_sub_descriptors': equivocator_descriptors
  (λ ei : sub_index selection,
     IM (`ei))
Hl: projT1 (l item) ∈ selection
Hpr_sub_item: match
  match
    equivocator_vlsm_transition_item_project
      (IM (projT1 (l item)))
      (composite_transition_item_projection
         (equivocator_IM
            (λ ei : sub_index selection,
               IM (`ei)))
         {|
           l :=
             composite_label_sub_projection
               (equivocator_IM IM)
               selection 
               (l item) Hl;
           input := input item;
           destination :=
             composite_state_sub_projection
               (equivocator_IM IM)
               selection
               (destination item);
           output := output item
         |})
      (final_descriptors
         (projT1 (l item)))
  with
  | Some (Some item', deqv') =>
      Some
        (Some
           {|
             l :=
               existT
                 (dexist
                    (projT1 (l item))
                    Hl) 
                 (l item');
             input := input item;
             destination :=
               equivocators_state_project
                 (λ ei : 
                     sub_index
                      selection,
                    IM (`ei))
                 (λ i : 
                     sub_index
                      selection,
                    final_descriptors
                      (`i))
                 (composite_state_sub_projection
                    (equivocator_IM IM)
                    selection
                    (destination item));
             output := output item
           |},
         equivocator_descriptors_update
           (λ ei : sub_index selection,
              IM (`ei))
           (λ i : sub_index selection,
              final_descriptors (`i))
           (dexist (projT1 (l item)) Hl)
           deqv')
  | Some (None, deqv') =>
      Some
        (None,
         equivocator_descriptors_update
           (λ ei : sub_index selection,
              IM (`ei))
           (λ i : sub_index selection,
              final_descriptors (`i))
           (dexist (projT1 (l item)) Hl)
           deqv')
  | None => None
  end
with
| Some (Some item', odescriptor) =>
    Some ([item'], odescriptor)
| Some (None, odescriptor) =>
    Some ([], odescriptor)
| None => None
end =
Some
  (pr_sub_item, final_sub_descriptors')
final_sub_descriptors' =
(λ i : sub_index selection, final_descriptors' (`i))
∧ finite_trace_sub_projection IM selection pr_item =
  pr_sub_item
    unfold
      composite_transition_item_projection,
      composite_transition_item_projection_from_eq,
      eq_rect_r,
      composite_state_sub_projection in Hpr_sub_item.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
item: composite_transition_item (equivocator_IM IM)
final_descriptors', final_descriptors: equivocator_descriptors IM
final_sub_descriptors:= λ i : sub_index selection,
  final_descriptors (`i): ∀ i : sub_index selection,
  MachineDescriptor (IM (`i))
pr_item: list (composite_transition_item IM)
ox: option (composite_transition_item IM)
final': equivocator_descriptors IM
oitem': option transition_item
descriptor': MachineDescriptor (IM (projT1 (l item)))
Heqpr_item_x: Some (oitem', descriptor') =
equivocator_vlsm_transition_item_project
  (IM (projT1 (l item)))
  {|
    l := projT2 (l item);
    input := input item;
    destination :=
      destination item
        (projT1 (l item));
    output := output item
  |}
  (final_descriptors (projT1 (l item)))
Hpr_item_x: match oitem' with
| Some item' =>
    Some
      (Some
         {|
           l :=
             existT (projT1 (l item))
               (l item');
           input := input item;
           destination :=
             equivocators_state_project
               IM final_descriptors
               (destination item);
           output := output item
         |},
       equivocator_descriptors_update IM
         final_descriptors
         (projT1 (l item)) descriptor')
| None =>
    Some
      (None,
       equivocator_descriptors_update IM
         final_descriptors
         (projT1 (l item)) descriptor')
end = Some (ox, final')
Hpr_item: match ox with
| Some item' => Some ([item'], final')
| None => Some ([], final')
end = Some (pr_item, final_descriptors')
pr_sub_item: list
  (composite_transition_item
     (λ ei : sub_index selection,
        IM (`ei)))
final_sub_descriptors': equivocator_descriptors
  (λ ei : sub_index selection,
     IM (`ei))
Hl: projT1 (l item) ∈ selection
Hpr_sub_item: match
  match
    equivocator_vlsm_transition_item_project
      (IM (projT1 (l item)))
      {|
        l :=
          eq_rect
            (projT1
               (l
                  {|
                    l :=
                      composite_label_sub_projection
                      (equivocator_IM
                      IM) selection
                      (l item) Hl;
                    input := input item;
                    destination :=
                      λ subi : 
                      sub_index
                      selection,
                      destination item
                      (`subi);
                    output :=
                      output item
                  |}))
            (λ y : sub_index selection,
               label
                 (equivocator_IM
                    (λ ei : 
                      sub_index
                      selection,
                      IM (`ei)) y))
            (projT2
               (l
                  {|
                    l :=
                      composite_label_sub_projection
                      (equivocator_IM
                      IM) selection
                      (l item) Hl;
                    input := input item;
                    destination :=
                      λ subi : 
                      sub_index
                      selection,
                      destination item
                      (`subi);
                    output :=
                      output item
                  |}))
            (projT1
               (l
                  {|
                    l :=
                      composite_label_sub_projection
                      (equivocator_IM
                      IM) selection
                      (l item) Hl;
                    input := input item;
                    destination :=
                      λ subi : 
                      sub_index
                      selection,
                      destination item
                      (`subi);
                    output :=
                      output item
                  |})) 
            (eq_sym eq_refl);
        input :=
          input
            {|
              l :=
                composite_label_sub_projection
                  (equivocator_IM IM)
                  selection 
                  (l item) Hl;
              input := input item;
              destination :=
                λ subi : 
                   sub_index selection,
                  destination item
                    (`subi);
              output := output item
            |};
        destination :=
          destination
            {|
              l :=
                composite_label_sub_projection
                  (equivocator_IM IM)
                  selection 
                  (l item) Hl;
              input := input item;
              destination :=
                λ subi : 
                   sub_index selection,
                  destination item
                    (`subi);
              output := output item
            |}
            (projT1
               (l
                  {|
                    l :=
                      composite_label_sub_projection
                      (equivocator_IM
                      IM) selection
                      (l item) Hl;
                    input := input item;
                    destination :=
                      λ subi : 
                      sub_index
                      selection,
                      destination item
                      (`subi);
                    output :=
                      output item
                  |}));
        output :=
          output
            {|
              l :=
                composite_label_sub_projection
                  (equivocator_IM IM)
                  selection 
                  (l item) Hl;
              input := input item;
              destination :=
                λ subi : 
                   sub_index selection,
                  destination item
                    (`subi);
              output := output item
            |}
      |}
      (final_descriptors
         (projT1 (l item)))
  with
  | Some (Some item', deqv') =>
      Some
        (Some
           {|
             l :=
               existT
                 (dexist
                    (projT1 (l item))
                    Hl) 
                 (l item');
             input := input item;
             destination :=
               equivocators_state_project
                 (λ ei : 
                     sub_index
                      selection,
                    IM (`ei))
                 (λ i : 
                     sub_index
                      selection,
                    final_descriptors
                      (`i))
                 (λ subi : 
                     sub_index
                      selection,
                    destination item
                      (`subi));
             output := output item
           |},
         equivocator_descriptors_update
           (λ ei : sub_index selection,
              IM (`ei))
           (λ i : sub_index selection,
              final_descriptors (`i))
           (dexist (projT1 (l item)) Hl)
           deqv')
  | Some (None, deqv') =>
      Some
        (None,
         equivocator_descriptors_update
           (λ ei : sub_index selection,
              IM (`ei))
           (λ i : sub_index selection,
              final_descriptors (`i))
           (dexist (projT1 (l item)) Hl)
           deqv')
  | None => None
  end
with
| Some (Some item', odescriptor) =>
    Some ([item'], odescriptor)
| Some (None, odescriptor) =>
    Some ([], odescriptor)
| None => None
end =
Some
  (pr_sub_item, final_sub_descriptors')
final_sub_descriptors' =
(λ i : sub_index selection, final_descriptors' (`i))
∧ finite_trace_sub_projection IM selection pr_item =
  pr_sub_item
    simpl in Hpr_sub_item.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
item: composite_transition_item (equivocator_IM IM)
final_descriptors', final_descriptors: equivocator_descriptors IM
final_sub_descriptors:= λ i : sub_index selection,
  final_descriptors (`i): ∀ i : sub_index selection,
  MachineDescriptor (IM (`i))
pr_item: list (composite_transition_item IM)
ox: option (composite_transition_item IM)
final': equivocator_descriptors IM
oitem': option transition_item
descriptor': MachineDescriptor (IM (projT1 (l item)))
Heqpr_item_x: Some (oitem', descriptor') =
equivocator_vlsm_transition_item_project
  (IM (projT1 (l item)))
  {|
    l := projT2 (l item);
    input := input item;
    destination :=
      destination item
        (projT1 (l item));
    output := output item
  |}
  (final_descriptors (projT1 (l item)))
Hpr_item_x: match oitem' with
| Some item' =>
    Some
      (Some
         {|
           l :=
             existT (projT1 (l item))
               (l item');
           input := input item;
           destination :=
             equivocators_state_project
               IM final_descriptors
               (destination item);
           output := output item
         |},
       equivocator_descriptors_update IM
         final_descriptors
         (projT1 (l item)) descriptor')
| None =>
    Some
      (None,
       equivocator_descriptors_update IM
         final_descriptors
         (projT1 (l item)) descriptor')
end = Some (ox, final')
Hpr_item: match ox with
| Some item' => Some ([item'], final')
| None => Some ([], final')
end = Some (pr_item, final_descriptors')
pr_sub_item: list
  (composite_transition_item
     (λ ei : sub_index selection,
        IM (`ei)))
final_sub_descriptors': equivocator_descriptors
  (λ ei : sub_index selection,
     IM (`ei))
Hl: projT1 (l item) ∈ selection
Hpr_sub_item: match
  match
    equivocator_vlsm_transition_item_project
      (IM (projT1 (l item)))
      {|
        l := projT2 (l item);
        input := input item;
        destination :=
          destination item
            (projT1 (l item));
        output := output item
      |}
      (final_descriptors
         (projT1 (l item)))
  with
  | Some (Some item', deqv') =>
      Some
        (Some
           {|
             l :=
               existT
                 (dexist
                    (projT1 (l item))
                    Hl) 
                 (l item');
             input := input item;
             destination :=
               equivocators_state_project
                 (λ ei : 
                     sub_index
                      selection,
                    IM (`ei))
                 (λ i : 
                     sub_index
                      selection,
                    final_descriptors
                      (`i))
                 (λ subi : 
                     sub_index
                      selection,
                    destination item
                      (`subi));
             output := output item
           |},
         equivocator_descriptors_update
           (λ ei : sub_index selection,
              IM (`ei))
           (λ i : sub_index selection,
              final_descriptors (`i))
           (dexist (projT1 (l item)) Hl)
           deqv')
  | Some (None, deqv') =>
      Some
        (None,
         equivocator_descriptors_update
           (λ ei : sub_index selection,
              IM (`ei))
           (λ i : sub_index selection,
              final_descriptors (`i))
           (dexist (projT1 (l item)) Hl)
           deqv')
  | None => None
  end
with
| Some (Some item', odescriptor) =>
    Some ([item'], odescriptor)
| Some (None, odescriptor) =>
    Some ([], odescriptor)
| None => None
end =
Some
  (pr_sub_item, final_sub_descriptors')
final_sub_descriptors' =
(λ i : sub_index selection, final_descriptors' (`i))
∧ finite_trace_sub_projection IM selection pr_item =
  pr_sub_item
    rewrite <-  Heqpr_item_x in Hpr_sub_item.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
item: composite_transition_item (equivocator_IM IM)
final_descriptors', final_descriptors: equivocator_descriptors IM
final_sub_descriptors:= λ i : sub_index selection,
  final_descriptors (`i): ∀ i : sub_index selection,
  MachineDescriptor (IM (`i))
pr_item: list (composite_transition_item IM)
ox: option (composite_transition_item IM)
final': equivocator_descriptors IM
oitem': option transition_item
descriptor': MachineDescriptor (IM (projT1 (l item)))
Heqpr_item_x: Some (oitem', descriptor') =
equivocator_vlsm_transition_item_project
  (IM (projT1 (l item)))
  {|
    l := projT2 (l item);
    input := input item;
    destination :=
      destination item
        (projT1 (l item));
    output := output item
  |}
  (final_descriptors (projT1 (l item)))
Hpr_item_x: match oitem' with
| Some item' =>
    Some
      (Some
         {|
           l :=
             existT (projT1 (l item))
               (l item');
           input := input item;
           destination :=
             equivocators_state_project
               IM final_descriptors
               (destination item);
           output := output item
         |},
       equivocator_descriptors_update IM
         final_descriptors
         (projT1 (l item)) descriptor')
| None =>
    Some
      (None,
       equivocator_descriptors_update IM
         final_descriptors
         (projT1 (l item)) descriptor')
end = Some (ox, final')
Hpr_item: match ox with
| Some item' => Some ([item'], final')
| None => Some ([], final')
end = Some (pr_item, final_descriptors')
pr_sub_item: list
  (composite_transition_item
     (λ ei : sub_index selection,
        IM (`ei)))
final_sub_descriptors': equivocator_descriptors
  (λ ei : sub_index selection,
     IM (`ei))
Hl: projT1 (l item) ∈ selection
Hpr_sub_item: match
  match oitem' with
  | Some item' =>
      Some
        (Some
           {|
             l :=
               existT
                 (dexist
                    (projT1 (l item))
                    Hl) 
                 (l item');
             input := input item;
             destination :=
               equivocators_state_project
                 (λ ei : 
                     sub_index
                      selection,
                    IM (`ei))
                 (λ i : 
                     sub_index
                      selection,
                    final_descriptors
                      (`i))
                 (λ subi : 
                     sub_index
                      selection,
                    destination item
                      (`subi));
             output := output item
           |},
         equivocator_descriptors_update
           (λ ei : sub_index selection,
              IM (`ei))
           (λ i : sub_index selection,
              final_descriptors (`i))
           (dexist (projT1 (l item)) Hl)
           descriptor')
  | None =>
      Some
        (None,
         equivocator_descriptors_update
           (λ ei : sub_index selection,
              IM (`ei))
           (λ i : sub_index selection,
              final_descriptors (`i))
           (dexist (projT1 (l item)) Hl)
           descriptor')
  end
with
| Some (Some item', odescriptor) =>
    Some ([item'], odescriptor)
| Some (None, odescriptor) =>
    Some ([], odescriptor)
| None => None
end =
Some
  (pr_sub_item, final_sub_descriptors')
final_sub_descriptors' =
(λ i : sub_index selection, final_descriptors' (`i))
∧ finite_trace_sub_projection IM selection pr_item =
  pr_sub_item clear Heqpr_item_x.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
item: composite_transition_item (equivocator_IM IM)
final_descriptors', final_descriptors: equivocator_descriptors IM
final_sub_descriptors:= λ i : sub_index selection,
  final_descriptors (`i): ∀ i : sub_index selection,
  MachineDescriptor (IM (`i))
pr_item: list (composite_transition_item IM)
ox: option (composite_transition_item IM)
final': equivocator_descriptors IM
oitem': option transition_item
descriptor': MachineDescriptor (IM (projT1 (l item)))
Hpr_item_x: match oitem' with
| Some item' =>
    Some
      (Some
         {|
           l :=
             existT (projT1 (l item))
               (l item');
           input := input item;
           destination :=
             equivocators_state_project
               IM final_descriptors
               (destination item);
           output := output item
         |},
       equivocator_descriptors_update IM
         final_descriptors
         (projT1 (l item)) descriptor')
| None =>
    Some
      (None,
       equivocator_descriptors_update IM
         final_descriptors
         (projT1 (l item)) descriptor')
end = Some (ox, final')
Hpr_item: match ox with
| Some item' => Some ([item'], final')
| None => Some ([], final')
end = Some (pr_item, final_descriptors')
pr_sub_item: list
  (composite_transition_item
     (λ ei : sub_index selection,
        IM (`ei)))
final_sub_descriptors': equivocator_descriptors
  (λ ei : sub_index selection,
     IM (`ei))
Hl: projT1 (l item) ∈ selection
Hpr_sub_item: match
  match oitem' with
  | Some item' =>
      Some
        (Some
           {|
             l :=
               existT
                 (dexist
                    (projT1 (l item))
                    Hl) 
                 (l item');
             input := input item;
             destination :=
               equivocators_state_project
                 (λ ei : 
                     sub_index
                      selection,
                    IM (`ei))
                 (λ i : 
                     sub_index
                      selection,
                    final_descriptors
                      (`i))
                 (λ subi : 
                     sub_index
                      selection,
                    destination item
                      (`subi));
             output := output item
           |},
         equivocator_descriptors_update
           (λ ei : sub_index selection,
              IM (`ei))
           (λ i : sub_index selection,
              final_descriptors (`i))
           (dexist (projT1 (l item)) Hl)
           descriptor')
  | None =>
      Some
        (None,
         equivocator_descriptors_update
           (λ ei : sub_index selection,
              IM (`ei))
           (λ i : sub_index selection,
              final_descriptors (`i))
           (dexist (projT1 (l item)) Hl)
           descriptor')
  end
with
| Some (Some item', odescriptor) =>
    Some ([item'], odescriptor)
| Some (None, odescriptor) =>
    Some ([], odescriptor)
| None => None
end =
Some
  (pr_sub_item, final_sub_descriptors')
final_sub_descriptors' =
(λ i : sub_index selection, final_descriptors' (`i))
∧ finite_trace_sub_projection IM selection pr_item =
  pr_sub_item
    split.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
item: composite_transition_item (equivocator_IM IM)
final_descriptors', final_descriptors: equivocator_descriptors IM
final_sub_descriptors:= λ i : sub_index selection,
  final_descriptors (`i): ∀ i : sub_index selection,
  MachineDescriptor (IM (`i))
pr_item: list (composite_transition_item IM)
ox: option (composite_transition_item IM)
final': equivocator_descriptors IM
oitem': option transition_item
descriptor': MachineDescriptor (IM (projT1 (l item)))
Hpr_item_x: match oitem' with
| Some item' =>
    Some
      (Some
         {|
           l :=
             existT (projT1 (l item))
               (l item');
           input := input item;
           destination :=
             equivocators_state_project
               IM final_descriptors
               (destination item);
           output := output item
         |},
       equivocator_descriptors_update IM
         final_descriptors
         (projT1 (l item)) descriptor')
| None =>
    Some
      (None,
       equivocator_descriptors_update IM
         final_descriptors
         (projT1 (l item)) descriptor')
end = Some (ox, final')
Hpr_item: match ox with
| Some item' => Some ([item'], final')
| None => Some ([], final')
end = Some (pr_item, final_descriptors')
pr_sub_item: list
  (composite_transition_item
     (λ ei : sub_index selection,
        IM (`ei)))
final_sub_descriptors': equivocator_descriptors
  (λ ei : sub_index selection,
     IM (`ei))
Hl: projT1 (l item) ∈ selection
Hpr_sub_item: match
  match oitem' with
  | Some item' =>
      Some
        (Some
           {|
             l :=
               existT
                 (dexist
                    (projT1 (l item))
                    Hl) 
                 (l item');
             input := input item;
             destination :=
               equivocators_state_project
                 (λ ei : 
                     sub_index
                      selection,
                    IM (`ei))
                 (λ i : 
                     sub_index
                      selection,
                    final_descriptors
                      (`i))
                 (λ subi : 
                     sub_index
                      selection,
                    destination item
                      (`subi));
             output := output item
           |},
         equivocator_descriptors_update
           (λ ei : sub_index selection,
              IM (`ei))
           (λ i : sub_index selection,
              final_descriptors (`i))
           (dexist (projT1 (l item)) Hl)
           descriptor')
  | None =>
      Some
        (None,
         equivocator_descriptors_update
           (λ ei : sub_index selection,
              IM (`ei))
           (λ i : sub_index selection,
              final_descriptors (`i))
           (dexist (projT1 (l item)) Hl)
           descriptor')
  end
with
| Some (Some item', odescriptor) =>
    Some ([item'], odescriptor)
| Some (None, odescriptor) =>
    Some ([], odescriptor)
| None => None
end =
Some
  (pr_sub_item, final_sub_descriptors')
final_sub_descriptors' =
(λ i : sub_index selection, final_descriptors' (`i))
message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
item: composite_transition_item (equivocator_IM IM)
final_descriptors', final_descriptors: equivocator_descriptors IM
final_sub_descriptors:= λ i : sub_index selection,
  final_descriptors (`i): ∀ i : sub_index selection,
  MachineDescriptor (IM (`i))
pr_item: list (composite_transition_item IM)
ox: option (composite_transition_item IM)
final': equivocator_descriptors IM
oitem': option transition_item
descriptor': MachineDescriptor (IM (projT1 (l item)))
Hpr_item_x: match oitem' with
| Some item' =>
    Some
      (Some
         {|
           l :=
             existT (projT1 (l item))
               (l item');
           input := input item;
           destination :=
             equivocators_state_project
               IM final_descriptors
               (destination item);
           output := output item
         |},
       equivocator_descriptors_update IM
         final_descriptors
         (projT1 (l item)) descriptor')
| None =>
    Some
      (None,
       equivocator_descriptors_update IM
         final_descriptors
         (projT1 (l item)) descriptor')
end = Some (ox, final')
Hpr_item: match ox with
| Some item' => Some ([item'], final')
| None => Some ([], final')
end = Some (pr_item, final_descriptors')
pr_sub_item: list
  (composite_transition_item
     (λ ei : sub_index selection,
        IM (`ei)))
final_sub_descriptors': equivocator_descriptors
  (λ ei : sub_index selection,
     IM (`ei))
Hl: projT1 (l item) ∈ selection
Hpr_sub_item: match
  match oitem' with
  | Some item' =>
      Some
        (Some
           {|
             l :=
               existT
                 (dexist
                    (projT1 (l item))
                    Hl) 
                 (l item');
             input := input item;
             destination :=
               equivocators_state_project
                 (λ ei : 
                     sub_index
                      selection,
                    IM (`ei))
                 (λ i : 
                     sub_index
                      selection,
                    final_descriptors
                      (`i))
                 (λ subi : 
                     sub_index
                      selection,
                    destination item
                      (`subi));
             output := output item
           |},
         equivocator_descriptors_update
           (λ ei : sub_index selection,
              IM (`ei))
           (λ i : sub_index selection,
              final_descriptors (`i))
           (dexist (projT1 (l item)) Hl)
           descriptor')
  | None =>
      Some
        (None,
         equivocator_descriptors_update
           (λ ei : sub_index selection,
              IM (`ei))
           (λ i : sub_index selection,
              final_descriptors (`i))
           (dexist (projT1 (l item)) Hl)
           descriptor')
  end
with
| Some (Some item', odescriptor) =>
    Some ([item'], odescriptor)
| Some (None, odescriptor) =>
    Some ([], odescriptor)
| None => None
end =
Some
  (pr_sub_item, final_sub_descriptors')
finite_trace_sub_projection IM selection pr_item =
pr_sub_item
    +message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
item: composite_transition_item (equivocator_IM IM)
final_descriptors', final_descriptors: equivocator_descriptors IM
final_sub_descriptors:= λ i : sub_index selection,
  final_descriptors (`i): ∀ i : sub_index selection,
  MachineDescriptor (IM (`i))
pr_item: list (composite_transition_item IM)
ox: option (composite_transition_item IM)
final': equivocator_descriptors IM
oitem': option transition_item
descriptor': MachineDescriptor (IM (projT1 (l item)))
Hpr_item_x: match oitem' with
| Some item' =>
    Some
      (Some
         {|
           l :=
             existT (projT1 (l item))
               (l item');
           input := input item;
           destination :=
             equivocators_state_project
               IM final_descriptors
               (destination item);
           output := output item
         |},
       equivocator_descriptors_update IM
         final_descriptors
         (projT1 (l item)) descriptor')
| None =>
    Some
      (None,
       equivocator_descriptors_update IM
         final_descriptors
         (projT1 (l item)) descriptor')
end = Some (ox, final')
Hpr_item: match ox with
| Some item' => Some ([item'], final')
| None => Some ([], final')
end = Some (pr_item, final_descriptors')
pr_sub_item: list
  (composite_transition_item
     (λ ei : sub_index selection,
        IM (`ei)))
final_sub_descriptors': equivocator_descriptors
  (λ ei : sub_index selection,
     IM (`ei))
Hl: projT1 (l item) ∈ selection
Hpr_sub_item: match
  match oitem' with
  | Some item' =>
      Some
        (Some
           {|
             l :=
               existT
                 (dexist
                    (projT1 (l item))
                    Hl) 
                 (l item');
             input := input item;
             destination :=
               equivocators_state_project
                 (λ ei : 
                     sub_index
                      selection,
                    IM (`ei))
                 (λ i : 
                     sub_index
                      selection,
                    final_descriptors
                      (`i))
                 (λ subi : 
                     sub_index
                      selection,
                    destination item
                      (`subi));
             output := output item
           |},
         equivocator_descriptors_update
           (λ ei : sub_index selection,
              IM (`ei))
           (λ i : sub_index selection,
              final_descriptors (`i))
           (dexist (projT1 (l item)) Hl)
           descriptor')
  | None =>
      Some
        (None,
         equivocator_descriptors_update
           (λ ei : sub_index selection,
              IM (`ei))
           (λ i : sub_index selection,
              final_descriptors (`i))
           (dexist (projT1 (l item)) Hl)
           descriptor')
  end
with
| Some (Some item', odescriptor) =>
    Some ([item'], odescriptor)
| Some (None, odescriptor) =>
    Some ([], odescriptor)
| None => None
end =
Some
  (pr_sub_item, final_sub_descriptors')
final_sub_descriptors' =
(λ i : sub_index selection, final_descriptors' (`i)) extensionality i.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
item: composite_transition_item (equivocator_IM IM)
final_descriptors', final_descriptors: equivocator_descriptors IM
final_sub_descriptors:= λ i : sub_index selection,
  final_descriptors (`i): ∀ i : sub_index selection,
  MachineDescriptor (IM (`i))
pr_item: list (composite_transition_item IM)
ox: option (composite_transition_item IM)
final': equivocator_descriptors IM
oitem': option transition_item
descriptor': MachineDescriptor (IM (projT1 (l item)))
Hpr_item_x: match oitem' with
| Some item' =>
    Some
      (Some
         {|
           l :=
             existT (projT1 (l item))
               (l item');
           input := input item;
           destination :=
             equivocators_state_project
               IM final_descriptors
               (destination item);
           output := output item
         |},
       equivocator_descriptors_update IM
         final_descriptors
         (projT1 (l item)) descriptor')
| None =>
    Some
      (None,
       equivocator_descriptors_update IM
         final_descriptors
         (projT1 (l item)) descriptor')
end = Some (ox, final')
Hpr_item: match ox with
| Some item' => Some ([item'], final')
| None => Some ([], final')
end = Some (pr_item, final_descriptors')
pr_sub_item: list
  (composite_transition_item
     (λ ei : sub_index selection,
        IM (`ei)))
final_sub_descriptors': equivocator_descriptors
  (λ ei : sub_index selection,
     IM (`ei))
Hl: projT1 (l item) ∈ selection
Hpr_sub_item: match
  match oitem' with
  | Some item' =>
      Some
        (Some
           {|
             l :=
               existT
                 (dexist
                    (projT1 (l item))
                    Hl) 
                 (l item');
             input := input item;
             destination :=
               equivocators_state_project
                 (λ ei : 
                     sub_index
                      selection,
                    IM (`ei))
                 (λ i : 
                     sub_index
                      selection,
                    final_descriptors
                      (`i))
                 (λ subi : 
                     sub_index
                      selection,
                    destination item
                      (`subi));
             output := output item
           |},
         equivocator_descriptors_update
           (λ ei : sub_index selection,
              IM (`ei))
           (λ i : sub_index selection,
              final_descriptors (`i))
           (dexist (projT1 (l item)) Hl)
           descriptor')
  | None =>
      Some
        (None,
         equivocator_descriptors_update
           (λ ei : sub_index selection,
              IM (`ei))
           (λ i : sub_index selection,
              final_descriptors (`i))
           (dexist (projT1 (l item)) Hl)
           descriptor')
  end
with
| Some (Some item', odescriptor) =>
    Some ([item'], odescriptor)
| Some (None, odescriptor) =>
    Some ([], odescriptor)
| None => None
end =
Some
  (pr_sub_item, final_sub_descriptors')
i: sub_index selection
final_sub_descriptors' i = final_descriptors' (`i)
      destruct oitem' as [item' |]
      ; inversion Hpr_sub_item; subst; clear Hpr_sub_item
      ; inversion Hpr_item_x; subst; clear Hpr_item_x
      ; inversion Hpr_item; subst; clear Hpr_item
      ; simpl
      ; destruct (decide ((proj1_sig i) = projT1 (l item))).message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
item: composite_transition_item (equivocator_IM IM)
final_descriptors: equivocator_descriptors IM
final_sub_descriptors:= λ i : sub_index selection,
  final_descriptors (`i): ∀ i : sub_index selection,
  MachineDescriptor (IM (`i))
item': transition_item
descriptor': MachineDescriptor (IM (projT1 (l item)))
Hl: projT1 (l item) ∈ selection
i: sub_index selection
e: `i = projT1 (l item)
equivocator_descriptors_update
  (λ ei : sub_index selection, IM (`ei))
  (λ i : sub_index selection, final_descriptors (`i))
  (dexist (projT1 (l item)) Hl) descriptor' i =
equivocator_descriptors_update IM final_descriptors
  (projT1 (l item)) descriptor' (`i)
message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
item: composite_transition_item (equivocator_IM IM)
final_descriptors: equivocator_descriptors IM
final_sub_descriptors:= λ i : sub_index selection,
  final_descriptors (`i): ∀ i : sub_index selection,
  MachineDescriptor (IM (`i))
item': transition_item
descriptor': MachineDescriptor (IM (projT1 (l item)))
Hl: projT1 (l item) ∈ selection
i: sub_index selection
n: `i ≠ projT1 (l item)
equivocator_descriptors_update
  (λ ei : sub_index selection, IM (`ei))
  (λ i : sub_index selection, final_descriptors (`i))
  (dexist (projT1 (l item)) Hl) descriptor' i =
equivocator_descriptors_update IM final_descriptors
  (projT1 (l item)) descriptor' 
  (`i)
message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
item: composite_transition_item (equivocator_IM IM)
final_descriptors: equivocator_descriptors IM
final_sub_descriptors:= λ i : sub_index selection,
  final_descriptors (`i): ∀ i : sub_index selection,
  MachineDescriptor (IM (`i))
descriptor': MachineDescriptor (IM (projT1 (l item)))
Hl: projT1 (l item) ∈ selection
i: sub_index selection
e: `i = projT1 (l item)
equivocator_descriptors_update
  (λ ei : sub_index selection, IM (`ei))
  (λ i : sub_index selection, final_descriptors (`i))
  (dexist (projT1 (l item)) Hl) descriptor' i =
equivocator_descriptors_update IM final_descriptors
  (projT1 (l item)) descriptor' 
  (`i)
message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
item: composite_transition_item (equivocator_IM IM)
final_descriptors: equivocator_descriptors IM
final_sub_descriptors:= λ i : sub_index selection,
  final_descriptors (`i): ∀ i : sub_index selection,
  MachineDescriptor (IM (`i))
descriptor': MachineDescriptor (IM (projT1 (l item)))
Hl: projT1 (l item) ∈ selection
i: sub_index selection
n: `i ≠ projT1 (l item)
equivocator_descriptors_update
  (λ ei : sub_index selection, IM (`ei))
  (λ i : sub_index selection, final_descriptors (`i))
  (dexist (projT1 (l item)) Hl) descriptor' i =
equivocator_descriptors_update IM final_descriptors
  (projT1 (l item)) descriptor' 
  (`i)
      *message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
item: composite_transition_item (equivocator_IM IM)
final_descriptors: equivocator_descriptors IM
final_sub_descriptors:= λ i : sub_index selection,
  final_descriptors (`i): ∀ i : sub_index selection,
  MachineDescriptor (IM (`i))
item': transition_item
descriptor': MachineDescriptor (IM (projT1 (l item)))
Hl: projT1 (l item) ∈ selection
i: sub_index selection
e: `i = projT1 (l item)
equivocator_descriptors_update
  (λ ei : sub_index selection, IM (`ei))
  (λ i : sub_index selection, final_descriptors (`i))
  (dexist (projT1 (l item)) Hl) descriptor' i =
equivocator_descriptors_update IM final_descriptors
  (projT1 (l item)) descriptor' (`i) rewrite equivocator_descriptors_update_eq_rew with (Heq := e).message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
item: composite_transition_item (equivocator_IM IM)
final_descriptors: equivocator_descriptors IM
final_sub_descriptors:= λ i : sub_index selection,
  final_descriptors (`i): ∀ i : sub_index selection,
  MachineDescriptor (IM (`i))
item': transition_item
descriptor': MachineDescriptor (IM (projT1 (l item)))
Hl: projT1 (l item) ∈ selection
i: sub_index selection
e: `i = projT1 (l item)
equivocator_descriptors_update
  (λ ei : sub_index selection, IM (`ei))
  (λ i : sub_index selection, final_descriptors (`i))
  (dexist (projT1 (l item)) Hl) descriptor' i =
eq_rect_r (λ i : index, MachineDescriptor (IM i))
  descriptor' e
        assert (e1 : i = (dexist (projT1 (l item)) Hl)) by (apply dsig_eq; done).message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
item: composite_transition_item (equivocator_IM IM)
final_descriptors: equivocator_descriptors IM
final_sub_descriptors:= λ i : sub_index selection,
  final_descriptors (`i): ∀ i : sub_index selection,
  MachineDescriptor (IM (`i))
item': transition_item
descriptor': MachineDescriptor (IM (projT1 (l item)))
Hl: projT1 (l item) ∈ selection
i: sub_index selection
e: `i = projT1 (l item)
e1: i = dexist (projT1 (l item)) Hl
equivocator_descriptors_update
  (λ ei : sub_index selection, IM (`ei))
  (λ i : sub_index selection, final_descriptors (`i))
  (dexist (projT1 (l item)) Hl) descriptor' i =
eq_rect_r (λ i : index, MachineDescriptor (IM i))
  descriptor' e
        subst i.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
item: composite_transition_item (equivocator_IM IM)
final_descriptors: equivocator_descriptors IM
final_sub_descriptors:= λ i : sub_index selection,
  final_descriptors (`i): ∀ i : sub_index selection,
  MachineDescriptor (IM (`i))
item': transition_item
descriptor': MachineDescriptor (IM (projT1 (l item)))
Hl: projT1 (l item) ∈ selection
e: `(dexist (projT1 (l item)) Hl) = projT1 (l item)
equivocator_descriptors_update
  (λ ei : sub_index selection, IM (`ei))
  (λ i : sub_index selection, final_descriptors (`i))
  (dexist (projT1 (l item)) Hl) descriptor'
  (dexist (projT1 (l item)) Hl) =
eq_rect_r (λ i : index, MachineDescriptor (IM i))
  descriptor' e
        rewrite equivocator_descriptors_update_eq_rew with (Heq := eq_refl).message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
item: composite_transition_item (equivocator_IM IM)
final_descriptors: equivocator_descriptors IM
final_sub_descriptors:= λ i : sub_index selection,
  final_descriptors (`i): ∀ i : sub_index selection,
  MachineDescriptor (IM (`i))
item': transition_item
descriptor': MachineDescriptor (IM (projT1 (l item)))
Hl: projT1 (l item) ∈ selection
e: `(dexist (projT1 (l item)) Hl) = projT1 (l item)
eq_rect_r
  (λ i : sub_index selection,
     MachineDescriptor (IM (`i))) descriptor' eq_refl =
eq_rect_r (λ i : index, MachineDescriptor (IM i))
  descriptor' e
        simpl in e.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
item: composite_transition_item (equivocator_IM IM)
final_descriptors: equivocator_descriptors IM
final_sub_descriptors:= λ i : sub_index selection,
  final_descriptors (`i): ∀ i : sub_index selection,
  MachineDescriptor (IM (`i))
item': transition_item
descriptor': MachineDescriptor (IM (projT1 (l item)))
Hl: projT1 (l item) ∈ selection
e: projT1 (l item) = projT1 (l item)
eq_rect_r
  (λ i : sub_index selection,
     MachineDescriptor (IM (`i))) descriptor' eq_refl =
eq_rect_r (λ i : index, MachineDescriptor (IM i))
  descriptor' e replace e with (@eq_refl _ (projT1 (l item))); [done |].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
item: composite_transition_item (equivocator_IM IM)
final_descriptors: equivocator_descriptors IM
final_sub_descriptors:= λ i : sub_index selection,
  final_descriptors (`i): ∀ i : sub_index selection,
  MachineDescriptor (IM (`i))
item': transition_item
descriptor': MachineDescriptor (IM (projT1 (l item)))
Hl: projT1 (l item) ∈ selection
e: projT1 (l item) = projT1 (l item)
eq_refl = e
        by apply Eqdep_dec.UIP_dec.
      *message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
item: composite_transition_item (equivocator_IM IM)
final_descriptors: equivocator_descriptors IM
final_sub_descriptors:= λ i : sub_index selection,
  final_descriptors (`i): ∀ i : sub_index selection,
  MachineDescriptor (IM (`i))
item': transition_item
descriptor': MachineDescriptor (IM (projT1 (l item)))
Hl: projT1 (l item) ∈ selection
i: sub_index selection
n: `i ≠ projT1 (l item)
equivocator_descriptors_update
  (λ ei : sub_index selection, IM (`ei))
  (λ i : sub_index selection, final_descriptors (`i))
  (dexist (projT1 (l item)) Hl) descriptor' i =
equivocator_descriptors_update IM final_descriptors
  (projT1 (l item)) descriptor' (`i) by rewrite! equivocator_descriptors_update_neq; [| | intros ->].
      *message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
item: composite_transition_item (equivocator_IM IM)
final_descriptors: equivocator_descriptors IM
final_sub_descriptors:= λ i : sub_index selection,
  final_descriptors (`i): ∀ i : sub_index selection,
  MachineDescriptor (IM (`i))
descriptor': MachineDescriptor (IM (projT1 (l item)))
Hl: projT1 (l item) ∈ selection
i: sub_index selection
e: `i = projT1 (l item)
equivocator_descriptors_update
  (λ ei : sub_index selection, IM (`ei))
  (λ i : sub_index selection, final_descriptors (`i))
  (dexist (projT1 (l item)) Hl) descriptor' i =
equivocator_descriptors_update IM final_descriptors
  (projT1 (l item)) descriptor' (`i) rewrite equivocator_descriptors_update_eq_rew with (Heq := e).message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
item: composite_transition_item (equivocator_IM IM)
final_descriptors: equivocator_descriptors IM
final_sub_descriptors:= λ i : sub_index selection,
  final_descriptors (`i): ∀ i : sub_index selection,
  MachineDescriptor (IM (`i))
descriptor': MachineDescriptor (IM (projT1 (l item)))
Hl: projT1 (l item) ∈ selection
i: sub_index selection
e: `i = projT1 (l item)
equivocator_descriptors_update
  (λ ei : sub_index selection, IM (`ei))
  (λ i : sub_index selection, final_descriptors (`i))
  (dexist (projT1 (l item)) Hl) descriptor' i =
eq_rect_r (λ i : index, MachineDescriptor (IM i))
  descriptor' e
        assert (e1 : i = (dexist (projT1 (l item)) Hl)) by (apply dsig_eq; done).message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
item: composite_transition_item (equivocator_IM IM)
final_descriptors: equivocator_descriptors IM
final_sub_descriptors:= λ i : sub_index selection,
  final_descriptors (`i): ∀ i : sub_index selection,
  MachineDescriptor (IM (`i))
descriptor': MachineDescriptor (IM (projT1 (l item)))
Hl: projT1 (l item) ∈ selection
i: sub_index selection
e: `i = projT1 (l item)
e1: i = dexist (projT1 (l item)) Hl
equivocator_descriptors_update
  (λ ei : sub_index selection, IM (`ei))
  (λ i : sub_index selection, final_descriptors (`i))
  (dexist (projT1 (l item)) Hl) descriptor' i =
eq_rect_r (λ i : index, MachineDescriptor (IM i))
  descriptor' e
        subst i.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
item: composite_transition_item (equivocator_IM IM)
final_descriptors: equivocator_descriptors IM
final_sub_descriptors:= λ i : sub_index selection,
  final_descriptors (`i): ∀ i : sub_index selection,
  MachineDescriptor (IM (`i))
descriptor': MachineDescriptor (IM (projT1 (l item)))
Hl: projT1 (l item) ∈ selection
e: `(dexist (projT1 (l item)) Hl) = projT1 (l item)
equivocator_descriptors_update
  (λ ei : sub_index selection, IM (`ei))
  (λ i : sub_index selection, final_descriptors (`i))
  (dexist (projT1 (l item)) Hl) descriptor'
  (dexist (projT1 (l item)) Hl) =
eq_rect_r (λ i : index, MachineDescriptor (IM i))
  descriptor' e
        rewrite equivocator_descriptors_update_eq_rew with (Heq := eq_refl).message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
item: composite_transition_item (equivocator_IM IM)
final_descriptors: equivocator_descriptors IM
final_sub_descriptors:= λ i : sub_index selection,
  final_descriptors (`i): ∀ i : sub_index selection,
  MachineDescriptor (IM (`i))
descriptor': MachineDescriptor (IM (projT1 (l item)))
Hl: projT1 (l item) ∈ selection
e: `(dexist (projT1 (l item)) Hl) = projT1 (l item)
eq_rect_r
  (λ i : sub_index selection,
     MachineDescriptor (IM (`i))) descriptor' eq_refl =
eq_rect_r (λ i : index, MachineDescriptor (IM i))
  descriptor' e
        simpl in e.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
item: composite_transition_item (equivocator_IM IM)
final_descriptors: equivocator_descriptors IM
final_sub_descriptors:= λ i : sub_index selection,
  final_descriptors (`i): ∀ i : sub_index selection,
  MachineDescriptor (IM (`i))
descriptor': MachineDescriptor (IM (projT1 (l item)))
Hl: projT1 (l item) ∈ selection
e: projT1 (l item) = projT1 (l item)
eq_rect_r
  (λ i : sub_index selection,
     MachineDescriptor (IM (`i))) descriptor' eq_refl =
eq_rect_r (λ i : index, MachineDescriptor (IM i))
  descriptor' e replace e with (@eq_refl _ (projT1 (l item))); [done |].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
item: composite_transition_item (equivocator_IM IM)
final_descriptors: equivocator_descriptors IM
final_sub_descriptors:= λ i : sub_index selection,
  final_descriptors (`i): ∀ i : sub_index selection,
  MachineDescriptor (IM (`i))
descriptor': MachineDescriptor (IM (projT1 (l item)))
Hl: projT1 (l item) ∈ selection
e: projT1 (l item) = projT1 (l item)
eq_refl = e
        by apply Eqdep_dec.UIP_dec.
      *message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
item: composite_transition_item (equivocator_IM IM)
final_descriptors: equivocator_descriptors IM
final_sub_descriptors:= λ i : sub_index selection,
  final_descriptors (`i): ∀ i : sub_index selection,
  MachineDescriptor (IM (`i))
descriptor': MachineDescriptor (IM (projT1 (l item)))
Hl: projT1 (l item) ∈ selection
i: sub_index selection
n: `i ≠ projT1 (l item)
equivocator_descriptors_update
  (λ ei : sub_index selection, IM (`ei))
  (λ i : sub_index selection, final_descriptors (`i))
  (dexist (projT1 (l item)) Hl) descriptor' i =
equivocator_descriptors_update IM final_descriptors
  (projT1 (l item)) descriptor' (`i) by rewrite! equivocator_descriptors_update_neq; [| | intros ->].
    +message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
item: composite_transition_item (equivocator_IM IM)
final_descriptors', final_descriptors: equivocator_descriptors IM
final_sub_descriptors:= λ i : sub_index selection,
  final_descriptors (`i): ∀ i : sub_index selection,
  MachineDescriptor (IM (`i))
pr_item: list (composite_transition_item IM)
ox: option (composite_transition_item IM)
final': equivocator_descriptors IM
oitem': option transition_item
descriptor': MachineDescriptor (IM (projT1 (l item)))
Hpr_item_x: match oitem' with
| Some item' =>
    Some
      (Some
         {|
           l :=
             existT (projT1 (l item))
               (l item');
           input := input item;
           destination :=
             equivocators_state_project
               IM final_descriptors
               (destination item);
           output := output item
         |},
       equivocator_descriptors_update IM
         final_descriptors
         (projT1 (l item)) descriptor')
| None =>
    Some
      (None,
       equivocator_descriptors_update IM
         final_descriptors
         (projT1 (l item)) descriptor')
end = Some (ox, final')
Hpr_item: match ox with
| Some item' => Some ([item'], final')
| None => Some ([], final')
end = Some (pr_item, final_descriptors')
pr_sub_item: list
  (composite_transition_item
     (λ ei : sub_index selection,
        IM (`ei)))
final_sub_descriptors': equivocator_descriptors
  (λ ei : sub_index selection,
     IM (`ei))
Hl: projT1 (l item) ∈ selection
Hpr_sub_item: match
  match oitem' with
  | Some item' =>
      Some
        (Some
           {|
             l :=
               existT
                 (dexist
                    (projT1 (l item))
                    Hl) 
                 (l item');
             input := input item;
             destination :=
               equivocators_state_project
                 (λ ei : 
                     sub_index
                      selection,
                    IM (`ei))
                 (λ i : 
                     sub_index
                      selection,
                    final_descriptors
                      (`i))
                 (λ subi : 
                     sub_index
                      selection,
                    destination item
                      (`subi));
             output := output item
           |},
         equivocator_descriptors_update
           (λ ei : sub_index selection,
              IM (`ei))
           (λ i : sub_index selection,
              final_descriptors (`i))
           (dexist (projT1 (l item)) Hl)
           descriptor')
  | None =>
      Some
        (None,
         equivocator_descriptors_update
           (λ ei : sub_index selection,
              IM (`ei))
           (λ i : sub_index selection,
              final_descriptors (`i))
           (dexist (projT1 (l item)) Hl)
           descriptor')
  end
with
| Some (Some item', odescriptor) =>
    Some ([item'], odescriptor)
| Some (None, odescriptor) =>
    Some ([], odescriptor)
| None => None
end =
Some
  (pr_sub_item, final_sub_descriptors')
finite_trace_sub_projection IM selection pr_item =
pr_sub_item destruct oitem' as [item' |]
      ; inversion Hpr_sub_item; subst; clear Hpr_sub_item
      ; inversion Hpr_item_x; subst; clear Hpr_item_x
      ; inversion Hpr_item; subst; clear Hpr_item
      ; simpl; [| done].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
item: composite_transition_item (equivocator_IM IM)
final_descriptors: equivocator_descriptors IM
final_sub_descriptors:= λ i : sub_index selection,
  final_descriptors (`i): ∀ i : sub_index selection,
  MachineDescriptor (IM (`i))
item': transition_item
descriptor': MachineDescriptor (IM (projT1 (l item)))
Hl: projT1 (l item) ∈ selection
match
  pre_VLSM_projection_transition_item_project
    (composite_type IM)
    (composite_type (sub_IM IM selection))
    (composite_label_sub_projection_option IM
       selection)
    (composite_state_sub_projection IM selection)
    {|
      l := existT (projT1 (l item)) (l item');
      input := input item;
      destination :=
        equivocators_state_project IM
          final_descriptors (destination item);
      output := output item
    |}
with
| Some y => [y]
| None => []
end =
[{|
   l := existT (dexist (projT1 (l item)) Hl) (l item');
   input := input item;
   destination :=
     equivocators_state_project
       (λ ei : sub_index selection, IM (`ei))
       (λ i : sub_index selection,
          final_descriptors (`i))
       (λ subi : sub_index selection,
          destination item (`subi));
   output := output item
 |}]
      unfold pre_VLSM_projection_transition_item_project,
        composite_label_sub_projection_option.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
item: composite_transition_item (equivocator_IM IM)
final_descriptors: equivocator_descriptors IM
final_sub_descriptors:= λ i : sub_index selection,
  final_descriptors (`i): ∀ i : sub_index selection,
  MachineDescriptor (IM (`i))
item': transition_item
descriptor': MachineDescriptor (IM (projT1 (l item)))
Hl: projT1 (l item) ∈ selection
match
  match
    match
      decide
        (projT1
           (l
              {|
                l :=
                  existT (projT1 (l item)) (l item');
                input := input item;
                destination :=
                  equivocators_state_project IM
                    final_descriptors
                    (destination item);
                output := output item
              |}) ∈ selection)
    with
    | left i_in =>
        Some
          (composite_label_sub_projection IM selection
             (l
                {|
                  l :=
                    existT (projT1 (l item)) (l item');
                  input := input item;
                  destination :=
                    equivocators_state_project IM
                      final_descriptors
                      (destination item);
                  output := output item
                |}) i_in)
    | right _ => None
    end
  with
  | Some lY =>
      Some
        {|
          l := lY;
          input :=
            input
              {|
                l :=
                  existT (projT1 (l item)) (l item');
                input := input item;
                destination :=
                  equivocators_state_project IM
                    final_descriptors
                    (destination item);
                output := output item
              |};
          destination :=
            composite_state_sub_projection IM
              selection
              (destination
                 {|
                   l :=
                     existT (projT1 (l item))
                       (l item');
                   input := input item;
                   destination :=
                     equivocators_state_project IM
                       final_descriptors
                       (destination item);
                   output := output item
                 |});
          output :=
            output
              {|
                l :=
                  existT (projT1 (l item)) (l item');
                input := input item;
                destination :=
                  equivocators_state_project IM
                    final_descriptors
                    (destination item);
                output := output item
              |}
        |}
  | None => None
  end
with
| Some y => [y]
| None => []
end =
[{|
   l := existT (dexist (projT1 (l item)) Hl) (l item');
   input := input item;
   destination :=
     equivocators_state_project
       (λ ei : sub_index selection, IM (`ei))
       (λ i : sub_index selection,
          final_descriptors (`i))
       (λ subi : sub_index selection,
          destination item (`subi));
   output := output item
 |}]
      simpl.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
item: composite_transition_item (equivocator_IM IM)
final_descriptors: equivocator_descriptors IM
final_sub_descriptors:= λ i : sub_index selection,
  final_descriptors (`i): ∀ i : sub_index selection,
  MachineDescriptor (IM (`i))
item': transition_item
descriptor': MachineDescriptor (IM (projT1 (l item)))
Hl: projT1 (l item) ∈ selection
match
  match
    match decide (projT1 (l item) ∈ selection) with
    | left i_in =>
        Some
          (composite_label_sub_projection IM selection
             (existT (projT1 (l item)) (l item')) i_in)
    | right _ => None
    end
  with
  | Some lY =>
      Some
        {|
          l := lY;
          input := input item;
          destination :=
            composite_state_sub_projection IM
              selection
              (equivocators_state_project IM
                 final_descriptors (destination item));
          output := output item
        |}
  | None => None
  end
with
| Some y => [y]
| None => []
end =
[{|
   l := existT (dexist (projT1 (l item)) Hl) (l item');
   input := input item;
   destination :=
     equivocators_state_project
       (λ ei : sub_index selection, IM (`ei))
       (λ i : sub_index selection,
          final_descriptors (`i))
       (λ subi : sub_index selection,
          destination item (`subi));
   output := output item
 |}]
      case_decide as _Hl; [| done].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
item: composite_transition_item (equivocator_IM IM)
final_descriptors: equivocator_descriptors IM
final_sub_descriptors:= λ i : sub_index selection,
  final_descriptors (`i): ∀ i : sub_index selection,
  MachineDescriptor (IM (`i))
item': transition_item
descriptor': MachineDescriptor (IM (projT1 (l item)))
Hl, _Hl: projT1 (l item) ∈ selection
[{|
   l :=
     composite_label_sub_projection IM selection
       (existT (projT1 (l item)) (l item')) _Hl;
   input := input item;
   destination :=
     composite_state_sub_projection IM selection
       (equivocators_state_project IM
          final_descriptors (destination item));
   output := output item
 |}] =
[{|
   l := existT (dexist (projT1 (l item)) Hl) (l item');
   input := input item;
   destination :=
     equivocators_state_project
       (λ ei : sub_index selection, IM (`ei))
       (λ i : sub_index selection,
          final_descriptors (`i))
       (λ subi : sub_index selection,
          destination item (`subi));
   output := output item
 |}]
      do 2 f_equal.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
item: composite_transition_item (equivocator_IM IM)
final_descriptors: equivocator_descriptors IM
final_sub_descriptors:= λ i : sub_index selection,
  final_descriptors (`i): ∀ i : sub_index selection,
  MachineDescriptor (IM (`i))
item': transition_item
descriptor': MachineDescriptor (IM (projT1 (l item)))
Hl, _Hl: projT1 (l item) ∈ selection
composite_label_sub_projection IM selection
  (existT (projT1 (l item)) (l item')) _Hl =
existT (dexist (projT1 (l item)) Hl) (l item')
      unfold composite_label_sub_projection.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
item: composite_transition_item (equivocator_IM IM)
final_descriptors: equivocator_descriptors IM
final_sub_descriptors:= λ i : sub_index selection,
  final_descriptors (`i): ∀ i : sub_index selection,
  MachineDescriptor (IM (`i))
item': transition_item
descriptor': MachineDescriptor (IM (projT1 (l item)))
Hl, _Hl: projT1 (l item) ∈ selection
existT
  (dexist
     (projT1 (existT (projT1 (l item)) (l item'))) _Hl)
  (projT2 (existT (projT1 (l item)) (l item'))) =
existT (dexist (projT1 (l item)) Hl) (l item')
      by apply
        (@dec_sig_sigT_eq _
          (sub_index_prop selection)
          (sub_index_prop_dec selection)
          (fun n => label (IM n))
          (projT1 (l item)) (l item') (l item') _Hl Hl).
  -message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
item: composite_transition_item (equivocator_IM IM)
final_descriptors', final_descriptors: equivocator_descriptors IM
final_sub_descriptors:= λ i : sub_index selection,
  final_descriptors (`i): ∀ i : sub_index selection,
  MachineDescriptor (IM (`i))
pr_item: list (composite_transition_item IM)
ox: option (composite_transition_item IM)
final': equivocator_descriptors IM
oitem': option transition_item
descriptor': MachineDescriptor (IM (projT1 (l item)))
Heqpr_item_x: Some (oitem', descriptor') =
equivocator_vlsm_transition_item_project
  (IM (projT1 (l item)))
  {|
    l := projT2 (l item);
    input := input item;
    destination :=
      destination item
        (projT1 (l item));
    output := output item
  |}
  (final_descriptors (projT1 (l item)))
Hpr_item_x: match oitem' with
| Some item' =>
    Some
      (Some
         {|
           l :=
             existT (projT1 (l item))
               (l item');
           input := input item;
           destination :=
             equivocators_state_project
               IM final_descriptors
               (destination item);
           output := output item
         |},
       equivocator_descriptors_update IM
         final_descriptors
         (projT1 (l item)) descriptor')
| None =>
    Some
      (None,
       equivocator_descriptors_update IM
         final_descriptors
         (projT1 (l item)) descriptor')
end = Some (ox, final')
Hpr_item: match ox with
| Some item' => Some ([item'], final')
| None => Some ([], final')
end = Some (pr_item, final_descriptors')
pr_sub_item: list
  (composite_transition_item
     (λ ei : sub_index selection,
        IM (`ei)))
final_sub_descriptors': equivocator_descriptors
  (λ ei : sub_index selection,
     IM (`ei))
Hl: projT1 (l item) ∉ selection
Hpr_sub_item: equivocators_trace_project
  (λ ei : sub_index selection, IM (`ei))
  final_sub_descriptors [] =
Some
  (pr_sub_item, final_sub_descriptors')
final_sub_descriptors' =
(λ i : sub_index selection, final_descriptors' (`i))
∧ finite_trace_sub_projection IM selection pr_item =
  pr_sub_item simpl in Hpr_sub_item.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
item: composite_transition_item (equivocator_IM IM)
final_descriptors', final_descriptors: equivocator_descriptors IM
final_sub_descriptors:= λ i : sub_index selection,
  final_descriptors (`i): ∀ i : sub_index selection,
  MachineDescriptor (IM (`i))
pr_item: list (composite_transition_item IM)
ox: option (composite_transition_item IM)
final': equivocator_descriptors IM
oitem': option transition_item
descriptor': MachineDescriptor (IM (projT1 (l item)))
Heqpr_item_x: Some (oitem', descriptor') =
equivocator_vlsm_transition_item_project
  (IM (projT1 (l item)))
  {|
    l := projT2 (l item);
    input := input item;
    destination :=
      destination item
        (projT1 (l item));
    output := output item
  |}
  (final_descriptors (projT1 (l item)))
Hpr_item_x: match oitem' with
| Some item' =>
    Some
      (Some
         {|
           l :=
             existT (projT1 (l item))
               (l item');
           input := input item;
           destination :=
             equivocators_state_project
               IM final_descriptors
               (destination item);
           output := output item
         |},
       equivocator_descriptors_update IM
         final_descriptors
         (projT1 (l item)) descriptor')
| None =>
    Some
      (None,
       equivocator_descriptors_update IM
         final_descriptors
         (projT1 (l item)) descriptor')
end = Some (ox, final')
Hpr_item: match ox with
| Some item' => Some ([item'], final')
| None => Some ([], final')
end = Some (pr_item, final_descriptors')
pr_sub_item: list
  (composite_transition_item
     (λ ei : sub_index selection,
        IM (`ei)))
final_sub_descriptors': equivocator_descriptors
  (λ ei : sub_index selection,
     IM (`ei))
Hl: projT1 (l item) ∉ selection
Hpr_sub_item: Some ([], final_sub_descriptors) =
Some
  (pr_sub_item, final_sub_descriptors')
final_sub_descriptors' =
(λ i : sub_index selection, final_descriptors' (`i))
∧ finite_trace_sub_projection IM selection pr_item =
  pr_sub_item unfold final_sub_descriptors in *.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
item: composite_transition_item (equivocator_IM IM)
final_descriptors', final_descriptors: equivocator_descriptors IM
final_sub_descriptors:= λ i : sub_index selection,
  final_descriptors (`i): ∀ i : sub_index selection,
  MachineDescriptor (IM (`i))
pr_item: list (composite_transition_item IM)
ox: option (composite_transition_item IM)
final': equivocator_descriptors IM
oitem': option transition_item
descriptor': MachineDescriptor (IM (projT1 (l item)))
Heqpr_item_x: Some (oitem', descriptor') =
equivocator_vlsm_transition_item_project
  (IM (projT1 (l item)))
  {|
    l := projT2 (l item);
    input := input item;
    destination :=
      destination item
        (projT1 (l item));
    output := output item
  |}
  (final_descriptors (projT1 (l item)))
Hpr_item_x: match oitem' with
| Some item' =>
    Some
      (Some
         {|
           l :=
             existT (projT1 (l item))
               (l item');
           input := input item;
           destination :=
             equivocators_state_project
               IM final_descriptors
               (destination item);
           output := output item
         |},
       equivocator_descriptors_update IM
         final_descriptors
         (projT1 (l item)) descriptor')
| None =>
    Some
      (None,
       equivocator_descriptors_update IM
         final_descriptors
         (projT1 (l item)) descriptor')
end = Some (ox, final')
Hpr_item: match ox with
| Some item' => Some ([item'], final')
| None => Some ([], final')
end = Some (pr_item, final_descriptors')
pr_sub_item: list
  (composite_transition_item
     (λ ei : sub_index selection,
        IM (`ei)))
final_sub_descriptors': equivocator_descriptors
  (λ ei : sub_index selection,
     IM (`ei))
Hl: projT1 (l item) ∉ selection
Hpr_sub_item: Some
  ([],
   λ i : sub_index selection,
     final_descriptors (`i)) =
Some
  (pr_sub_item, final_sub_descriptors')
final_sub_descriptors' =
(λ i : sub_index selection, final_descriptors' (`i))
∧ finite_trace_sub_projection IM selection pr_item =
  pr_sub_item
    inversion Hpr_sub_item.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
item: composite_transition_item (equivocator_IM IM)
final_descriptors', final_descriptors: equivocator_descriptors IM
final_sub_descriptors:= λ i : sub_index selection,
  final_descriptors (`i): ∀ i : sub_index selection,
  MachineDescriptor (IM (`i))
pr_item: list (composite_transition_item IM)
ox: option (composite_transition_item IM)
final': equivocator_descriptors IM
oitem': option transition_item
descriptor': MachineDescriptor (IM (projT1 (l item)))
Heqpr_item_x: Some (oitem', descriptor') =
equivocator_vlsm_transition_item_project
  (IM (projT1 (l item)))
  {|
    l := projT2 (l item);
    input := input item;
    destination :=
      destination item
        (projT1 (l item));
    output := output item
  |}
  (final_descriptors (projT1 (l item)))
Hpr_item_x: match oitem' with
| Some item' =>
    Some
      (Some
         {|
           l :=
             existT (projT1 (l item))
               (l item');
           input := input item;
           destination :=
             equivocators_state_project
               IM final_descriptors
               (destination item);
           output := output item
         |},
       equivocator_descriptors_update IM
         final_descriptors
         (projT1 (l item)) descriptor')
| None =>
    Some
      (None,
       equivocator_descriptors_update IM
         final_descriptors
         (projT1 (l item)) descriptor')
end = Some (ox, final')
Hpr_item: match ox with
| Some item' => Some ([item'], final')
| None => Some ([], final')
end = Some (pr_item, final_descriptors')
pr_sub_item: list
  (composite_transition_item
     (λ ei : sub_index selection,
        IM (`ei)))
final_sub_descriptors': equivocator_descriptors
  (λ ei : sub_index selection,
     IM (`ei))
Hl: projT1 (l item) ∉ selection
Hpr_sub_item: Some
  ([],
   λ i : sub_index selection,
     final_descriptors (`i)) =
Some
  (pr_sub_item, final_sub_descriptors')
H2: [] = pr_sub_item
H3: (λ i : sub_index selection,
   final_descriptors (`i)) =
final_sub_descriptors'
(λ i : sub_index selection, final_descriptors (`i)) =
(λ i : sub_index selection, final_descriptors' (`i))
∧ finite_trace_sub_projection IM selection pr_item =
  [] subst.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
item: composite_transition_item (equivocator_IM IM)
final_descriptors', final_descriptors: equivocator_descriptors IM
final_sub_descriptors:= λ i : sub_index selection,
  final_descriptors (`i): ∀ i : sub_index selection,
  MachineDescriptor (IM (`i))
pr_item: list (composite_transition_item IM)
ox: option (composite_transition_item IM)
final': equivocator_descriptors IM
oitem': option transition_item
descriptor': MachineDescriptor (IM (projT1 (l item)))
Heqpr_item_x: Some (oitem', descriptor') =
equivocator_vlsm_transition_item_project
  (IM (projT1 (l item)))
  {|
    l := projT2 (l item);
    input := input item;
    destination :=
      destination item
        (projT1 (l item));
    output := output item
  |}
  (final_descriptors (projT1 (l item)))
Hpr_item_x: match oitem' with
| Some item' =>
    Some
      (Some
         {|
           l :=
             existT (projT1 (l item))
               (l item');
           input := input item;
           destination :=
             equivocators_state_project
               IM final_descriptors
               (destination item);
           output := output item
         |},
       equivocator_descriptors_update IM
         final_descriptors
         (projT1 (l item)) descriptor')
| None =>
    Some
      (None,
       equivocator_descriptors_update IM
         final_descriptors
         (projT1 (l item)) descriptor')
end = Some (ox, final')
Hpr_item: match ox with
| Some item' => Some ([item'], final')
| None => Some ([], final')
end = Some (pr_item, final_descriptors')
Hl: projT1 (l item) ∉ selection
Hpr_sub_item: Some
  ([],
   λ i : sub_index selection,
     final_descriptors (`i)) =
Some
  ([],
   λ i : sub_index selection,
     final_descriptors (`i))
(λ i : sub_index selection, final_descriptors (`i)) =
(λ i : sub_index selection, final_descriptors' (`i))
∧ finite_trace_sub_projection IM selection pr_item =
  [] clear Hpr_sub_item.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
item: composite_transition_item (equivocator_IM IM)
final_descriptors', final_descriptors: equivocator_descriptors IM
final_sub_descriptors:= λ i : sub_index selection,
  final_descriptors (`i): ∀ i : sub_index selection,
  MachineDescriptor (IM (`i))
pr_item: list (composite_transition_item IM)
ox: option (composite_transition_item IM)
final': equivocator_descriptors IM
oitem': option transition_item
descriptor': MachineDescriptor (IM (projT1 (l item)))
Heqpr_item_x: Some (oitem', descriptor') =
equivocator_vlsm_transition_item_project
  (IM (projT1 (l item)))
  {|
    l := projT2 (l item);
    input := input item;
    destination :=
      destination item
        (projT1 (l item));
    output := output item
  |}
  (final_descriptors (projT1 (l item)))
Hpr_item_x: match oitem' with
| Some item' =>
    Some
      (Some
         {|
           l :=
             existT (projT1 (l item))
               (l item');
           input := input item;
           destination :=
             equivocators_state_project
               IM final_descriptors
               (destination item);
           output := output item
         |},
       equivocator_descriptors_update IM
         final_descriptors
         (projT1 (l item)) descriptor')
| None =>
    Some
      (None,
       equivocator_descriptors_update IM
         final_descriptors
         (projT1 (l item)) descriptor')
end = Some (ox, final')
Hpr_item: match ox with
| Some item' => Some ([item'], final')
| None => Some ([], final')
end = Some (pr_item, final_descriptors')
Hl: projT1 (l item) ∉ selection
(λ i : sub_index selection, final_descriptors (`i)) =
(λ i : sub_index selection, final_descriptors' (`i))
∧ finite_trace_sub_projection IM selection pr_item =
  []
    split.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
item: composite_transition_item (equivocator_IM IM)
final_descriptors', final_descriptors: equivocator_descriptors IM
final_sub_descriptors:= λ i : sub_index selection,
  final_descriptors (`i): ∀ i : sub_index selection,
  MachineDescriptor (IM (`i))
pr_item: list (composite_transition_item IM)
ox: option (composite_transition_item IM)
final': equivocator_descriptors IM
oitem': option transition_item
descriptor': MachineDescriptor (IM (projT1 (l item)))
Heqpr_item_x: Some (oitem', descriptor') =
equivocator_vlsm_transition_item_project
  (IM (projT1 (l item)))
  {|
    l := projT2 (l item);
    input := input item;
    destination :=
      destination item
        (projT1 (l item));
    output := output item
  |}
  (final_descriptors (projT1 (l item)))
Hpr_item_x: match oitem' with
| Some item' =>
    Some
      (Some
         {|
           l :=
             existT (projT1 (l item))
               (l item');
           input := input item;
           destination :=
             equivocators_state_project
               IM final_descriptors
               (destination item);
           output := output item
         |},
       equivocator_descriptors_update IM
         final_descriptors
         (projT1 (l item)) descriptor')
| None =>
    Some
      (None,
       equivocator_descriptors_update IM
         final_descriptors
         (projT1 (l item)) descriptor')
end = Some (ox, final')
Hpr_item: match ox with
| Some item' => Some ([item'], final')
| None => Some ([], final')
end = Some (pr_item, final_descriptors')
Hl: projT1 (l item) ∉ selection
(λ i : sub_index selection, final_descriptors (`i)) =
(λ i : sub_index selection, final_descriptors' (`i))
message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
item: composite_transition_item (equivocator_IM IM)
final_descriptors', final_descriptors: equivocator_descriptors IM
final_sub_descriptors:= λ i : sub_index selection,
  final_descriptors (`i): ∀ i : sub_index selection,
  MachineDescriptor (IM (`i))
pr_item: list (composite_transition_item IM)
ox: option (composite_transition_item IM)
final': equivocator_descriptors IM
oitem': option transition_item
descriptor': MachineDescriptor (IM (projT1 (l item)))
Heqpr_item_x: Some (oitem', descriptor') =
equivocator_vlsm_transition_item_project
  (IM (projT1 (l item)))
  {|
    l := projT2 (l item);
    input := input item;
    destination :=
      destination item
        (projT1 (l item));
    output := output item
  |}
  (final_descriptors (projT1 (l item)))
Hpr_item_x: match oitem' with
| Some item' =>
    Some
      (Some
         {|
           l :=
             existT (projT1 (l item))
               (l item');
           input := input item;
           destination :=
             equivocators_state_project
               IM final_descriptors
               (destination item);
           output := output item
         |},
       equivocator_descriptors_update IM
         final_descriptors
         (projT1 (l item)) descriptor')
| None =>
    Some
      (None,
       equivocator_descriptors_update IM
         final_descriptors
         (projT1 (l item)) descriptor')
end = Some (ox, final')
Hpr_item: match ox with
| Some item' => Some ([item'], final')
| None => Some ([], final')
end = Some (pr_item, final_descriptors')
Hl: projT1 (l item) ∉ selection
finite_trace_sub_projection IM selection pr_item = []
    +message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
item: composite_transition_item (equivocator_IM IM)
final_descriptors', final_descriptors: equivocator_descriptors IM
final_sub_descriptors:= λ i : sub_index selection,
  final_descriptors (`i): ∀ i : sub_index selection,
  MachineDescriptor (IM (`i))
pr_item: list (composite_transition_item IM)
ox: option (composite_transition_item IM)
final': equivocator_descriptors IM
oitem': option transition_item
descriptor': MachineDescriptor (IM (projT1 (l item)))
Heqpr_item_x: Some (oitem', descriptor') =
equivocator_vlsm_transition_item_project
  (IM (projT1 (l item)))
  {|
    l := projT2 (l item);
    input := input item;
    destination :=
      destination item
        (projT1 (l item));
    output := output item
  |}
  (final_descriptors (projT1 (l item)))
Hpr_item_x: match oitem' with
| Some item' =>
    Some
      (Some
         {|
           l :=
             existT (projT1 (l item))
               (l item');
           input := input item;
           destination :=
             equivocators_state_project
               IM final_descriptors
               (destination item);
           output := output item
         |},
       equivocator_descriptors_update IM
         final_descriptors
         (projT1 (l item)) descriptor')
| None =>
    Some
      (None,
       equivocator_descriptors_update IM
         final_descriptors
         (projT1 (l item)) descriptor')
end = Some (ox, final')
Hpr_item: match ox with
| Some item' => Some ([item'], final')
| None => Some ([], final')
end = Some (pr_item, final_descriptors')
Hl: projT1 (l item) ∉ selection
(λ i : sub_index selection, final_descriptors (`i)) =
(λ i : sub_index selection, final_descriptors' (`i)) extensionality i.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
item: composite_transition_item (equivocator_IM IM)
final_descriptors', final_descriptors: equivocator_descriptors IM
final_sub_descriptors:= λ i : sub_index selection,
  final_descriptors (`i): ∀ i : sub_index selection,
  MachineDescriptor (IM (`i))
pr_item: list (composite_transition_item IM)
ox: option (composite_transition_item IM)
final': equivocator_descriptors IM
oitem': option transition_item
descriptor': MachineDescriptor (IM (projT1 (l item)))
Heqpr_item_x: Some (oitem', descriptor') =
equivocator_vlsm_transition_item_project
  (IM (projT1 (l item)))
  {|
    l := projT2 (l item);
    input := input item;
    destination :=
      destination item
        (projT1 (l item));
    output := output item
  |}
  (final_descriptors (projT1 (l item)))
Hpr_item_x: match oitem' with
| Some item' =>
    Some
      (Some
         {|
           l :=
             existT (projT1 (l item))
               (l item');
           input := input item;
           destination :=
             equivocators_state_project
               IM final_descriptors
               (destination item);
           output := output item
         |},
       equivocator_descriptors_update IM
         final_descriptors
         (projT1 (l item)) descriptor')
| None =>
    Some
      (None,
       equivocator_descriptors_update IM
         final_descriptors
         (projT1 (l item)) descriptor')
end = Some (ox, final')
Hpr_item: match ox with
| Some item' => Some ([item'], final')
| None => Some ([], final')
end = Some (pr_item, final_descriptors')
Hl: projT1 (l item) ∉ selection
i: sub_index selection
final_descriptors (`i) = final_descriptors' (`i)
      assert (Hnot : proj1_sig i <> projT1 (l item)).message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
item: composite_transition_item (equivocator_IM IM)
final_descriptors', final_descriptors: equivocator_descriptors IM
final_sub_descriptors:= λ i : sub_index selection,
  final_descriptors (`i): ∀ i : sub_index selection,
  MachineDescriptor (IM (`i))
pr_item: list (composite_transition_item IM)
ox: option (composite_transition_item IM)
final': equivocator_descriptors IM
oitem': option transition_item
descriptor': MachineDescriptor (IM (projT1 (l item)))
Heqpr_item_x: Some (oitem', descriptor') =
equivocator_vlsm_transition_item_project
  (IM (projT1 (l item)))
  {|
    l := projT2 (l item);
    input := input item;
    destination :=
      destination item
        (projT1 (l item));
    output := output item
  |}
  (final_descriptors (projT1 (l item)))
Hpr_item_x: match oitem' with
| Some item' =>
    Some
      (Some
         {|
           l :=
             existT (projT1 (l item))
               (l item');
           input := input item;
           destination :=
             equivocators_state_project
               IM final_descriptors
               (destination item);
           output := output item
         |},
       equivocator_descriptors_update IM
         final_descriptors
         (projT1 (l item)) descriptor')
| None =>
    Some
      (None,
       equivocator_descriptors_update IM
         final_descriptors
         (projT1 (l item)) descriptor')
end = Some (ox, final')
Hpr_item: match ox with
| Some item' => Some ([item'], final')
| None => Some ([], final')
end = Some (pr_item, final_descriptors')
Hl: projT1 (l item) ∉ selection
i: sub_index selection
`i ≠ projT1 (l item)
message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
item: composite_transition_item (equivocator_IM IM)
final_descriptors', final_descriptors: equivocator_descriptors IM
final_sub_descriptors:= λ i : sub_index selection,
  final_descriptors (`i): ∀ i : sub_index selection,
  MachineDescriptor (IM (`i))
pr_item: list (composite_transition_item IM)
ox: option (composite_transition_item IM)
final': equivocator_descriptors IM
oitem': option transition_item
descriptor': MachineDescriptor (IM (projT1 (l item)))
Heqpr_item_x: Some (oitem', descriptor') =
equivocator_vlsm_transition_item_project
  (IM (projT1 (l item)))
  {|
    l := projT2 (l item);
    input := input item;
    destination :=
      destination item
        (projT1 (l item));
    output := output item
  |}
  (final_descriptors (projT1 (l item)))
Hpr_item_x: match oitem' with
| Some item' =>
    Some
      (Some
         {|
           l :=
             existT (projT1 (l item))
               (l item');
           input := input item;
           destination :=
             equivocators_state_project
               IM final_descriptors
               (destination item);
           output := output item
         |},
       equivocator_descriptors_update IM
         final_descriptors
         (projT1 (l item)) descriptor')
| None =>
    Some
      (None,
       equivocator_descriptors_update IM
         final_descriptors
         (projT1 (l item)) descriptor')
end = Some (ox, final')
Hpr_item: match ox with
| Some item' => Some ([item'], final')
| None => Some ([], final')
end = Some (pr_item, final_descriptors')
Hl: projT1 (l item) ∉ selection
i: sub_index selection
Hnot: `i ≠ projT1 (l item)
final_descriptors (`i) = final_descriptors' (`i)
      {message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
item: composite_transition_item (equivocator_IM IM)
final_descriptors', final_descriptors: equivocator_descriptors IM
final_sub_descriptors:= λ i : sub_index selection,
  final_descriptors (`i): ∀ i : sub_index selection,
  MachineDescriptor (IM (`i))
pr_item: list (composite_transition_item IM)
ox: option (composite_transition_item IM)
final': equivocator_descriptors IM
oitem': option transition_item
descriptor': MachineDescriptor (IM (projT1 (l item)))
Heqpr_item_x: Some (oitem', descriptor') =
equivocator_vlsm_transition_item_project
  (IM (projT1 (l item)))
  {|
    l := projT2 (l item);
    input := input item;
    destination :=
      destination item
        (projT1 (l item));
    output := output item
  |}
  (final_descriptors (projT1 (l item)))
Hpr_item_x: match oitem' with
| Some item' =>
    Some
      (Some
         {|
           l :=
             existT (projT1 (l item))
               (l item');
           input := input item;
           destination :=
             equivocators_state_project
               IM final_descriptors
               (destination item);
           output := output item
         |},
       equivocator_descriptors_update IM
         final_descriptors
         (projT1 (l item)) descriptor')
| None =>
    Some
      (None,
       equivocator_descriptors_update IM
         final_descriptors
         (projT1 (l item)) descriptor')
end = Some (ox, final')
Hpr_item: match ox with
| Some item' => Some ([item'], final')
| None => Some ([], final')
end = Some (pr_item, final_descriptors')
Hl: projT1 (l item) ∉ selection
i: sub_index selection
`i ≠ projT1 (l item) intro Hnot.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
item: composite_transition_item (equivocator_IM IM)
final_descriptors', final_descriptors: equivocator_descriptors IM
final_sub_descriptors:= λ i : sub_index selection,
  final_descriptors (`i): ∀ i : sub_index selection,
  MachineDescriptor (IM (`i))
pr_item: list (composite_transition_item IM)
ox: option (composite_transition_item IM)
final': equivocator_descriptors IM
oitem': option transition_item
descriptor': MachineDescriptor (IM (projT1 (l item)))
Heqpr_item_x: Some (oitem', descriptor') =
equivocator_vlsm_transition_item_project
  (IM (projT1 (l item)))
  {|
    l := projT2 (l item);
    input := input item;
    destination :=
      destination item
        (projT1 (l item));
    output := output item
  |}
  (final_descriptors (projT1 (l item)))
Hpr_item_x: match oitem' with
| Some item' =>
    Some
      (Some
         {|
           l :=
             existT (projT1 (l item))
               (l item');
           input := input item;
           destination :=
             equivocators_state_project
               IM final_descriptors
               (destination item);
           output := output item
         |},
       equivocator_descriptors_update IM
         final_descriptors
         (projT1 (l item)) descriptor')
| None =>
    Some
      (None,
       equivocator_descriptors_update IM
         final_descriptors
         (projT1 (l item)) descriptor')
end = Some (ox, final')
Hpr_item: match ox with
| Some item' => Some ([item'], final')
| None => Some ([], final')
end = Some (pr_item, final_descriptors')
Hl: projT1 (l item) ∉ selection
i: sub_index selection
Hnot: `i = projT1 (l item)
False contradict Hl.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
item: composite_transition_item (equivocator_IM IM)
final_descriptors', final_descriptors: equivocator_descriptors IM
final_sub_descriptors:= λ i : sub_index selection,
  final_descriptors (`i): ∀ i : sub_index selection,
  MachineDescriptor (IM (`i))
pr_item: list (composite_transition_item IM)
ox: option (composite_transition_item IM)
final': equivocator_descriptors IM
oitem': option transition_item
descriptor': MachineDescriptor (IM (projT1 (l item)))
Heqpr_item_x: Some (oitem', descriptor') =
equivocator_vlsm_transition_item_project
  (IM (projT1 (l item)))
  {|
    l := projT2 (l item);
    input := input item;
    destination :=
      destination item
        (projT1 (l item));
    output := output item
  |}
  (final_descriptors (projT1 (l item)))
Hpr_item_x: match oitem' with
| Some item' =>
    Some
      (Some
         {|
           l :=
             existT (projT1 (l item))
               (l item');
           input := input item;
           destination :=
             equivocators_state_project
               IM final_descriptors
               (destination item);
           output := output item
         |},
       equivocator_descriptors_update IM
         final_descriptors
         (projT1 (l item)) descriptor')
| None =>
    Some
      (None,
       equivocator_descriptors_update IM
         final_descriptors
         (projT1 (l item)) descriptor')
end = Some (ox, final')
Hpr_item: match ox with
| Some item' => Some ([item'], final')
| None => Some ([], final')
end = Some (pr_item, final_descriptors')
i: sub_index selection
Hnot: `i = projT1 (l item)
projT1 (l item) ∈ selection destruct i.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
item: composite_transition_item (equivocator_IM IM)
final_descriptors', final_descriptors: equivocator_descriptors IM
final_sub_descriptors:= λ i : sub_index selection,
  final_descriptors (`i): ∀ i : sub_index selection,
  MachineDescriptor (IM (`i))
pr_item: list (composite_transition_item IM)
ox: option (composite_transition_item IM)
final': equivocator_descriptors IM
oitem': option transition_item
descriptor': MachineDescriptor (IM (projT1 (l item)))
Heqpr_item_x: Some (oitem', descriptor') =
equivocator_vlsm_transition_item_project
  (IM (projT1 (l item)))
  {|
    l := projT2 (l item);
    input := input item;
    destination :=
      destination item
        (projT1 (l item));
    output := output item
  |}
  (final_descriptors (projT1 (l item)))
Hpr_item_x: match oitem' with
| Some item' =>
    Some
      (Some
         {|
           l :=
             existT (projT1 (l item))
               (l item');
           input := input item;
           destination :=
             equivocators_state_project
               IM final_descriptors
               (destination item);
           output := output item
         |},
       equivocator_descriptors_update IM
         final_descriptors
         (projT1 (l item)) descriptor')
| None =>
    Some
      (None,
       equivocator_descriptors_update IM
         final_descriptors
         (projT1 (l item)) descriptor')
end = Some (ox, final')
Hpr_item: match ox with
| Some item' => Some ([item'], final')
| None => Some ([], final')
end = Some (pr_item, final_descriptors')
x: index
i: bool_decide (sub_index_prop selection x)
Hnot: `(x ↾ i) = projT1 (l item)
projT1 (l item) ∈ selection simpl in Hnot.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
item: composite_transition_item (equivocator_IM IM)
final_descriptors', final_descriptors: equivocator_descriptors IM
final_sub_descriptors:= λ i : sub_index selection,
  final_descriptors (`i): ∀ i : sub_index selection,
  MachineDescriptor (IM (`i))
pr_item: list (composite_transition_item IM)
ox: option (composite_transition_item IM)
final': equivocator_descriptors IM
oitem': option transition_item
descriptor': MachineDescriptor (IM (projT1 (l item)))
Heqpr_item_x: Some (oitem', descriptor') =
equivocator_vlsm_transition_item_project
  (IM (projT1 (l item)))
  {|
    l := projT2 (l item);
    input := input item;
    destination :=
      destination item
        (projT1 (l item));
    output := output item
  |}
  (final_descriptors (projT1 (l item)))
Hpr_item_x: match oitem' with
| Some item' =>
    Some
      (Some
         {|
           l :=
             existT (projT1 (l item))
               (l item');
           input := input item;
           destination :=
             equivocators_state_project
               IM final_descriptors
               (destination item);
           output := output item
         |},
       equivocator_descriptors_update IM
         final_descriptors
         (projT1 (l item)) descriptor')
| None =>
    Some
      (None,
       equivocator_descriptors_update IM
         final_descriptors
         (projT1 (l item)) descriptor')
end = Some (ox, final')
Hpr_item: match ox with
| Some item' => Some ([item'], final')
| None => Some ([], final')
end = Some (pr_item, final_descriptors')
x: index
i: bool_decide (sub_index_prop selection x)
Hnot: x = projT1 (l item)
projT1 (l item) ∈ selection subst.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
item: composite_transition_item (equivocator_IM IM)
final_descriptors', final_descriptors: equivocator_descriptors IM
final_sub_descriptors:= λ i : sub_index selection,
  final_descriptors (`i): ∀ i : sub_index selection,
  MachineDescriptor (IM (`i))
pr_item: list (composite_transition_item IM)
ox: option (composite_transition_item IM)
final': equivocator_descriptors IM
oitem': option transition_item
descriptor': MachineDescriptor (IM (projT1 (l item)))
Heqpr_item_x: Some (oitem', descriptor') =
equivocator_vlsm_transition_item_project
  (IM (projT1 (l item)))
  {|
    l := projT2 (l item);
    input := input item;
    destination :=
      destination item
        (projT1 (l item));
    output := output item
  |}
  (final_descriptors (projT1 (l item)))
Hpr_item_x: match oitem' with
| Some item' =>
    Some
      (Some
         {|
           l :=
             existT (projT1 (l item))
               (l item');
           input := input item;
           destination :=
             equivocators_state_project
               IM final_descriptors
               (destination item);
           output := output item
         |},
       equivocator_descriptors_update IM
         final_descriptors
         (projT1 (l item)) descriptor')
| None =>
    Some
      (None,
       equivocator_descriptors_update IM
         final_descriptors
         (projT1 (l item)) descriptor')
end = Some (ox, final')
Hpr_item: match ox with
| Some item' => Some ([item'], final')
| None => Some ([], final')
end = Some (pr_item, final_descriptors')
i: bool_decide
  (sub_index_prop selection (projT1 (l item)))
projT1 (l item) ∈ selection
        by apply bool_decide_spec in i.
      }message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
item: composite_transition_item (equivocator_IM IM)
final_descriptors', final_descriptors: equivocator_descriptors IM
final_sub_descriptors:= λ i : sub_index selection,
  final_descriptors (`i): ∀ i : sub_index selection,
  MachineDescriptor (IM (`i))
pr_item: list (composite_transition_item IM)
ox: option (composite_transition_item IM)
final': equivocator_descriptors IM
oitem': option transition_item
descriptor': MachineDescriptor (IM (projT1 (l item)))
Heqpr_item_x: Some (oitem', descriptor') =
equivocator_vlsm_transition_item_project
  (IM (projT1 (l item)))
  {|
    l := projT2 (l item);
    input := input item;
    destination :=
      destination item
        (projT1 (l item));
    output := output item
  |}
  (final_descriptors (projT1 (l item)))
Hpr_item_x: match oitem' with
| Some item' =>
    Some
      (Some
         {|
           l :=
             existT (projT1 (l item))
               (l item');
           input := input item;
           destination :=
             equivocators_state_project
               IM final_descriptors
               (destination item);
           output := output item
         |},
       equivocator_descriptors_update IM
         final_descriptors
         (projT1 (l item)) descriptor')
| None =>
    Some
      (None,
       equivocator_descriptors_update IM
         final_descriptors
         (projT1 (l item)) descriptor')
end = Some (ox, final')
Hpr_item: match ox with
| Some item' => Some ([item'], final')
| None => Some ([], final')
end = Some (pr_item, final_descriptors')
Hl: projT1 (l item) ∉ selection
i: sub_index selection
Hnot: `i ≠ projT1 (l item)
final_descriptors (`i) = final_descriptors' (`i)
      by destruct oitem' as [item' |]
      ; inversion Hpr_item_x; subst; clear Hpr_item_x
      ; inversion Hpr_item; subst; clear Hpr_item
      ; state_update_simpl.
    +message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
item: composite_transition_item (equivocator_IM IM)
final_descriptors', final_descriptors: equivocator_descriptors IM
final_sub_descriptors:= λ i : sub_index selection,
  final_descriptors (`i): ∀ i : sub_index selection,
  MachineDescriptor (IM (`i))
pr_item: list (composite_transition_item IM)
ox: option (composite_transition_item IM)
final': equivocator_descriptors IM
oitem': option transition_item
descriptor': MachineDescriptor (IM (projT1 (l item)))
Heqpr_item_x: Some (oitem', descriptor') =
equivocator_vlsm_transition_item_project
  (IM (projT1 (l item)))
  {|
    l := projT2 (l item);
    input := input item;
    destination :=
      destination item
        (projT1 (l item));
    output := output item
  |}
  (final_descriptors (projT1 (l item)))
Hpr_item_x: match oitem' with
| Some item' =>
    Some
      (Some
         {|
           l :=
             existT (projT1 (l item))
               (l item');
           input := input item;
           destination :=
             equivocators_state_project
               IM final_descriptors
               (destination item);
           output := output item
         |},
       equivocator_descriptors_update IM
         final_descriptors
         (projT1 (l item)) descriptor')
| None =>
    Some
      (None,
       equivocator_descriptors_update IM
         final_descriptors
         (projT1 (l item)) descriptor')
end = Some (ox, final')
Hpr_item: match ox with
| Some item' => Some ([item'], final')
| None => Some ([], final')
end = Some (pr_item, final_descriptors')
Hl: projT1 (l item) ∉ selection
finite_trace_sub_projection IM selection pr_item = [] destruct oitem' as [item' |]
      ; inversion Hpr_item_x; subst; clear Hpr_item_x
      ; inversion Hpr_item; subst; clear Hpr_item
      ; simpl; [| done].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
item: composite_transition_item (equivocator_IM IM)
final_descriptors: equivocator_descriptors IM
final_sub_descriptors:= λ i : sub_index selection,
  final_descriptors (`i): ∀ i : sub_index selection,
  MachineDescriptor (IM (`i))
item': transition_item
descriptor': MachineDescriptor (IM (projT1 (l item)))
Heqpr_item_x: Some (Some item', descriptor') =
equivocator_vlsm_transition_item_project
  (IM (projT1 (l item)))
  {|
    l := projT2 (l item);
    input := input item;
    destination :=
      destination item
        (projT1 (l item));
    output := output item
  |}
  (final_descriptors (projT1 (l item)))
Hl: projT1 (l item) ∉ selection
match
  pre_VLSM_projection_transition_item_project
    (composite_type IM)
    (composite_type (sub_IM IM selection))
    (composite_label_sub_projection_option IM
       selection)
    (composite_state_sub_projection IM selection)
    {|
      l := existT (projT1 (l item)) (l item');
      input := input item;
      destination :=
        equivocators_state_project IM
          final_descriptors (destination item);
      output := output item
    |}
with
| Some y => [y]
| None => []
end = []
      unfold from_sub_projection.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
item: composite_transition_item (equivocator_IM IM)
final_descriptors: equivocator_descriptors IM
final_sub_descriptors:= λ i : sub_index selection,
  final_descriptors (`i): ∀ i : sub_index selection,
  MachineDescriptor (IM (`i))
item': transition_item
descriptor': MachineDescriptor (IM (projT1 (l item)))
Heqpr_item_x: Some (Some item', descriptor') =
equivocator_vlsm_transition_item_project
  (IM (projT1 (l item)))
  {|
    l := projT2 (l item);
    input := input item;
    destination :=
      destination item
        (projT1 (l item));
    output := output item
  |}
  (final_descriptors (projT1 (l item)))
Hl: projT1 (l item) ∉ selection
match
  pre_VLSM_projection_transition_item_project
    (composite_type IM)
    (composite_type (sub_IM IM selection))
    (composite_label_sub_projection_option IM
       selection)
    (composite_state_sub_projection IM selection)
    {|
      l := existT (projT1 (l item)) (l item');
      input := input item;
      destination :=
        equivocators_state_project IM
          final_descriptors (destination item);
      output := output item
    |}
with
| Some y => [y]
| None => []
end = [] simpl.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
item: composite_transition_item (equivocator_IM IM)
final_descriptors: equivocator_descriptors IM
final_sub_descriptors:= λ i : sub_index selection,
  final_descriptors (`i): ∀ i : sub_index selection,
  MachineDescriptor (IM (`i))
item': transition_item
descriptor': MachineDescriptor (IM (projT1 (l item)))
Heqpr_item_x: Some (Some item', descriptor') =
equivocator_vlsm_transition_item_project
  (IM (projT1 (l item)))
  {|
    l := projT2 (l item);
    input := input item;
    destination :=
      destination item
        (projT1 (l item));
    output := output item
  |}
  (final_descriptors (projT1 (l item)))
Hl: projT1 (l item) ∉ selection
match
  pre_VLSM_projection_transition_item_project
    (composite_type IM)
    (composite_type (sub_IM IM selection))
    (composite_label_sub_projection_option IM
       selection)
    (composite_state_sub_projection IM selection)
    {|
      l := existT (projT1 (l item)) (l item');
      input := input item;
      destination :=
        equivocators_state_project IM
          final_descriptors (destination item);
      output := output item
    |}
with
| Some y => [y]
| None => []
end = []
      unfold pre_VLSM_projection_transition_item_project,
        composite_label_sub_projection_option.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
item: composite_transition_item (equivocator_IM IM)
final_descriptors: equivocator_descriptors IM
final_sub_descriptors:= λ i : sub_index selection,
  final_descriptors (`i): ∀ i : sub_index selection,
  MachineDescriptor (IM (`i))
item': transition_item
descriptor': MachineDescriptor (IM (projT1 (l item)))
Heqpr_item_x: Some (Some item', descriptor') =
equivocator_vlsm_transition_item_project
  (IM (projT1 (l item)))
  {|
    l := projT2 (l item);
    input := input item;
    destination :=
      destination item
        (projT1 (l item));
    output := output item
  |}
  (final_descriptors (projT1 (l item)))
Hl: projT1 (l item) ∉ selection
match
  match
    match
      decide
        (projT1
           (l
              {|
                l :=
                  existT (projT1 (l item)) (l item');
                input := input item;
                destination :=
                  equivocators_state_project IM
                    final_descriptors
                    (destination item);
                output := output item
              |}) ∈ selection)
    with
    | left i_in =>
        Some
          (composite_label_sub_projection IM selection
             (l
                {|
                  l :=
                    existT (projT1 (l item)) (l item');
                  input := input item;
                  destination :=
                    equivocators_state_project IM
                      final_descriptors
                      (destination item);
                  output := output item
                |}) i_in)
    | right _ => None
    end
  with
  | Some lY =>
      Some
        {|
          l := lY;
          input :=
            input
              {|
                l :=
                  existT (projT1 (l item)) (l item');
                input := input item;
                destination :=
                  equivocators_state_project IM
                    final_descriptors
                    (destination item);
                output := output item
              |};
          destination :=
            composite_state_sub_projection IM
              selection
              (destination
                 {|
                   l :=
                     existT (projT1 (l item))
                       (l item');
                   input := input item;
                   destination :=
                     equivocators_state_project IM
                       final_descriptors
                       (destination item);
                   output := output item
                 |});
          output :=
            output
              {|
                l :=
                  existT (projT1 (l item)) (l item');
                input := input item;
                destination :=
                  equivocators_state_project IM
                    final_descriptors
                    (destination item);
                output := output item
              |}
        |}
  | None => None
  end
with
| Some y => [y]
| None => []
end = []
      by case_decide.
Qed.

Lemma equivocators_trace_project_finite_trace_sub_projection_commute
  (final_descriptors initial_descriptors : equivocator_descriptors IM)
  (initial_sub_descriptors : equivocator_descriptors (sub_IM IM selection))
  (tr : list (composite_transition_item (equivocator_IM IM)))
  (trX : list (composite_transition_item IM))
  (tr_subX : list (composite_transition_item (sub_IM IM selection)))
  (final_sub_descriptors := fun i : sub_index selection => final_descriptors (proj1_sig i))
  (Hproject_tr : equivocators_trace_project IM final_descriptors tr = Some (trX, initial_descriptors))
  (Hproject_sub_tr :
    equivocators_trace_project (sub_IM IM selection) final_sub_descriptors
      (finite_trace_sub_projection (equivocator_IM IM) selection tr)
    = Some (tr_subX, initial_sub_descriptors))
  : initial_sub_descriptors = (fun i => initial_descriptors (proj1_sig i)) /\
    finite_trace_sub_projection IM selection trX = tr_subX.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
final_descriptors, initial_descriptors: equivocator_descriptors IM
initial_sub_descriptors: equivocator_descriptors
  (sub_IM IM selection)
tr: list
  (composite_transition_item (equivocator_IM IM))
trX: list (composite_transition_item IM)
tr_subX: list
  (composite_transition_item
     (sub_IM IM selection))
final_sub_descriptors:= λ i : sub_index selection,
  final_descriptors (`i): ∀ i : sub_index selection,
  MachineDescriptor (IM (`i))
Hproject_tr: equivocators_trace_project IM
  final_descriptors tr =
Some (trX, initial_descriptors)
Hproject_sub_tr: equivocators_trace_project
  (sub_IM IM selection)
  final_sub_descriptors
  (finite_trace_sub_projection
     (equivocator_IM IM) selection
     tr) =
Some
  (tr_subX, initial_sub_descriptors)
initial_sub_descriptors =
(λ i : sub_index selection, initial_descriptors (`i))
∧ finite_trace_sub_projection IM selection trX =
  tr_subX
Proof.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
final_descriptors, initial_descriptors: equivocator_descriptors IM
initial_sub_descriptors: equivocator_descriptors
  (sub_IM IM selection)
tr: list
  (composite_transition_item (equivocator_IM IM))
trX: list (composite_transition_item IM)
tr_subX: list
  (composite_transition_item
     (sub_IM IM selection))
final_sub_descriptors:= λ i : sub_index selection,
  final_descriptors (`i): ∀ i : sub_index selection,
  MachineDescriptor (IM (`i))
Hproject_tr: equivocators_trace_project IM
  final_descriptors tr =
Some (trX, initial_descriptors)
Hproject_sub_tr: equivocators_trace_project
  (sub_IM IM selection)
  final_sub_descriptors
  (finite_trace_sub_projection
     (equivocator_IM IM) selection
     tr) =
Some
  (tr_subX, initial_sub_descriptors)
initial_sub_descriptors =
(λ i : sub_index selection, initial_descriptors (`i))
∧ finite_trace_sub_projection IM selection trX =
  tr_subX
  generalize dependent tr_subX.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
final_descriptors, initial_descriptors: equivocator_descriptors IM
initial_sub_descriptors: equivocator_descriptors
  (sub_IM IM selection)
tr: list
  (composite_transition_item (equivocator_IM IM))
trX: list (composite_transition_item IM)
final_sub_descriptors:= λ i : sub_index selection,
  final_descriptors (`i): ∀ i : sub_index selection,
  MachineDescriptor (IM (`i))
Hproject_tr: equivocators_trace_project IM
  final_descriptors tr =
Some (trX, initial_descriptors)
∀ tr_subX : list
              (composite_transition_item
                 (sub_IM IM selection)),
  equivocators_trace_project (sub_IM IM selection)
    final_sub_descriptors
    (finite_trace_sub_projection (equivocator_IM IM)
       selection tr) =
  Some (tr_subX, initial_sub_descriptors)
  → initial_sub_descriptors =
    (λ i : sub_index selection,
       initial_descriptors (`i))
    ∧ finite_trace_sub_projection IM selection trX =
      tr_subX generalize dependent trX.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
final_descriptors, initial_descriptors: equivocator_descriptors IM
initial_sub_descriptors: equivocator_descriptors
  (sub_IM IM selection)
tr: list
  (composite_transition_item (equivocator_IM IM))
final_sub_descriptors:= λ i : sub_index selection,
  final_descriptors (`i): ∀ i : sub_index selection,
  MachineDescriptor (IM (`i))
∀ trX : list (composite_transition_item IM),
  equivocators_trace_project IM final_descriptors tr =
  Some (trX, initial_descriptors)
  → ∀ tr_subX : list
                  (composite_transition_item
                     (sub_IM IM selection)),
      equivocators_trace_project (sub_IM IM selection)
        final_sub_descriptors
        (finite_trace_sub_projection
           (equivocator_IM IM) selection tr) =
      Some (tr_subX, initial_sub_descriptors)
      → initial_sub_descriptors =
        (λ i : sub_index selection,
           initial_descriptors (`i))
        ∧ finite_trace_sub_projection IM selection trX =
          tr_subX
  generalize dependent final_descriptors.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
initial_descriptors: equivocator_descriptors IM
initial_sub_descriptors: equivocator_descriptors
  (sub_IM IM selection)
tr: list
  (composite_transition_item (equivocator_IM IM))
∀ (final_descriptors : equivocator_descriptors IM) 
  (final_sub_descriptors :=
   λ i : sub_index selection, final_descriptors (`i)) 
  (trX : list (composite_transition_item IM)),
  equivocators_trace_project IM final_descriptors tr =
  Some (trX, initial_descriptors)
  → ∀ tr_subX : list
                  (composite_transition_item
                     (sub_IM IM selection)),
      equivocators_trace_project (sub_IM IM selection)
        final_sub_descriptors
        (finite_trace_sub_projection
           (equivocator_IM IM) selection tr) =
      Some (tr_subX, initial_sub_descriptors)
      → initial_sub_descriptors =
        (λ i : sub_index selection,
           initial_descriptors (`i))
        ∧ finite_trace_sub_projection IM selection trX =
          tr_subX
  induction tr using rev_ind; intros.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
initial_descriptors: equivocator_descriptors IM
initial_sub_descriptors: equivocator_descriptors
  (sub_IM IM selection)
final_descriptors: equivocator_descriptors IM
final_sub_descriptors:= λ i : sub_index selection,
  final_descriptors (`i): ∀ i : sub_index selection,
  MachineDescriptor (IM (`i))
trX: list (composite_transition_item IM)
Hproject_tr: equivocators_trace_project IM
  final_descriptors [] =
Some (trX, initial_descriptors)
tr_subX: list
  (composite_transition_item
     (sub_IM IM selection))
Hproject_sub_tr: equivocators_trace_project
  (sub_IM IM selection)
  final_sub_descriptors
  (finite_trace_sub_projection
     (equivocator_IM IM) selection
     []) =
Some
  (tr_subX, initial_sub_descriptors)
initial_sub_descriptors =
(λ i : sub_index selection, initial_descriptors (`i))
∧ finite_trace_sub_projection IM selection trX =
  tr_subX
message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
initial_descriptors: equivocator_descriptors IM
initial_sub_descriptors: equivocator_descriptors
  (sub_IM IM selection)
x: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
IHtr: ∀ (final_descriptors : 
   equivocator_descriptors IM) 
  (final_sub_descriptors :=
   λ i : sub_index selection,
     final_descriptors (`i)) 
  (trX : list (composite_transition_item IM)),
  equivocators_trace_project IM
    final_descriptors tr =
  Some (trX, initial_descriptors)
  → ∀ tr_subX : list
                  (composite_transition_item
                     (sub_IM IM selection)),
      equivocators_trace_project
        (sub_IM IM selection)
        final_sub_descriptors
        (finite_trace_sub_projection
           (equivocator_IM IM) selection tr) =
      Some (tr_subX, initial_sub_descriptors)
      → initial_sub_descriptors =
        (λ i : sub_index selection,
           initial_descriptors (`i))
        ∧ finite_trace_sub_projection IM
            selection trX = tr_subX
final_descriptors: equivocator_descriptors IM
final_sub_descriptors:= λ i : sub_index selection,
  final_descriptors (`i): ∀ i : sub_index selection,
  MachineDescriptor (IM (`i))
trX: list (composite_transition_item IM)
Hproject_tr: equivocators_trace_project IM
  final_descriptors (tr ++ [x]) =
Some (trX, initial_descriptors)
tr_subX: list
  (composite_transition_item
     (sub_IM IM selection))
Hproject_sub_tr: equivocators_trace_project
  (sub_IM IM selection)
  final_sub_descriptors
  (finite_trace_sub_projection
     (equivocator_IM IM) selection
     (tr ++ [x])) =
Some
  (tr_subX, initial_sub_descriptors)
initial_sub_descriptors =
(λ i : sub_index selection, initial_descriptors (`i))
∧ finite_trace_sub_projection IM selection trX =
  tr_subX
  -message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
initial_descriptors: equivocator_descriptors IM
initial_sub_descriptors: equivocator_descriptors
  (sub_IM IM selection)
final_descriptors: equivocator_descriptors IM
final_sub_descriptors:= λ i : sub_index selection,
  final_descriptors (`i): ∀ i : sub_index selection,
  MachineDescriptor (IM (`i))
trX: list (composite_transition_item IM)
Hproject_tr: equivocators_trace_project IM
  final_descriptors [] =
Some (trX, initial_descriptors)
tr_subX: list
  (composite_transition_item
     (sub_IM IM selection))
Hproject_sub_tr: equivocators_trace_project
  (sub_IM IM selection)
  final_sub_descriptors
  (finite_trace_sub_projection
     (equivocator_IM IM) selection
     []) =
Some
  (tr_subX, initial_sub_descriptors)
initial_sub_descriptors =
(λ i : sub_index selection, initial_descriptors (`i))
∧ finite_trace_sub_projection IM selection trX =
  tr_subX by inversion Hproject_tr; inversion Hproject_sub_tr; subst.
  -message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
initial_descriptors: equivocator_descriptors IM
initial_sub_descriptors: equivocator_descriptors
  (sub_IM IM selection)
x: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
IHtr: ∀ (final_descriptors : 
   equivocator_descriptors IM) 
  (final_sub_descriptors :=
   λ i : sub_index selection,
     final_descriptors (`i)) 
  (trX : list (composite_transition_item IM)),
  equivocators_trace_project IM
    final_descriptors tr =
  Some (trX, initial_descriptors)
  → ∀ tr_subX : list
                  (composite_transition_item
                     (sub_IM IM selection)),
      equivocators_trace_project
        (sub_IM IM selection)
        final_sub_descriptors
        (finite_trace_sub_projection
           (equivocator_IM IM) selection tr) =
      Some (tr_subX, initial_sub_descriptors)
      → initial_sub_descriptors =
        (λ i : sub_index selection,
           initial_descriptors (`i))
        ∧ finite_trace_sub_projection IM
            selection trX = tr_subX
final_descriptors: equivocator_descriptors IM
final_sub_descriptors:= λ i : sub_index selection,
  final_descriptors (`i): ∀ i : sub_index selection,
  MachineDescriptor (IM (`i))
trX: list (composite_transition_item IM)
Hproject_tr: equivocators_trace_project IM
  final_descriptors (tr ++ [x]) =
Some (trX, initial_descriptors)
tr_subX: list
  (composite_transition_item
     (sub_IM IM selection))
Hproject_sub_tr: equivocators_trace_project
  (sub_IM IM selection)
  final_sub_descriptors
  (finite_trace_sub_projection
     (equivocator_IM IM) selection
     (tr ++ [x])) =
Some
  (tr_subX, initial_sub_descriptors)
initial_sub_descriptors =
(λ i : sub_index selection, initial_descriptors (`i))
∧ finite_trace_sub_projection IM selection trX =
  tr_subX unfold equivocators_trace_project in Hproject_tr.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
initial_descriptors: equivocator_descriptors IM
initial_sub_descriptors: equivocator_descriptors
  (sub_IM IM selection)
x: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
IHtr: ∀ (final_descriptors : 
   equivocator_descriptors IM) 
  (final_sub_descriptors :=
   λ i : sub_index selection,
     final_descriptors (`i)) 
  (trX : list (composite_transition_item IM)),
  equivocators_trace_project IM
    final_descriptors tr =
  Some (trX, initial_descriptors)
  → ∀ tr_subX : list
                  (composite_transition_item
                     (sub_IM IM selection)),
      equivocators_trace_project
        (sub_IM IM selection)
        final_sub_descriptors
        (finite_trace_sub_projection
           (equivocator_IM IM) selection tr) =
      Some (tr_subX, initial_sub_descriptors)
      → initial_sub_descriptors =
        (λ i : sub_index selection,
           initial_descriptors (`i))
        ∧ finite_trace_sub_projection IM
            selection trX = tr_subX
final_descriptors: equivocator_descriptors IM
final_sub_descriptors:= λ i : sub_index selection,
  final_descriptors (`i): ∀ i : sub_index selection,
  MachineDescriptor (IM (`i))
trX: list (composite_transition_item IM)
Hproject_tr: foldr
  (equivocators_trace_project_folder IM)
  (Some ([], final_descriptors))
  (tr ++ [x]) =
Some (trX, initial_descriptors)
tr_subX: list
  (composite_transition_item
     (sub_IM IM selection))
Hproject_sub_tr: equivocators_trace_project
  (sub_IM IM selection)
  final_sub_descriptors
  (finite_trace_sub_projection
     (equivocator_IM IM) selection
     (tr ++ [x])) =
Some
  (tr_subX, initial_sub_descriptors)
initial_sub_descriptors =
(λ i : sub_index selection, initial_descriptors (`i))
∧ finite_trace_sub_projection IM selection trX =
  tr_subX
    rewrite fold_right_app in Hproject_tr.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
initial_descriptors: equivocator_descriptors IM
initial_sub_descriptors: equivocator_descriptors
  (sub_IM IM selection)
x: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
IHtr: ∀ (final_descriptors : 
   equivocator_descriptors IM) 
  (final_sub_descriptors :=
   λ i : sub_index selection,
     final_descriptors (`i)) 
  (trX : list (composite_transition_item IM)),
  equivocators_trace_project IM
    final_descriptors tr =
  Some (trX, initial_descriptors)
  → ∀ tr_subX : list
                  (composite_transition_item
                     (sub_IM IM selection)),
      equivocators_trace_project
        (sub_IM IM selection)
        final_sub_descriptors
        (finite_trace_sub_projection
           (equivocator_IM IM) selection tr) =
      Some (tr_subX, initial_sub_descriptors)
      → initial_sub_descriptors =
        (λ i : sub_index selection,
           initial_descriptors (`i))
        ∧ finite_trace_sub_projection IM
            selection trX = tr_subX
final_descriptors: equivocator_descriptors IM
final_sub_descriptors:= λ i : sub_index selection,
  final_descriptors (`i): ∀ i : sub_index selection,
  MachineDescriptor (IM (`i))
trX: list (composite_transition_item IM)
Hproject_tr: foldr
  (equivocators_trace_project_folder IM)
  (foldr
     (equivocators_trace_project_folder
        IM)
     (Some ([], final_descriptors)) [x])
  tr = Some (trX, initial_descriptors)
tr_subX: list
  (composite_transition_item
     (sub_IM IM selection))
Hproject_sub_tr: equivocators_trace_project
  (sub_IM IM selection)
  final_sub_descriptors
  (finite_trace_sub_projection
     (equivocator_IM IM) selection
     (tr ++ [x])) =
Some
  (tr_subX, initial_sub_descriptors)
initial_sub_descriptors =
(λ i : sub_index selection, initial_descriptors (`i))
∧ finite_trace_sub_projection IM selection trX =
  tr_subX
    match type of Hproject_tr with
    | fold_right _ ?i _ = _ => destruct i as [(projectx, final_descriptors') |] eqn: Hproject_x
    end
    ; [| by rewrite equivocators_trace_project_fold_None in Hproject_tr; inversion Hproject_tr].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
initial_descriptors: equivocator_descriptors IM
initial_sub_descriptors: equivocator_descriptors
  (sub_IM IM selection)
x: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
IHtr: ∀ (final_descriptors : 
   equivocator_descriptors IM) 
  (final_sub_descriptors :=
   λ i : sub_index selection,
     final_descriptors (`i)) 
  (trX : list (composite_transition_item IM)),
  equivocators_trace_project IM
    final_descriptors tr =
  Some (trX, initial_descriptors)
  → ∀ tr_subX : list
                  (composite_transition_item
                     (sub_IM IM selection)),
      equivocators_trace_project
        (sub_IM IM selection)
        final_sub_descriptors
        (finite_trace_sub_projection
           (equivocator_IM IM) selection tr) =
      Some (tr_subX, initial_sub_descriptors)
      → initial_sub_descriptors =
        (λ i : sub_index selection,
           initial_descriptors (`i))
        ∧ finite_trace_sub_projection IM
            selection trX = tr_subX
final_descriptors: equivocator_descriptors IM
final_sub_descriptors:= λ i : sub_index selection,
  final_descriptors (`i): ∀ i : sub_index selection,
  MachineDescriptor (IM (`i))
trX, projectx: list (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hproject_x: foldr
  (equivocators_trace_project_folder IM)
  (Some ([], final_descriptors)) [x] =
Some (projectx, final_descriptors')
Hproject_tr: foldr
  (equivocators_trace_project_folder IM)
  (Some (projectx, final_descriptors'))
  tr = Some (trX, initial_descriptors)
tr_subX: list
  (composite_transition_item
     (sub_IM IM selection))
Hproject_sub_tr: equivocators_trace_project
  (sub_IM IM selection)
  final_sub_descriptors
  (finite_trace_sub_projection
     (equivocator_IM IM) selection
     (tr ++ [x])) =
Some
  (tr_subX, initial_sub_descriptors)
initial_sub_descriptors =
(λ i : sub_index selection, initial_descriptors (`i))
∧ finite_trace_sub_projection IM selection trX =
  tr_subX
    apply equivocators_trace_project_folder_additive_iff in Hproject_tr.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
initial_descriptors: equivocator_descriptors IM
initial_sub_descriptors: equivocator_descriptors
  (sub_IM IM selection)
x: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
IHtr: ∀ (final_descriptors : 
   equivocator_descriptors IM) 
  (final_sub_descriptors :=
   λ i : sub_index selection,
     final_descriptors (`i)) 
  (trX : list (composite_transition_item IM)),
  equivocators_trace_project IM
    final_descriptors tr =
  Some (trX, initial_descriptors)
  → ∀ tr_subX : list
                  (composite_transition_item
                     (sub_IM IM selection)),
      equivocators_trace_project
        (sub_IM IM selection)
        final_sub_descriptors
        (finite_trace_sub_projection
           (equivocator_IM IM) selection tr) =
      Some (tr_subX, initial_sub_descriptors)
      → initial_sub_descriptors =
        (λ i : sub_index selection,
           initial_descriptors (`i))
        ∧ finite_trace_sub_projection IM
            selection trX = tr_subX
final_descriptors: equivocator_descriptors IM
final_sub_descriptors:= λ i : sub_index selection,
  final_descriptors (`i): ∀ i : sub_index selection,
  MachineDescriptor (IM (`i))
trX, projectx: list (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hproject_x: foldr
  (equivocators_trace_project_folder IM)
  (Some ([], final_descriptors)) [x] =
Some (projectx, final_descriptors')
Hproject_tr: ∃ trX0 : list
           (composite_transition_item IM),
  foldr
    (equivocators_trace_project_folder
       IM)
    (Some ([], final_descriptors')) tr =
  Some (trX0, initial_descriptors)
  ∧ trX = trX0 ++ projectx
tr_subX: list
  (composite_transition_item
     (sub_IM IM selection))
Hproject_sub_tr: equivocators_trace_project
  (sub_IM IM selection)
  final_sub_descriptors
  (finite_trace_sub_projection
     (equivocator_IM IM) selection
     (tr ++ [x])) =
Some
  (tr_subX, initial_sub_descriptors)
initial_sub_descriptors =
(λ i : sub_index selection, initial_descriptors (`i))
∧ finite_trace_sub_projection IM selection trX =
  tr_subX
    destruct Hproject_tr as [trX0 [HtrX0 HtrX]].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
initial_descriptors: equivocator_descriptors IM
initial_sub_descriptors: equivocator_descriptors
  (sub_IM IM selection)
x: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
IHtr: ∀ (final_descriptors : 
   equivocator_descriptors IM) 
  (final_sub_descriptors :=
   λ i : sub_index selection,
     final_descriptors (`i)) 
  (trX : list (composite_transition_item IM)),
  equivocators_trace_project IM
    final_descriptors tr =
  Some (trX, initial_descriptors)
  → ∀ tr_subX : list
                  (composite_transition_item
                     (sub_IM IM selection)),
      equivocators_trace_project
        (sub_IM IM selection)
        final_sub_descriptors
        (finite_trace_sub_projection
           (equivocator_IM IM) selection tr) =
      Some (tr_subX, initial_sub_descriptors)
      → initial_sub_descriptors =
        (λ i : sub_index selection,
           initial_descriptors (`i))
        ∧ finite_trace_sub_projection IM
            selection trX = tr_subX
final_descriptors: equivocator_descriptors IM
final_sub_descriptors:= λ i : sub_index selection,
  final_descriptors (`i): ∀ i : sub_index selection,
  MachineDescriptor (IM (`i))
trX, projectx: list (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hproject_x: foldr
  (equivocators_trace_project_folder IM)
  (Some ([], final_descriptors)) [x] =
Some (projectx, final_descriptors')
trX0: list (composite_transition_item IM)
HtrX0: foldr (equivocators_trace_project_folder IM)
  (Some ([], final_descriptors')) tr =
Some (trX0, initial_descriptors)
HtrX: trX = trX0 ++ projectx
tr_subX: list
  (composite_transition_item
     (sub_IM IM selection))
Hproject_sub_tr: equivocators_trace_project
  (sub_IM IM selection)
  final_sub_descriptors
  (finite_trace_sub_projection
     (equivocator_IM IM) selection
     (tr ++ [x])) =
Some
  (tr_subX, initial_sub_descriptors)
initial_sub_descriptors =
(λ i : sub_index selection, initial_descriptors (`i))
∧ finite_trace_sub_projection IM selection trX =
  tr_subX
    specialize (IHtr _ _ HtrX0).message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
initial_descriptors: equivocator_descriptors IM
initial_sub_descriptors: equivocator_descriptors
  (sub_IM IM selection)
x: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
final_descriptors': equivocator_descriptors IM
trX0: list (composite_transition_item IM)
IHtr: ∀ tr_subX : list
              (composite_transition_item
                 (sub_IM IM selection)),
  equivocators_trace_project
    (sub_IM IM selection)
    (λ i : sub_index selection,
       final_descriptors' (`i))
    (finite_trace_sub_projection
       (equivocator_IM IM) selection tr) =
  Some (tr_subX, initial_sub_descriptors)
  → initial_sub_descriptors =
    (λ i : sub_index selection,
       initial_descriptors (`i))
    ∧ finite_trace_sub_projection IM selection
        trX0 = tr_subX
final_descriptors: equivocator_descriptors IM
final_sub_descriptors:= λ i : sub_index selection,
  final_descriptors (`i): ∀ i : sub_index selection,
  MachineDescriptor (IM (`i))
trX, projectx: list (composite_transition_item IM)
Hproject_x: foldr
  (equivocators_trace_project_folder IM)
  (Some ([], final_descriptors)) [x] =
Some (projectx, final_descriptors')
HtrX0: foldr (equivocators_trace_project_folder IM)
  (Some ([], final_descriptors')) tr =
Some (trX0, initial_descriptors)
HtrX: trX = trX0 ++ projectx
tr_subX: list
  (composite_transition_item
     (sub_IM IM selection))
Hproject_sub_tr: equivocators_trace_project
  (sub_IM IM selection)
  final_sub_descriptors
  (finite_trace_sub_projection
     (equivocator_IM IM) selection
     (tr ++ [x])) =
Some
  (tr_subX, initial_sub_descriptors)
initial_sub_descriptors =
(λ i : sub_index selection, initial_descriptors (`i))
∧ finite_trace_sub_projection IM selection trX =
  tr_subX
    rewrite finite_trace_sub_projection_app in Hproject_sub_tr.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
initial_descriptors: equivocator_descriptors IM
initial_sub_descriptors: equivocator_descriptors
  (sub_IM IM selection)
x: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
final_descriptors': equivocator_descriptors IM
trX0: list (composite_transition_item IM)
IHtr: ∀ tr_subX : list
              (composite_transition_item
                 (sub_IM IM selection)),
  equivocators_trace_project
    (sub_IM IM selection)
    (λ i : sub_index selection,
       final_descriptors' (`i))
    (finite_trace_sub_projection
       (equivocator_IM IM) selection tr) =
  Some (tr_subX, initial_sub_descriptors)
  → initial_sub_descriptors =
    (λ i : sub_index selection,
       initial_descriptors (`i))
    ∧ finite_trace_sub_projection IM selection
        trX0 = tr_subX
final_descriptors: equivocator_descriptors IM
final_sub_descriptors:= λ i : sub_index selection,
  final_descriptors (`i): ∀ i : sub_index selection,
  MachineDescriptor (IM (`i))
trX, projectx: list (composite_transition_item IM)
Hproject_x: foldr
  (equivocators_trace_project_folder IM)
  (Some ([], final_descriptors)) [x] =
Some (projectx, final_descriptors')
HtrX0: foldr (equivocators_trace_project_folder IM)
  (Some ([], final_descriptors')) tr =
Some (trX0, initial_descriptors)
HtrX: trX = trX0 ++ projectx
tr_subX: list
  (composite_transition_item
     (sub_IM IM selection))
Hproject_sub_tr: equivocators_trace_project
  (sub_IM IM selection)
  final_sub_descriptors
  (finite_trace_sub_projection
     (equivocator_IM IM) selection
     tr ++
   finite_trace_sub_projection
     (equivocator_IM IM) selection
     [x]) =
Some
  (tr_subX, initial_sub_descriptors)
initial_sub_descriptors =
(λ i : sub_index selection, initial_descriptors (`i))
∧ finite_trace_sub_projection IM selection trX =
  tr_subX
    apply equivocators_trace_project_app_iff in Hproject_sub_tr.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
initial_descriptors: equivocator_descriptors IM
initial_sub_descriptors: equivocator_descriptors
  (sub_IM IM selection)
x: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
final_descriptors': equivocator_descriptors IM
trX0: list (composite_transition_item IM)
IHtr: ∀ tr_subX : list
              (composite_transition_item
                 (sub_IM IM selection)),
  equivocators_trace_project
    (sub_IM IM selection)
    (λ i : sub_index selection,
       final_descriptors' (`i))
    (finite_trace_sub_projection
       (equivocator_IM IM) selection tr) =
  Some (tr_subX, initial_sub_descriptors)
  → initial_sub_descriptors =
    (λ i : sub_index selection,
       initial_descriptors (`i))
    ∧ finite_trace_sub_projection IM selection
        trX0 = tr_subX
final_descriptors: equivocator_descriptors IM
final_sub_descriptors:= λ i : sub_index selection,
  final_descriptors (`i): ∀ i : sub_index selection,
  MachineDescriptor (IM (`i))
trX, projectx: list (composite_transition_item IM)
Hproject_x: foldr
  (equivocators_trace_project_folder IM)
  (Some ([], final_descriptors)) [x] =
Some (projectx, final_descriptors')
HtrX0: foldr (equivocators_trace_project_folder IM)
  (Some ([], final_descriptors')) tr =
Some (trX0, initial_descriptors)
HtrX: trX = trX0 ++ projectx
tr_subX: list
  (composite_transition_item
     (sub_IM IM selection))
Hproject_sub_tr: ∃ (preX
   sufX : list
            (composite_transition_item
               (λ i : sub_index
                      selection,
                  IM (`i)))) 
  (eqv_descriptors' : equivocator_descriptors
                      (λ i : 
                      sub_index
                      selection,
                      IM (`i))),
  equivocators_trace_project
    (λ i : sub_index selection,
       IM (`i))
    final_sub_descriptors
    (finite_trace_sub_projection
       (equivocator_IM IM) selection
       [x]) =
  Some (sufX, eqv_descriptors')
  ∧ equivocators_trace_project
      (λ i : sub_index selection,
         IM (`i)) eqv_descriptors'
      (finite_trace_sub_projection
         (equivocator_IM IM)
         selection tr) =
    Some
      (preX, initial_sub_descriptors)
    ∧ tr_subX = preX ++ sufX
initial_sub_descriptors =
(λ i : sub_index selection, initial_descriptors (`i))
∧ finite_trace_sub_projection IM selection trX =
  tr_subX
    destruct Hproject_sub_tr as [tr_subX' [project_sub_x [final_sub_descriptors'
      [Hproject_sub_x [Htr_subX' Heqtr_subX]]]]].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
initial_descriptors: equivocator_descriptors IM
initial_sub_descriptors: equivocator_descriptors
  (sub_IM IM selection)
x: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
final_descriptors': equivocator_descriptors IM
trX0: list (composite_transition_item IM)
IHtr: ∀ tr_subX : list
              (composite_transition_item
                 (sub_IM IM selection)),
  equivocators_trace_project
    (sub_IM IM selection)
    (λ i : sub_index selection,
       final_descriptors' (`i))
    (finite_trace_sub_projection
       (equivocator_IM IM) selection tr) =
  Some (tr_subX, initial_sub_descriptors)
  → initial_sub_descriptors =
    (λ i : sub_index selection,
       initial_descriptors (`i))
    ∧ finite_trace_sub_projection IM selection
        trX0 = tr_subX
final_descriptors: equivocator_descriptors IM
final_sub_descriptors:= λ i : sub_index selection,
  final_descriptors (`i): ∀ i : sub_index selection,
  MachineDescriptor (IM (`i))
trX, projectx: list (composite_transition_item IM)
Hproject_x: foldr
  (equivocators_trace_project_folder IM)
  (Some ([], final_descriptors)) [x] =
Some (projectx, final_descriptors')
HtrX0: foldr (equivocators_trace_project_folder IM)
  (Some ([], final_descriptors')) tr =
Some (trX0, initial_descriptors)
HtrX: trX = trX0 ++ projectx
tr_subX: list
  (composite_transition_item
     (sub_IM IM selection))
tr_subX', project_sub_x: list
  (composite_transition_item
     (λ i : sub_index selection,
        IM (`i)))
final_sub_descriptors': equivocator_descriptors
  (λ i : sub_index selection,
     IM (`i))
Hproject_sub_x: equivocators_trace_project
  (λ i : sub_index selection, IM (`i))
  final_sub_descriptors
  (finite_trace_sub_projection
     (equivocator_IM IM) selection
     [x]) =
Some
  (project_sub_x,
   final_sub_descriptors')
Htr_subX': equivocators_trace_project
  (λ i : sub_index selection, IM (`i))
  final_sub_descriptors'
  (finite_trace_sub_projection
     (equivocator_IM IM) selection tr) =
Some (tr_subX', initial_sub_descriptors)
Heqtr_subX: tr_subX = tr_subX' ++ project_sub_x
initial_sub_descriptors =
(λ i : sub_index selection, initial_descriptors (`i))
∧ finite_trace_sub_projection IM selection trX =
  tr_subX
    specialize
      (equivocators_trace_project_finite_trace_sub_projection_item_commute
        x _ _ _ Hproject_x _ _ Hproject_sub_x)
      as Hfinal_sub'.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
initial_descriptors: equivocator_descriptors IM
initial_sub_descriptors: equivocator_descriptors
  (sub_IM IM selection)
x: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
final_descriptors': equivocator_descriptors IM
trX0: list (composite_transition_item IM)
IHtr: ∀ tr_subX : list
              (composite_transition_item
                 (sub_IM IM selection)),
  equivocators_trace_project
    (sub_IM IM selection)
    (λ i : sub_index selection,
       final_descriptors' (`i))
    (finite_trace_sub_projection
       (equivocator_IM IM) selection tr) =
  Some (tr_subX, initial_sub_descriptors)
  → initial_sub_descriptors =
    (λ i : sub_index selection,
       initial_descriptors (`i))
    ∧ finite_trace_sub_projection IM selection
        trX0 = tr_subX
final_descriptors: equivocator_descriptors IM
final_sub_descriptors:= λ i : sub_index selection,
  final_descriptors (`i): ∀ i : sub_index selection,
  MachineDescriptor (IM (`i))
trX, projectx: list (composite_transition_item IM)
Hproject_x: foldr
  (equivocators_trace_project_folder IM)
  (Some ([], final_descriptors)) [x] =
Some (projectx, final_descriptors')
HtrX0: foldr (equivocators_trace_project_folder IM)
  (Some ([], final_descriptors')) tr =
Some (trX0, initial_descriptors)
HtrX: trX = trX0 ++ projectx
tr_subX: list
  (composite_transition_item
     (sub_IM IM selection))
tr_subX', project_sub_x: list
  (composite_transition_item
     (λ i : sub_index selection,
        IM (`i)))
final_sub_descriptors': equivocator_descriptors
  (λ i : sub_index selection,
     IM (`i))
Hproject_sub_x: equivocators_trace_project
  (λ i : sub_index selection, IM (`i))
  final_sub_descriptors
  (finite_trace_sub_projection
     (equivocator_IM IM) selection
     [x]) =
Some
  (project_sub_x,
   final_sub_descriptors')
Htr_subX': equivocators_trace_project
  (λ i : sub_index selection, IM (`i))
  final_sub_descriptors'
  (finite_trace_sub_projection
     (equivocator_IM IM) selection tr) =
Some (tr_subX', initial_sub_descriptors)
Heqtr_subX: tr_subX = tr_subX' ++ project_sub_x
Hfinal_sub': final_sub_descriptors' =
(λ i : sub_index selection,
   final_descriptors' (`i))
∧ finite_trace_sub_projection IM
    selection projectx = project_sub_x
initial_sub_descriptors =
(λ i : sub_index selection, initial_descriptors (`i))
∧ finite_trace_sub_projection IM selection trX =
  tr_subX

    destruct Hfinal_sub' as [Hfinal_sub' Hpr_sub_x].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
initial_descriptors: equivocator_descriptors IM
initial_sub_descriptors: equivocator_descriptors
  (sub_IM IM selection)
x: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
final_descriptors': equivocator_descriptors IM
trX0: list (composite_transition_item IM)
IHtr: ∀ tr_subX : list
              (composite_transition_item
                 (sub_IM IM selection)),
  equivocators_trace_project
    (sub_IM IM selection)
    (λ i : sub_index selection,
       final_descriptors' (`i))
    (finite_trace_sub_projection
       (equivocator_IM IM) selection tr) =
  Some (tr_subX, initial_sub_descriptors)
  → initial_sub_descriptors =
    (λ i : sub_index selection,
       initial_descriptors (`i))
    ∧ finite_trace_sub_projection IM selection
        trX0 = tr_subX
final_descriptors: equivocator_descriptors IM
final_sub_descriptors:= λ i : sub_index selection,
  final_descriptors (`i): ∀ i : sub_index selection,
  MachineDescriptor (IM (`i))
trX, projectx: list (composite_transition_item IM)
Hproject_x: foldr
  (equivocators_trace_project_folder IM)
  (Some ([], final_descriptors)) [x] =
Some (projectx, final_descriptors')
HtrX0: foldr (equivocators_trace_project_folder IM)
  (Some ([], final_descriptors')) tr =
Some (trX0, initial_descriptors)
HtrX: trX = trX0 ++ projectx
tr_subX: list
  (composite_transition_item
     (sub_IM IM selection))
tr_subX', project_sub_x: list
  (composite_transition_item
     (λ i : sub_index selection,
        IM (`i)))
final_sub_descriptors': equivocator_descriptors
  (λ i : sub_index selection,
     IM (`i))
Hproject_sub_x: equivocators_trace_project
  (λ i : sub_index selection, IM (`i))
  final_sub_descriptors
  (finite_trace_sub_projection
     (equivocator_IM IM) selection
     [x]) =
Some
  (project_sub_x,
   final_sub_descriptors')
Htr_subX': equivocators_trace_project
  (λ i : sub_index selection, IM (`i))
  final_sub_descriptors'
  (finite_trace_sub_projection
     (equivocator_IM IM) selection tr) =
Some (tr_subX', initial_sub_descriptors)
Heqtr_subX: tr_subX = tr_subX' ++ project_sub_x
Hfinal_sub': final_sub_descriptors' =
(λ i : sub_index selection,
   final_descriptors' (`i))
Hpr_sub_x: finite_trace_sub_projection IM selection
  projectx = project_sub_x
initial_sub_descriptors =
(λ i : sub_index selection, initial_descriptors (`i))
∧ finite_trace_sub_projection IM selection trX =
  tr_subX
    subst final_sub_descriptors'.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
initial_descriptors: equivocator_descriptors IM
initial_sub_descriptors: equivocator_descriptors
  (sub_IM IM selection)
x: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
final_descriptors': equivocator_descriptors IM
trX0: list (composite_transition_item IM)
IHtr: ∀ tr_subX : list
              (composite_transition_item
                 (sub_IM IM selection)),
  equivocators_trace_project
    (sub_IM IM selection)
    (λ i : sub_index selection,
       final_descriptors' (`i))
    (finite_trace_sub_projection
       (equivocator_IM IM) selection tr) =
  Some (tr_subX, initial_sub_descriptors)
  → initial_sub_descriptors =
    (λ i : sub_index selection,
       initial_descriptors (`i))
    ∧ finite_trace_sub_projection IM selection
        trX0 = tr_subX
final_descriptors: equivocator_descriptors IM
final_sub_descriptors:= λ i : sub_index selection,
  final_descriptors (`i): ∀ i : sub_index selection,
  MachineDescriptor (IM (`i))
trX, projectx: list (composite_transition_item IM)
Hproject_x: foldr
  (equivocators_trace_project_folder IM)
  (Some ([], final_descriptors)) [x] =
Some (projectx, final_descriptors')
HtrX0: foldr (equivocators_trace_project_folder IM)
  (Some ([], final_descriptors')) tr =
Some (trX0, initial_descriptors)
HtrX: trX = trX0 ++ projectx
tr_subX: list
  (composite_transition_item
     (sub_IM IM selection))
tr_subX', project_sub_x: list
  (composite_transition_item
     (λ i : sub_index selection,
        IM (`i)))
Htr_subX': equivocators_trace_project
  (λ i : sub_index selection, IM (`i))
  (λ i : sub_index selection,
     final_descriptors' (`i))
  (finite_trace_sub_projection
     (equivocator_IM IM) selection tr) =
Some (tr_subX', initial_sub_descriptors)
Hproject_sub_x: equivocators_trace_project
  (λ i : sub_index selection, IM (`i))
  final_sub_descriptors
  (finite_trace_sub_projection
     (equivocator_IM IM) selection
     [x]) =
Some
  (project_sub_x,
   λ i : sub_index selection,
     final_descriptors' (`i))
Heqtr_subX: tr_subX = tr_subX' ++ project_sub_x
Hpr_sub_x: finite_trace_sub_projection IM selection
  projectx = project_sub_x
initial_sub_descriptors =
(λ i : sub_index selection, initial_descriptors (`i))
∧ finite_trace_sub_projection IM selection trX =
  tr_subX
    specialize (IHtr _ Htr_subX').message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
initial_descriptors: equivocator_descriptors IM
initial_sub_descriptors: equivocator_descriptors
  (sub_IM IM selection)
x: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
final_descriptors': equivocator_descriptors IM
trX0: list (composite_transition_item IM)
tr_subX': list
  (composite_transition_item
     (λ i : sub_index selection, IM (`i)))
IHtr: initial_sub_descriptors =
(λ i : sub_index selection,
   initial_descriptors (`i))
∧ finite_trace_sub_projection IM selection trX0 =
  tr_subX'
final_descriptors: equivocator_descriptors IM
final_sub_descriptors:= λ i : sub_index selection,
  final_descriptors (`i): ∀ i : sub_index selection,
  MachineDescriptor (IM (`i))
trX, projectx: list (composite_transition_item IM)
Hproject_x: foldr
  (equivocators_trace_project_folder IM)
  (Some ([], final_descriptors)) [x] =
Some (projectx, final_descriptors')
HtrX0: foldr (equivocators_trace_project_folder IM)
  (Some ([], final_descriptors')) tr =
Some (trX0, initial_descriptors)
HtrX: trX = trX0 ++ projectx
tr_subX: list
  (composite_transition_item
     (sub_IM IM selection))
project_sub_x: list
  (composite_transition_item
     (λ i : sub_index selection,
        IM (`i)))
Htr_subX': equivocators_trace_project
  (λ i : sub_index selection, IM (`i))
  (λ i : sub_index selection,
     final_descriptors' (`i))
  (finite_trace_sub_projection
     (equivocator_IM IM) selection tr) =
Some (tr_subX', initial_sub_descriptors)
Hproject_sub_x: equivocators_trace_project
  (λ i : sub_index selection, IM (`i))
  final_sub_descriptors
  (finite_trace_sub_projection
     (equivocator_IM IM) selection
     [x]) =
Some
  (project_sub_x,
   λ i : sub_index selection,
     final_descriptors' (`i))
Heqtr_subX: tr_subX = tr_subX' ++ project_sub_x
Hpr_sub_x: finite_trace_sub_projection IM selection
  projectx = project_sub_x
initial_sub_descriptors =
(λ i : sub_index selection, initial_descriptors (`i))
∧ finite_trace_sub_projection IM selection trX =
  tr_subX
    destruct IHtr as [Heqv_initial Hpr_trXi'].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
initial_descriptors: equivocator_descriptors IM
initial_sub_descriptors: equivocator_descriptors
  (sub_IM IM selection)
x: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
final_descriptors': equivocator_descriptors IM
trX0: list (composite_transition_item IM)
tr_subX': list
  (composite_transition_item
     (λ i : sub_index selection, IM (`i)))
Heqv_initial: initial_sub_descriptors =
(λ i : sub_index selection,
   initial_descriptors (`i))
Hpr_trXi': finite_trace_sub_projection IM selection
  trX0 = tr_subX'
final_descriptors: equivocator_descriptors IM
final_sub_descriptors:= λ i : sub_index selection,
  final_descriptors (`i): ∀ i : sub_index selection,
  MachineDescriptor (IM (`i))
trX, projectx: list (composite_transition_item IM)
Hproject_x: foldr
  (equivocators_trace_project_folder IM)
  (Some ([], final_descriptors)) [x] =
Some (projectx, final_descriptors')
HtrX0: foldr (equivocators_trace_project_folder IM)
  (Some ([], final_descriptors')) tr =
Some (trX0, initial_descriptors)
HtrX: trX = trX0 ++ projectx
tr_subX: list
  (composite_transition_item
     (sub_IM IM selection))
project_sub_x: list
  (composite_transition_item
     (λ i : sub_index selection,
        IM (`i)))
Htr_subX': equivocators_trace_project
  (λ i : sub_index selection, IM (`i))
  (λ i : sub_index selection,
     final_descriptors' (`i))
  (finite_trace_sub_projection
     (equivocator_IM IM) selection tr) =
Some (tr_subX', initial_sub_descriptors)
Hproject_sub_x: equivocators_trace_project
  (λ i : sub_index selection, IM (`i))
  final_sub_descriptors
  (finite_trace_sub_projection
     (equivocator_IM IM) selection
     [x]) =
Some
  (project_sub_x,
   λ i : sub_index selection,
     final_descriptors' (`i))
Heqtr_subX: tr_subX = tr_subX' ++ project_sub_x
Hpr_sub_x: finite_trace_sub_projection IM selection
  projectx = project_sub_x
initial_sub_descriptors =
(λ i : sub_index selection, initial_descriptors (`i))
∧ finite_trace_sub_projection IM selection trX =
  tr_subX
    split; [done |].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
initial_descriptors: equivocator_descriptors IM
initial_sub_descriptors: equivocator_descriptors
  (sub_IM IM selection)
x: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
final_descriptors': equivocator_descriptors IM
trX0: list (composite_transition_item IM)
tr_subX': list
  (composite_transition_item
     (λ i : sub_index selection, IM (`i)))
Heqv_initial: initial_sub_descriptors =
(λ i : sub_index selection,
   initial_descriptors (`i))
Hpr_trXi': finite_trace_sub_projection IM selection
  trX0 = tr_subX'
final_descriptors: equivocator_descriptors IM
final_sub_descriptors:= λ i : sub_index selection,
  final_descriptors (`i): ∀ i : sub_index selection,
  MachineDescriptor (IM (`i))
trX, projectx: list (composite_transition_item IM)
Hproject_x: foldr
  (equivocators_trace_project_folder IM)
  (Some ([], final_descriptors)) [x] =
Some (projectx, final_descriptors')
HtrX0: foldr (equivocators_trace_project_folder IM)
  (Some ([], final_descriptors')) tr =
Some (trX0, initial_descriptors)
HtrX: trX = trX0 ++ projectx
tr_subX: list
  (composite_transition_item
     (sub_IM IM selection))
project_sub_x: list
  (composite_transition_item
     (λ i : sub_index selection,
        IM (`i)))
Htr_subX': equivocators_trace_project
  (λ i : sub_index selection, IM (`i))
  (λ i : sub_index selection,
     final_descriptors' (`i))
  (finite_trace_sub_projection
     (equivocator_IM IM) selection tr) =
Some (tr_subX', initial_sub_descriptors)
Hproject_sub_x: equivocators_trace_project
  (λ i : sub_index selection, IM (`i))
  final_sub_descriptors
  (finite_trace_sub_projection
     (equivocator_IM IM) selection
     [x]) =
Some
  (project_sub_x,
   λ i : sub_index selection,
     final_descriptors' (`i))
Heqtr_subX: tr_subX = tr_subX' ++ project_sub_x
Hpr_sub_x: finite_trace_sub_projection IM selection
  projectx = project_sub_x
finite_trace_sub_projection IM selection trX = tr_subX
    subst.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
initial_descriptors: equivocator_descriptors IM
x: composite_transition_item (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
final_descriptors': equivocator_descriptors IM
trX0: list (composite_transition_item IM)
final_descriptors: equivocator_descriptors IM
final_sub_descriptors:= λ i : sub_index selection,
  final_descriptors (`i): ∀ i : sub_index selection,
  MachineDescriptor (IM (`i))
projectx: list (composite_transition_item IM)
Hproject_x: foldr
  (equivocators_trace_project_folder IM)
  (Some ([], final_descriptors)) [x] =
Some (projectx, final_descriptors')
HtrX0: foldr (equivocators_trace_project_folder IM)
  (Some ([], final_descriptors')) tr =
Some (trX0, initial_descriptors)
Htr_subX': equivocators_trace_project
  (λ i : sub_index selection, IM (`i))
  (λ i : sub_index selection,
     final_descriptors' (`i))
  (finite_trace_sub_projection
     (equivocator_IM IM) selection tr) =
Some
  (finite_trace_sub_projection IM
     selection trX0,
   λ i : sub_index selection,
     initial_descriptors (`i))
Hproject_sub_x: equivocators_trace_project
  (λ i : sub_index selection, IM (`i))
  final_sub_descriptors
  (finite_trace_sub_projection
     (equivocator_IM IM) selection
     [x]) =
Some
  (finite_trace_sub_projection IM
     selection projectx,
   λ i : sub_index selection,
     final_descriptors' (`i))
finite_trace_sub_projection IM selection
  (trX0 ++ projectx) =
finite_trace_sub_projection IM selection trX0 ++
finite_trace_sub_projection IM selection projectx
    by apply finite_trace_sub_projection_app.
Qed.

Section sec_seeded_equivocators_valid_trace_project.

Context
  (seed : message -> Prop)
  (SeededXE := seeded_equivocators_no_equivocation_vlsm IM selection seed)
  (SubFreeE := free_composite_vlsm (sub_IM (equivocator_IM IM) selection))
  (SubPreFreeE := preloaded_with_all_messages_vlsm SubFreeE)
  (SubFree := free_composite_vlsm (sub_IM IM selection))
  (SeededX := preloaded_vlsm SubFree seed)
  .

Lemma seeded_equivocators_initial_message
  (m : message)
  (Hem : initial_message_prop SeededXE m)
  : initial_message_prop SeededX m.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
m: message
Hem: initial_message_prop m
initial_message_prop m
Proof.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
m: message
Hem: initial_message_prop m
initial_message_prop m
  destruct Hem as [[eqv [emi Hem]] | Hseed].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
m: message
eqv: sub_index selection
emi: initial_message
  (sub_IM (equivocator_IM IM) selection eqv)
Hem: `emi = m
initial_message_prop m
message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
m: message
Hseed: seed m
initial_message_prop m
  -message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
m: message
eqv: sub_index selection
emi: initial_message
  (sub_IM (equivocator_IM IM) selection eqv)
Hem: `emi = m
initial_message_prop m by left; exists eqv, emi.
  -message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
m: message
Hseed: seed m
initial_message_prop m by right.
Qed.

Lemma seeded_no_equivocation_incl_preloaded
  : VLSM_incl SeededXE SubPreFreeE.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
VLSM_incl SeededXE SubPreFreeE
Proof.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
VLSM_incl SeededXE SubPreFreeE
  by apply seeded_no_equivocation_incl_preloaded.
Qed.

Lemma seeded_equivocators_valid_trace_project
  (is : composite_state (sub_IM (equivocator_IM IM) selection))
  (tr : list (composite_transition_item (sub_IM (equivocator_IM IM) selection)))
  (Htr : finite_valid_trace SeededXE is tr)
  (final_state := finite_trace_last is tr)
  (IM' := fun i : sub_index selection => IM (`i))
  (final_descriptors : equivocator_descriptors IM')
  (Hproper : proper_equivocator_descriptors IM' final_descriptors final_state)
  : exists
    (trX : list (composite_transition_item IM'))
    (initial_descriptors : equivocator_descriptors IM')
    (isX := equivocators_state_project IM' initial_descriptors is)
    (final_stateX := finite_trace_last isX trX),
    proper_equivocator_descriptors IM' initial_descriptors is /\
    equivocators_trace_project IM' final_descriptors tr = Some (trX, initial_descriptors) /\
    equivocators_state_project IM' final_descriptors final_state = final_stateX /\
    finite_valid_trace SeededX isX trX.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
is: composite_state
  (sub_IM (equivocator_IM IM) selection)
tr: list
  (composite_transition_item
     (sub_IM (equivocator_IM IM) selection))
Htr: finite_valid_trace SeededXE is tr
final_state:= finite_trace_last is tr: state
  (composite_type
     (sub_IM (equivocator_IM IM)
        selection))
IM':= λ i : sub_index selection, IM (`i): sub_index selection → VLSM message
final_descriptors: equivocator_descriptors IM'
Hproper: proper_equivocator_descriptors IM'
  final_descriptors final_state
∃ (trX : list (composite_transition_item IM')) (initial_descriptors : 
                                                equivocator_descriptors
                                                 IM'),
  let isX :=
    equivocators_state_project IM' initial_descriptors
      is in
  let final_stateX := finite_trace_last isX trX in
  proper_equivocator_descriptors IM'
    initial_descriptors is
  ∧ equivocators_trace_project IM' final_descriptors
      tr = Some (trX, initial_descriptors)
    ∧ equivocators_state_project IM' final_descriptors
        final_state = final_stateX
      ∧ finite_valid_trace SeededX isX trX
Proof.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
is: composite_state
  (sub_IM (equivocator_IM IM) selection)
tr: list
  (composite_transition_item
     (sub_IM (equivocator_IM IM) selection))
Htr: finite_valid_trace SeededXE is tr
final_state:= finite_trace_last is tr: state
  (composite_type
     (sub_IM (equivocator_IM IM)
        selection))
IM':= λ i : sub_index selection, IM (`i): sub_index selection → VLSM message
final_descriptors: equivocator_descriptors IM'
Hproper: proper_equivocator_descriptors IM'
  final_descriptors final_state
∃ (trX : list (composite_transition_item IM')) (initial_descriptors : 
                                                equivocator_descriptors
                                                 IM'),
  let isX :=
    equivocators_state_project IM' initial_descriptors
      is in
  let final_stateX := finite_trace_last isX trX in
  proper_equivocator_descriptors IM'
    initial_descriptors is
  ∧ equivocators_trace_project IM' final_descriptors
      tr = Some (trX, initial_descriptors)
    ∧ equivocators_state_project IM' final_descriptors
        final_state = final_stateX
      ∧ finite_valid_trace SeededX isX trX
  assert (Htr_to : finite_valid_trace_init_to SeededXE is final_state tr).message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
is: composite_state
  (sub_IM (equivocator_IM IM) selection)
tr: list
  (composite_transition_item
     (sub_IM (equivocator_IM IM) selection))
Htr: finite_valid_trace SeededXE is tr
final_state:= finite_trace_last is tr: state
  (composite_type
     (sub_IM (equivocator_IM IM)
        selection))
IM':= λ i : sub_index selection, IM (`i): sub_index selection → VLSM message
final_descriptors: equivocator_descriptors IM'
Hproper: proper_equivocator_descriptors IM'
  final_descriptors final_state
finite_valid_trace_init_to SeededXE is final_state tr
message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
is: composite_state
  (sub_IM (equivocator_IM IM) selection)
tr: list
  (composite_transition_item
     (sub_IM (equivocator_IM IM) selection))
Htr: finite_valid_trace SeededXE is tr
final_state:= finite_trace_last is tr: state
  (composite_type
     (sub_IM (equivocator_IM IM)
        selection))
IM':= λ i : sub_index selection, IM (`i): sub_index selection → VLSM message
final_descriptors: equivocator_descriptors IM'
Hproper: proper_equivocator_descriptors IM'
  final_descriptors final_state
Htr_to: finite_valid_trace_init_to SeededXE is
  final_state tr
∃ (trX : list (composite_transition_item IM')) 
  (initial_descriptors : 
   equivocator_descriptors IM'),
  let isX :=
    equivocators_state_project IM' initial_descriptors
      is in
  let final_stateX := finite_trace_last isX trX in
  proper_equivocator_descriptors IM'
    initial_descriptors is
  ∧ equivocators_trace_project IM' final_descriptors
      tr = Some (trX, initial_descriptors)
    ∧ equivocators_state_project IM' final_descriptors
        final_state = final_stateX
      ∧ finite_valid_trace SeededX isX trX
  {message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
is: composite_state
  (sub_IM (equivocator_IM IM) selection)
tr: list
  (composite_transition_item
     (sub_IM (equivocator_IM IM) selection))
Htr: finite_valid_trace SeededXE is tr
final_state:= finite_trace_last is tr: state
  (composite_type
     (sub_IM (equivocator_IM IM)
        selection))
IM':= λ i : sub_index selection, IM (`i): sub_index selection → VLSM message
final_descriptors: equivocator_descriptors IM'
Hproper: proper_equivocator_descriptors IM'
  final_descriptors final_state
finite_valid_trace_init_to SeededXE is final_state tr destruct Htr as [Htr Hinit].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
is: composite_state
  (sub_IM (equivocator_IM IM) selection)
tr: list
  (composite_transition_item
     (sub_IM (equivocator_IM IM) selection))
Htr: finite_valid_trace_from SeededXE is tr
Hinit: initial_state_prop is
final_state:= finite_trace_last is tr: state
  (composite_type
     (sub_IM (equivocator_IM IM)
        selection))
IM':= λ i : sub_index selection, IM (`i): sub_index selection → VLSM message
final_descriptors: equivocator_descriptors IM'
Hproper: proper_equivocator_descriptors IM'
  final_descriptors final_state
finite_valid_trace_init_to SeededXE is final_state tr split; [| done].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
is: composite_state
  (sub_IM (equivocator_IM IM) selection)
tr: list
  (composite_transition_item
     (sub_IM (equivocator_IM IM) selection))
Htr: finite_valid_trace_from SeededXE is tr
Hinit: initial_state_prop is
final_state:= finite_trace_last is tr: state
  (composite_type
     (sub_IM (equivocator_IM IM)
        selection))
IM':= λ i : sub_index selection, IM (`i): sub_index selection → VLSM message
final_descriptors: equivocator_descriptors IM'
Hproper: proper_equivocator_descriptors IM'
  final_descriptors final_state
finite_valid_trace_from_to SeededXE is final_state tr
    by apply finite_valid_trace_from_add_last.
  }message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
is: composite_state
  (sub_IM (equivocator_IM IM) selection)
tr: list
  (composite_transition_item
     (sub_IM (equivocator_IM IM) selection))
Htr: finite_valid_trace SeededXE is tr
final_state:= finite_trace_last is tr: state
  (composite_type
     (sub_IM (equivocator_IM IM)
        selection))
IM':= λ i : sub_index selection, IM (`i): sub_index selection → VLSM message
final_descriptors: equivocator_descriptors IM'
Hproper: proper_equivocator_descriptors IM'
  final_descriptors final_state
Htr_to: finite_valid_trace_init_to SeededXE is
  final_state tr
∃ (trX : list (composite_transition_item IM')) (initial_descriptors : 
                                                equivocator_descriptors
                                                 IM'),
  let isX :=
    equivocators_state_project IM' initial_descriptors
      is in
  let final_stateX := finite_trace_last isX trX in
  proper_equivocator_descriptors IM'
    initial_descriptors is
  ∧ equivocators_trace_project IM' final_descriptors
      tr = Some (trX, initial_descriptors)
    ∧ equivocators_state_project IM' final_descriptors
        final_state = final_stateX
      ∧ finite_valid_trace SeededX isX trX
  assert (Hpre_tr_to : finite_constrained_trace_init_to SubFreeE is final_state tr).message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
is: composite_state
  (sub_IM (equivocator_IM IM) selection)
tr: list
  (composite_transition_item
     (sub_IM (equivocator_IM IM) selection))
Htr: finite_valid_trace SeededXE is tr
final_state:= finite_trace_last is tr: state
  (composite_type
     (sub_IM (equivocator_IM IM)
        selection))
IM':= λ i : sub_index selection, IM (`i): sub_index selection → VLSM message
final_descriptors: equivocator_descriptors IM'
Hproper: proper_equivocator_descriptors IM'
  final_descriptors final_state
Htr_to: finite_valid_trace_init_to SeededXE is
  final_state tr
finite_constrained_trace_init_to SubFreeE is
  final_state tr
message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
is: composite_state
  (sub_IM (equivocator_IM IM) selection)
tr: list
  (composite_transition_item
     (sub_IM (equivocator_IM IM) selection))
Htr: finite_valid_trace SeededXE is tr
final_state:= finite_trace_last is tr: state
  (composite_type
     (sub_IM (equivocator_IM IM)
        selection))
IM':= λ i : sub_index selection, IM (`i): sub_index selection → VLSM message
final_descriptors: equivocator_descriptors IM'
Hproper: proper_equivocator_descriptors IM'
  final_descriptors final_state
Htr_to: finite_valid_trace_init_to SeededXE is
  final_state tr
Hpre_tr_to: finite_constrained_trace_init_to SubFreeE
  is final_state tr
∃ (trX : list (composite_transition_item IM')) 
  (initial_descriptors : 
   equivocator_descriptors IM'),
  let isX :=
    equivocators_state_project IM' initial_descriptors
      is in
  let final_stateX := finite_trace_last isX trX in
  proper_equivocator_descriptors IM'
    initial_descriptors is
  ∧ equivocators_trace_project IM' final_descriptors
      tr = Some (trX, initial_descriptors)
    ∧ equivocators_state_project IM' final_descriptors
        final_state = final_stateX
      ∧ finite_valid_trace SeededX isX trX
  {message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
is: composite_state
  (sub_IM (equivocator_IM IM) selection)
tr: list
  (composite_transition_item
     (sub_IM (equivocator_IM IM) selection))
Htr: finite_valid_trace SeededXE is tr
final_state:= finite_trace_last is tr: state
  (composite_type
     (sub_IM (equivocator_IM IM)
        selection))
IM':= λ i : sub_index selection, IM (`i): sub_index selection → VLSM message
final_descriptors: equivocator_descriptors IM'
Hproper: proper_equivocator_descriptors IM'
  final_descriptors final_state
Htr_to: finite_valid_trace_init_to SeededXE is
  final_state tr
finite_constrained_trace_init_to SubFreeE is
  final_state tr
    revert Htr_to; apply VLSM_incl_finite_valid_trace_init_to.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
is: composite_state
  (sub_IM (equivocator_IM IM) selection)
tr: list
  (composite_transition_item
     (sub_IM (equivocator_IM IM) selection))
Htr: finite_valid_trace SeededXE is tr
final_state:= finite_trace_last is tr: state
  (composite_type
     (sub_IM (equivocator_IM IM)
        selection))
IM':= λ i : sub_index selection, IM (`i): sub_index selection → VLSM message
final_descriptors: equivocator_descriptors IM'
Hproper: proper_equivocator_descriptors IM'
  final_descriptors final_state
VLSM_incl_part
  (preloaded_vlsm_machine
     (composite_vlsm
        (sub_IM (equivocator_IM IM) selection)
        (no_equivocations_additional_constraint_with_preloaded
           (sub_IM (equivocator_IM IM) selection)
           (free_constraint
              (sub_IM (equivocator_IM IM) selection))
           seed)) seed)
  (preloaded_vlsm_machine SubFreeE
     (λ _ : message, True))
    by apply seeded_no_equivocation_incl_preloaded.
  }message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
is: composite_state
  (sub_IM (equivocator_IM IM) selection)
tr: list
  (composite_transition_item
     (sub_IM (equivocator_IM IM) selection))
Htr: finite_valid_trace SeededXE is tr
final_state:= finite_trace_last is tr: state
  (composite_type
     (sub_IM (equivocator_IM IM)
        selection))
IM':= λ i : sub_index selection, IM (`i): sub_index selection → VLSM message
final_descriptors: equivocator_descriptors IM'
Hproper: proper_equivocator_descriptors IM'
  final_descriptors final_state
Htr_to: finite_valid_trace_init_to SeededXE is
  final_state tr
Hpre_tr_to: finite_constrained_trace_init_to SubFreeE
  is final_state tr
∃ (trX : list (composite_transition_item IM')) (initial_descriptors : 
                                                equivocator_descriptors
                                                 IM'),
  let isX :=
    equivocators_state_project IM' initial_descriptors
      is in
  let final_stateX := finite_trace_last isX trX in
  proper_equivocator_descriptors IM'
    initial_descriptors is
  ∧ equivocators_trace_project IM' final_descriptors
      tr = Some (trX, initial_descriptors)
    ∧ equivocators_state_project IM' final_descriptors
        final_state = final_stateX
      ∧ finite_valid_trace SeededX isX trX
  pose proof (pre_equivocators_valid_trace_project IM' _ _ _
    Hpre_tr_to final_descriptors Hproper) as Hex.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
is: composite_state
  (sub_IM (equivocator_IM IM) selection)
tr: list
  (composite_transition_item
     (sub_IM (equivocator_IM IM) selection))
Htr: finite_valid_trace SeededXE is tr
final_state:= finite_trace_last is tr: state
  (composite_type
     (sub_IM (equivocator_IM IM)
        selection))
IM':= λ i : sub_index selection, IM (`i): sub_index selection → VLSM message
final_descriptors: equivocator_descriptors IM'
Hproper: proper_equivocator_descriptors IM'
  final_descriptors final_state
Htr_to: finite_valid_trace_init_to SeededXE is
  final_state tr
Hpre_tr_to: finite_constrained_trace_init_to SubFreeE
  is final_state tr
Hex: ∃ initial_descriptors : 
   equivocator_descriptors IM',
  proper_equivocator_descriptors IM'
    initial_descriptors is
  ∧ (let isX :=
       equivocators_state_project IM'
         initial_descriptors is in
     let final_stateX :=
       equivocators_state_project IM'
         final_descriptors final_state in
     ∃ trX : list
               (composite_transition_item IM'),
       equivocators_trace_project IM'
         final_descriptors tr =
       Some (trX, initial_descriptors)
       ∧ finite_constrained_trace_init_to
           (free_composite_vlsm IM') isX
           final_stateX trX)
∃ (trX : list (composite_transition_item IM')) (initial_descriptors : 
                                                equivocator_descriptors
                                                 IM'),
  let isX :=
    equivocators_state_project IM' initial_descriptors
      is in
  let final_stateX := finite_trace_last isX trX in
  proper_equivocator_descriptors IM'
    initial_descriptors is
  ∧ equivocators_trace_project IM' final_descriptors
      tr = Some (trX, initial_descriptors)
    ∧ equivocators_state_project IM' final_descriptors
        final_state = final_stateX
      ∧ finite_valid_trace SeededX isX trX
  destruct Hex as [initial_descriptors [Hproper_initial [trX [Hpr_trX Hpre_trX]]]].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
is: composite_state
  (sub_IM (equivocator_IM IM) selection)
tr: list
  (composite_transition_item
     (sub_IM (equivocator_IM IM) selection))
Htr: finite_valid_trace SeededXE is tr
final_state:= finite_trace_last is tr: state
  (composite_type
     (sub_IM (equivocator_IM IM)
        selection))
IM':= λ i : sub_index selection, IM (`i): sub_index selection → VLSM message
final_descriptors: equivocator_descriptors IM'
Hproper: proper_equivocator_descriptors IM'
  final_descriptors final_state
Htr_to: finite_valid_trace_init_to SeededXE is
  final_state tr
Hpre_tr_to: finite_constrained_trace_init_to SubFreeE
  is final_state tr
initial_descriptors: equivocator_descriptors IM'
Hproper_initial: proper_equivocator_descriptors IM'
  initial_descriptors is
trX: list (composite_transition_item IM')
Hpr_trX: equivocators_trace_project IM'
  final_descriptors tr =
Some (trX, initial_descriptors)
Hpre_trX: finite_constrained_trace_init_to
  (free_composite_vlsm IM')
  (equivocators_state_project IM'
     initial_descriptors is)
  (equivocators_state_project IM'
     final_descriptors final_state) trX
∃ (trX : list (composite_transition_item IM')) (initial_descriptors : 
                                                equivocator_descriptors
                                                 IM'),
  let isX :=
    equivocators_state_project IM' initial_descriptors
      is in
  let final_stateX := finite_trace_last isX trX in
  proper_equivocator_descriptors IM'
    initial_descriptors is
  ∧ equivocators_trace_project IM' final_descriptors
      tr = Some (trX, initial_descriptors)
    ∧ equivocators_state_project IM' final_descriptors
        final_state = final_stateX
      ∧ finite_valid_trace SeededX isX trX
  exists trX, initial_descriptors.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
is: composite_state
  (sub_IM (equivocator_IM IM) selection)
tr: list
  (composite_transition_item
     (sub_IM (equivocator_IM IM) selection))
Htr: finite_valid_trace SeededXE is tr
final_state:= finite_trace_last is tr: state
  (composite_type
     (sub_IM (equivocator_IM IM)
        selection))
IM':= λ i : sub_index selection, IM (`i): sub_index selection → VLSM message
final_descriptors: equivocator_descriptors IM'
Hproper: proper_equivocator_descriptors IM'
  final_descriptors final_state
Htr_to: finite_valid_trace_init_to SeededXE is
  final_state tr
Hpre_tr_to: finite_constrained_trace_init_to SubFreeE
  is final_state tr
initial_descriptors: equivocator_descriptors IM'
Hproper_initial: proper_equivocator_descriptors IM'
  initial_descriptors is
trX: list (composite_transition_item IM')
Hpr_trX: equivocators_trace_project IM'
  final_descriptors tr =
Some (trX, initial_descriptors)
Hpre_trX: finite_constrained_trace_init_to
  (free_composite_vlsm IM')
  (equivocators_state_project IM'
     initial_descriptors is)
  (equivocators_state_project IM'
     final_descriptors final_state) trX
let isX :=
  equivocators_state_project IM' initial_descriptors
    is in
let final_stateX := finite_trace_last isX trX in
proper_equivocator_descriptors IM' initial_descriptors
  is
∧ equivocators_trace_project IM' final_descriptors tr =
  Some (trX, initial_descriptors)
  ∧ equivocators_state_project IM' final_descriptors
      final_state = final_stateX
    ∧ finite_valid_trace SeededX isX trX
  split; [done |].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
is: composite_state
  (sub_IM (equivocator_IM IM) selection)
tr: list
  (composite_transition_item
     (sub_IM (equivocator_IM IM) selection))
Htr: finite_valid_trace SeededXE is tr
final_state:= finite_trace_last is tr: state
  (composite_type
     (sub_IM (equivocator_IM IM)
        selection))
IM':= λ i : sub_index selection, IM (`i): sub_index selection → VLSM message
final_descriptors: equivocator_descriptors IM'
Hproper: proper_equivocator_descriptors IM'
  final_descriptors final_state
Htr_to: finite_valid_trace_init_to SeededXE is
  final_state tr
Hpre_tr_to: finite_constrained_trace_init_to SubFreeE
  is final_state tr
initial_descriptors: equivocator_descriptors IM'
Hproper_initial: proper_equivocator_descriptors IM'
  initial_descriptors is
trX: list (composite_transition_item IM')
Hpr_trX: equivocators_trace_project IM'
  final_descriptors tr =
Some (trX, initial_descriptors)
Hpre_trX: finite_constrained_trace_init_to
  (free_composite_vlsm IM')
  (equivocators_state_project IM'
     initial_descriptors is)
  (equivocators_state_project IM'
     final_descriptors final_state) trX
equivocators_trace_project IM' final_descriptors tr =
Some (trX, initial_descriptors)
∧ equivocators_state_project IM' final_descriptors
    final_state =
  finite_trace_last
    (equivocators_state_project IM'
       initial_descriptors is) trX
  ∧ finite_valid_trace SeededX
      (equivocators_state_project IM'
         initial_descriptors is) trX split; [done |].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
is: composite_state
  (sub_IM (equivocator_IM IM) selection)
tr: list
  (composite_transition_item
     (sub_IM (equivocator_IM IM) selection))
Htr: finite_valid_trace SeededXE is tr
final_state:= finite_trace_last is tr: state
  (composite_type
     (sub_IM (equivocator_IM IM)
        selection))
IM':= λ i : sub_index selection, IM (`i): sub_index selection → VLSM message
final_descriptors: equivocator_descriptors IM'
Hproper: proper_equivocator_descriptors IM'
  final_descriptors final_state
Htr_to: finite_valid_trace_init_to SeededXE is
  final_state tr
Hpre_tr_to: finite_constrained_trace_init_to SubFreeE
  is final_state tr
initial_descriptors: equivocator_descriptors IM'
Hproper_initial: proper_equivocator_descriptors IM'
  initial_descriptors is
trX: list (composite_transition_item IM')
Hpr_trX: equivocators_trace_project IM'
  final_descriptors tr =
Some (trX, initial_descriptors)
Hpre_trX: finite_constrained_trace_init_to
  (free_composite_vlsm IM')
  (equivocators_state_project IM'
     initial_descriptors is)
  (equivocators_state_project IM'
     final_descriptors final_state) trX
equivocators_state_project IM' final_descriptors
  final_state =
finite_trace_last
  (equivocators_state_project IM' initial_descriptors
     is) trX
∧ finite_valid_trace SeededX
    (equivocators_state_project IM'
       initial_descriptors is) trX
  apply finite_valid_trace_init_to_last in Hpre_trX as Hfinal_stateX.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
is: composite_state
  (sub_IM (equivocator_IM IM) selection)
tr: list
  (composite_transition_item
     (sub_IM (equivocator_IM IM) selection))
Htr: finite_valid_trace SeededXE is tr
final_state:= finite_trace_last is tr: state
  (composite_type
     (sub_IM (equivocator_IM IM)
        selection))
IM':= λ i : sub_index selection, IM (`i): sub_index selection → VLSM message
final_descriptors: equivocator_descriptors IM'
Hproper: proper_equivocator_descriptors IM'
  final_descriptors final_state
Htr_to: finite_valid_trace_init_to SeededXE is
  final_state tr
Hpre_tr_to: finite_constrained_trace_init_to SubFreeE
  is final_state tr
initial_descriptors: equivocator_descriptors IM'
Hproper_initial: proper_equivocator_descriptors IM'
  initial_descriptors is
trX: list (composite_transition_item IM')
Hpr_trX: equivocators_trace_project IM'
  final_descriptors tr =
Some (trX, initial_descriptors)
Hpre_trX: finite_constrained_trace_init_to
  (free_composite_vlsm IM')
  (equivocators_state_project IM'
     initial_descriptors is)
  (equivocators_state_project IM'
     final_descriptors final_state) trX
Hfinal_stateX: finite_trace_last
  (equivocators_state_project IM'
     initial_descriptors is) trX =
equivocators_state_project IM'
  final_descriptors final_state
equivocators_state_project IM' final_descriptors
  final_state =
finite_trace_last
  (equivocators_state_project IM' initial_descriptors
     is) trX
∧ finite_valid_trace SeededX
    (equivocators_state_project IM'
       initial_descriptors is) trX
  symmetry in Hfinal_stateX.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
is: composite_state
  (sub_IM (equivocator_IM IM) selection)
tr: list
  (composite_transition_item
     (sub_IM (equivocator_IM IM) selection))
Htr: finite_valid_trace SeededXE is tr
final_state:= finite_trace_last is tr: state
  (composite_type
     (sub_IM (equivocator_IM IM)
        selection))
IM':= λ i : sub_index selection, IM (`i): sub_index selection → VLSM message
final_descriptors: equivocator_descriptors IM'
Hproper: proper_equivocator_descriptors IM'
  final_descriptors final_state
Htr_to: finite_valid_trace_init_to SeededXE is
  final_state tr
Hpre_tr_to: finite_constrained_trace_init_to SubFreeE
  is final_state tr
initial_descriptors: equivocator_descriptors IM'
Hproper_initial: proper_equivocator_descriptors IM'
  initial_descriptors is
trX: list (composite_transition_item IM')
Hpr_trX: equivocators_trace_project IM'
  final_descriptors tr =
Some (trX, initial_descriptors)
Hpre_trX: finite_constrained_trace_init_to
  (free_composite_vlsm IM')
  (equivocators_state_project IM'
     initial_descriptors is)
  (equivocators_state_project IM'
     final_descriptors final_state) trX
Hfinal_stateX: equivocators_state_project IM'
  final_descriptors final_state =
finite_trace_last
  (equivocators_state_project IM'
     initial_descriptors is) trX
equivocators_state_project IM' final_descriptors
  final_state =
finite_trace_last
  (equivocators_state_project IM' initial_descriptors
     is) trX
∧ finite_valid_trace SeededX
    (equivocators_state_project IM'
       initial_descriptors is) trX
  split; [done |].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
is: composite_state
  (sub_IM (equivocator_IM IM) selection)
tr: list
  (composite_transition_item
     (sub_IM (equivocator_IM IM) selection))
Htr: finite_valid_trace SeededXE is tr
final_state:= finite_trace_last is tr: state
  (composite_type
     (sub_IM (equivocator_IM IM)
        selection))
IM':= λ i : sub_index selection, IM (`i): sub_index selection → VLSM message
final_descriptors: equivocator_descriptors IM'
Hproper: proper_equivocator_descriptors IM'
  final_descriptors final_state
Htr_to: finite_valid_trace_init_to SeededXE is
  final_state tr
Hpre_tr_to: finite_constrained_trace_init_to SubFreeE
  is final_state tr
initial_descriptors: equivocator_descriptors IM'
Hproper_initial: proper_equivocator_descriptors IM'
  initial_descriptors is
trX: list (composite_transition_item IM')
Hpr_trX: equivocators_trace_project IM'
  final_descriptors tr =
Some (trX, initial_descriptors)
Hpre_trX: finite_constrained_trace_init_to
  (free_composite_vlsm IM')
  (equivocators_state_project IM'
     initial_descriptors is)
  (equivocators_state_project IM'
     final_descriptors final_state) trX
Hfinal_stateX: equivocators_state_project IM'
  final_descriptors final_state =
finite_trace_last
  (equivocators_state_project IM'
     initial_descriptors is) trX
finite_valid_trace SeededX
  (equivocators_state_project IM' initial_descriptors
     is) trX
  clear -SubPreFreeE Htr Hproper Hpr_trX.message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
is: composite_state
  (sub_IM (equivocator_IM IM) selection)
tr: list
  (composite_transition_item
     (sub_IM (equivocator_IM IM) selection))
Htr: finite_valid_trace SeededXE is tr
final_state:= finite_trace_last is tr: state
  (composite_type
     (sub_IM (equivocator_IM IM)
        selection))
IM':= λ i : sub_index selection, IM (`i): sub_index selection → VLSM message
final_descriptors: equivocator_descriptors IM'
Hproper: proper_equivocator_descriptors IM'
  final_descriptors final_state
initial_descriptors: equivocator_descriptors IM'
trX: list (composite_transition_item IM')
Hpr_trX: equivocators_trace_project IM'
  final_descriptors tr =
Some (trX, initial_descriptors)
finite_valid_trace SeededX
  (equivocators_state_project IM' initial_descriptors
     is) trX
  remember (length tr) as len_tr.message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
is: composite_state
  (sub_IM (equivocator_IM IM) selection)
tr: list
  (composite_transition_item
     (sub_IM (equivocator_IM IM) selection))
Htr: finite_valid_trace SeededXE is tr
final_state:= finite_trace_last is tr: state
  (composite_type
     (sub_IM (equivocator_IM IM)
        selection))
IM':= λ i : sub_index selection, IM (`i): sub_index selection → VLSM message
final_descriptors: equivocator_descriptors IM'
Hproper: proper_equivocator_descriptors IM'
  final_descriptors final_state
initial_descriptors: equivocator_descriptors IM'
trX: list (composite_transition_item IM')
Hpr_trX: equivocators_trace_project IM'
  final_descriptors tr =
Some (trX, initial_descriptors)
len_tr: nat
Heqlen_tr: len_tr = length tr
finite_valid_trace SeededX
  (equivocators_state_project IM' initial_descriptors
     is) trX
  generalize dependent trX.message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
is: composite_state
  (sub_IM (equivocator_IM IM) selection)
tr: list
  (composite_transition_item
     (sub_IM (equivocator_IM IM) selection))
Htr: finite_valid_trace SeededXE is tr
final_state:= finite_trace_last is tr: state
  (composite_type
     (sub_IM (equivocator_IM IM)
        selection))
IM':= λ i : sub_index selection, IM (`i): sub_index selection → VLSM message
final_descriptors: equivocator_descriptors IM'
Hproper: proper_equivocator_descriptors IM'
  final_descriptors final_state
initial_descriptors: equivocator_descriptors IM'
len_tr: nat
Heqlen_tr: len_tr = length tr
∀ trX : list (composite_transition_item IM'),
  equivocators_trace_project IM' final_descriptors tr =
  Some (trX, initial_descriptors)
  → finite_valid_trace SeededX
      (equivocators_state_project IM'
         initial_descriptors is) trX
  generalize dependent initial_descriptors.message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
is: composite_state
  (sub_IM (equivocator_IM IM) selection)
tr: list
  (composite_transition_item
     (sub_IM (equivocator_IM IM) selection))
Htr: finite_valid_trace SeededXE is tr
final_state:= finite_trace_last is tr: state
  (composite_type
     (sub_IM (equivocator_IM IM)
        selection))
IM':= λ i : sub_index selection, IM (`i): sub_index selection → VLSM message
final_descriptors: equivocator_descriptors IM'
Hproper: proper_equivocator_descriptors IM'
  final_descriptors final_state
len_tr: nat
Heqlen_tr: len_tr = length tr
∀ (initial_descriptors : equivocator_descriptors IM') 
  (trX : list (composite_transition_item IM')),
  equivocators_trace_project IM' final_descriptors tr =
  Some (trX, initial_descriptors)
  → finite_valid_trace SeededX
      (equivocators_state_project IM'
         initial_descriptors is) trX
  generalize dependent final_descriptors.message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
is: composite_state
  (sub_IM (equivocator_IM IM) selection)
tr: list
  (composite_transition_item
     (sub_IM (equivocator_IM IM) selection))
Htr: finite_valid_trace SeededXE is tr
final_state:= finite_trace_last is tr: state
  (composite_type
     (sub_IM (equivocator_IM IM)
        selection))
IM':= λ i : sub_index selection, IM (`i): sub_index selection → VLSM message
len_tr: nat
Heqlen_tr: len_tr = length tr
∀ final_descriptors : equivocator_descriptors IM',
  proper_equivocator_descriptors IM' final_descriptors
    final_state
  → ∀ (initial_descriptors : equivocator_descriptors
                               IM') (trX : list
                                             (composite_transition_item
                                                IM')),
      equivocators_trace_project IM' final_descriptors
        tr = Some (trX, initial_descriptors)
      → finite_valid_trace SeededX
          (equivocators_state_project IM'
             initial_descriptors is) trX generalize dependent tr.message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
is: composite_state
  (sub_IM (equivocator_IM IM) selection)
IM':= λ i : sub_index selection, IM (`i): sub_index selection → VLSM message
len_tr: nat
∀ tr : list
         (composite_transition_item
            (sub_IM (equivocator_IM IM) selection)),
  finite_valid_trace SeededXE is tr
  → let final_state := finite_trace_last is tr in
    len_tr = length tr
    → ∀ final_descriptors : equivocator_descriptors
                              IM',
        proper_equivocator_descriptors IM'
          final_descriptors final_state
        → ∀ (initial_descriptors : equivocator_descriptors
                                     IM') (trX : list
                                                 (composite_transition_item
                                                 IM')),
            equivocators_trace_project IM'
              final_descriptors tr =
            Some (trX, initial_descriptors)
            → finite_valid_trace SeededX
                (equivocators_state_project IM'
                   initial_descriptors is) trX
  induction len_tr using (well_founded_induction Wf_nat.lt_wf); intros.message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
is: composite_state
  (sub_IM (equivocator_IM IM) selection)
IM':= λ i : sub_index selection, IM (`i): sub_index selection → VLSM message
len_tr: nat
H: ∀ y : nat,
  y < len_tr
  → ∀ tr : list
             (composite_transition_item
                (sub_IM 
                   (equivocator_IM IM) selection)),
      finite_valid_trace SeededXE is tr
      → let final_state := 
          finite_trace_last is tr in
        y = length tr
        → ∀ final_descriptors : 
             equivocator_descriptors IM',
            proper_equivocator_descriptors IM'
              final_descriptors final_state
            → ∀ (initial_descriptors : 
                 equivocator_descriptors IM') 
                (trX : 
                 list
                   (composite_transition_item IM')),
                equivocators_trace_project IM'
                  final_descriptors tr =
                Some (trX, initial_descriptors)
                → finite_valid_trace SeededX
                    (equivocators_state_project
                      IM' initial_descriptors is)
                    trX
tr: list
  (composite_transition_item
     (sub_IM (equivocator_IM IM) selection))
Htr: finite_valid_trace SeededXE is tr
final_state:= finite_trace_last is tr: state
  (composite_type
     (sub_IM (equivocator_IM IM)
        selection))
Heqlen_tr: len_tr = length tr
final_descriptors: equivocator_descriptors IM'
Hproper: proper_equivocator_descriptors IM'
  final_descriptors final_state
initial_descriptors: equivocator_descriptors IM'
trX: list (composite_transition_item IM')
Hpr_trX: equivocators_trace_project IM'
  final_descriptors tr =
Some (trX, initial_descriptors)
finite_valid_trace SeededX
  (equivocators_state_project IM' initial_descriptors
     is) trX
  subst len_tr.message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
is: composite_state
  (sub_IM (equivocator_IM IM) selection)
IM':= λ i : sub_index selection, IM (`i): sub_index selection → VLSM message
tr: list
  (composite_transition_item
     (sub_IM (equivocator_IM IM) selection))
H: ∀ y : nat,
  y < length tr
  → ∀ tr : list
             (composite_transition_item
                (sub_IM 
                   (equivocator_IM IM) selection)),
      finite_valid_trace SeededXE is tr
      → let final_state := 
          finite_trace_last is tr in
        y = length tr
        → ∀ final_descriptors : 
             equivocator_descriptors IM',
            proper_equivocator_descriptors IM'
              final_descriptors final_state
            → ∀ (initial_descriptors : 
                 equivocator_descriptors IM') 
                (trX : 
                 list
                   (composite_transition_item IM')),
                equivocators_trace_project IM'
                  final_descriptors tr =
                Some (trX, initial_descriptors)
                → finite_valid_trace SeededX
                    (equivocators_state_project
                      IM' initial_descriptors is)
                    trX
Htr: finite_valid_trace SeededXE is tr
final_state:= finite_trace_last is tr: state
  (composite_type
     (sub_IM (equivocator_IM IM)
        selection))
final_descriptors: equivocator_descriptors IM'
Hproper: proper_equivocator_descriptors IM'
  final_descriptors final_state
initial_descriptors: equivocator_descriptors IM'
trX: list (composite_transition_item IM')
Hpr_trX: equivocators_trace_project IM'
  final_descriptors tr =
Some (trX, initial_descriptors)
finite_valid_trace SeededX
  (equivocators_state_project IM' initial_descriptors
     is) trX
  destruct_list_last tr tr' lst Htr_lst.message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
is: composite_state
  (sub_IM (equivocator_IM IM) selection)
IM':= λ i : sub_index selection, IM (`i): sub_index selection → VLSM message
tr: list
  (composite_transition_item
     (sub_IM (equivocator_IM IM) selection))
H: ∀ y : nat,
  y < length []
  → ∀ tr : list
             (composite_transition_item
                (sub_IM 
                   (equivocator_IM IM) selection)),
      finite_valid_trace SeededXE is tr
      → let final_state := 
          finite_trace_last is tr in
        y = length tr
        → ∀ final_descriptors : 
             equivocator_descriptors IM',
            proper_equivocator_descriptors IM'
              final_descriptors final_state
            → ∀ (initial_descriptors : 
                 equivocator_descriptors IM') 
                (trX : 
                 list
                   (composite_transition_item IM')),
                equivocators_trace_project IM'
                  final_descriptors tr =
                Some (trX, initial_descriptors)
                → finite_valid_trace SeededX
                    (equivocators_state_project
                      IM' initial_descriptors is)
                    trX
Htr: finite_valid_trace SeededXE is []
final_state:= finite_trace_last is tr: state
  (composite_type
     (sub_IM (equivocator_IM IM)
        selection))
final_descriptors: equivocator_descriptors IM'
Hproper: proper_equivocator_descriptors IM'
  final_descriptors final_state
initial_descriptors: equivocator_descriptors IM'
trX: list (composite_transition_item IM')
Hpr_trX: equivocators_trace_project IM'
  final_descriptors [] =
Some (trX, initial_descriptors)
Htr_lst: tr = []
finite_valid_trace SeededX
  (equivocators_state_project IM' initial_descriptors
     is) trX
message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
is: composite_state
  (sub_IM (equivocator_IM IM) selection)
IM':= λ i : sub_index selection, IM (`i): sub_index selection → VLSM message
tr, tr': list
  (composite_transition_item
     (sub_IM (equivocator_IM IM) selection))
lst: composite_transition_item
  (sub_IM (equivocator_IM IM) selection)
H: ∀ y : nat,
  y < length (tr' ++ [lst])
  → ∀ tr : list
             (composite_transition_item
                (sub_IM 
                   (equivocator_IM IM) selection)),
      finite_valid_trace SeededXE is tr
      → let final_state := 
          finite_trace_last is tr in
        y = length tr
        → ∀ final_descriptors : 
             equivocator_descriptors IM',
            proper_equivocator_descriptors IM'
              final_descriptors final_state
            → ∀ (initial_descriptors : 
                 equivocator_descriptors IM') 
                (trX : 
                 list
                   (composite_transition_item IM')),
                equivocators_trace_project IM'
                  final_descriptors tr =
                Some (trX, initial_descriptors)
                → finite_valid_trace SeededX
                    (equivocators_state_project
                      IM' initial_descriptors is)
                    trX
Htr: finite_valid_trace SeededXE is (tr' ++ [lst])
final_state:= finite_trace_last is tr: state
  (composite_type
     (sub_IM (equivocator_IM IM)
        selection))
final_descriptors: equivocator_descriptors IM'
Hproper: proper_equivocator_descriptors IM'
  final_descriptors final_state
initial_descriptors: equivocator_descriptors IM'
trX: list (composite_transition_item IM')
Hpr_trX: equivocators_trace_project IM'
  final_descriptors (tr' ++ [lst]) =
Some (trX, initial_descriptors)
Htr_lst: tr = tr' ++ [lst]
finite_valid_trace SeededX
  (equivocators_state_project IM' initial_descriptors
     is) trX
  -message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
is: composite_state
  (sub_IM (equivocator_IM IM) selection)
IM':= λ i : sub_index selection, IM (`i): sub_index selection → VLSM message
tr: list
  (composite_transition_item
     (sub_IM (equivocator_IM IM) selection))
H: ∀ y : nat,
  y < length []
  → ∀ tr : list
             (composite_transition_item
                (sub_IM 
                   (equivocator_IM IM) selection)),
      finite_valid_trace SeededXE is tr
      → let final_state := 
          finite_trace_last is tr in
        y = length tr
        → ∀ final_descriptors : 
             equivocator_descriptors IM',
            proper_equivocator_descriptors IM'
              final_descriptors final_state
            → ∀ (initial_descriptors : 
                 equivocator_descriptors IM') 
                (trX : 
                 list
                   (composite_transition_item IM')),
                equivocators_trace_project IM'
                  final_descriptors tr =
                Some (trX, initial_descriptors)
                → finite_valid_trace SeededX
                    (equivocators_state_project
                      IM' initial_descriptors is)
                    trX
Htr: finite_valid_trace SeededXE is []
final_state:= finite_trace_last is tr: state
  (composite_type
     (sub_IM (equivocator_IM IM)
        selection))
final_descriptors: equivocator_descriptors IM'
Hproper: proper_equivocator_descriptors IM'
  final_descriptors final_state
initial_descriptors: equivocator_descriptors IM'
trX: list (composite_transition_item IM')
Hpr_trX: equivocators_trace_project IM'
  final_descriptors [] =
Some (trX, initial_descriptors)
Htr_lst: tr = []
finite_valid_trace SeededX
  (equivocators_state_project IM' initial_descriptors
     is) trX clear H.message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
is: composite_state
  (sub_IM (equivocator_IM IM) selection)
IM':= λ i : sub_index selection, IM (`i): sub_index selection → VLSM message
tr: list
  (composite_transition_item
     (sub_IM (equivocator_IM IM) selection))
Htr: finite_valid_trace SeededXE is []
final_state:= finite_trace_last is tr: state
  (composite_type
     (sub_IM (equivocator_IM IM)
        selection))
final_descriptors: equivocator_descriptors IM'
Hproper: proper_equivocator_descriptors IM'
  final_descriptors final_state
initial_descriptors: equivocator_descriptors IM'
trX: list (composite_transition_item IM')
Hpr_trX: equivocators_trace_project IM'
  final_descriptors [] =
Some (trX, initial_descriptors)
Htr_lst: tr = []
finite_valid_trace SeededX
  (equivocators_state_project IM' initial_descriptors
     is) trX subst.message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
is: composite_state
  (sub_IM (equivocator_IM IM) selection)
IM':= λ i : sub_index selection, IM (`i): sub_index selection → VLSM message
Htr: finite_valid_trace SeededXE is []
final_state:= finite_trace_last is []: state
  (composite_type
     (sub_IM (equivocator_IM IM)
        selection))
final_descriptors: equivocator_descriptors IM'
Hproper: proper_equivocator_descriptors IM'
  final_descriptors final_state
initial_descriptors: equivocator_descriptors IM'
trX: list (composite_transition_item IM')
Hpr_trX: equivocators_trace_project IM'
  final_descriptors [] =
Some (trX, initial_descriptors)
finite_valid_trace SeededX
  (equivocators_state_project IM' initial_descriptors
     is) trX subst final_state.message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
is: composite_state
  (sub_IM (equivocator_IM IM) selection)
IM':= λ i : sub_index selection, IM (`i): sub_index selection → VLSM message
Htr: finite_valid_trace SeededXE is []
final_descriptors: equivocator_descriptors IM'
Hproper: proper_equivocator_descriptors IM'
  final_descriptors
  (finite_trace_last is [])
initial_descriptors: equivocator_descriptors IM'
trX: list (composite_transition_item IM')
Hpr_trX: equivocators_trace_project IM'
  final_descriptors [] =
Some (trX, initial_descriptors)
finite_valid_trace SeededX
  (equivocators_state_project IM' initial_descriptors
     is) trX simpl in *.message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
is: composite_state
  (sub_IM (equivocator_IM IM) selection)
IM':= λ i : sub_index selection, IM (`i): sub_index selection → VLSM message
Htr: finite_valid_trace SeededXE is []
final_descriptors: equivocator_descriptors IM'
Hproper: proper_equivocator_descriptors IM'
  final_descriptors
  (finite_trace_last is [])
initial_descriptors: equivocator_descriptors IM'
trX: list (composite_transition_item IM')
Hpr_trX: Some ([], final_descriptors) =
Some (trX, initial_descriptors)
finite_valid_trace SeededX
  (equivocators_state_project IM' initial_descriptors
     is) trX inversion Hpr_trX.message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
is: composite_state
  (sub_IM (equivocator_IM IM) selection)
IM':= λ i : sub_index selection, IM (`i): sub_index selection → VLSM message
Htr: finite_valid_trace SeededXE is []
final_descriptors: equivocator_descriptors IM'
Hproper: proper_equivocator_descriptors IM'
  final_descriptors
  (finite_trace_last is [])
initial_descriptors: equivocator_descriptors IM'
trX: list (composite_transition_item IM')
Hpr_trX: Some ([], final_descriptors) =
Some (trX, initial_descriptors)
H1: [] = trX
H2: final_descriptors = initial_descriptors
finite_valid_trace SeededX
  (equivocators_state_project IM' initial_descriptors
     is) [] subst.message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
is: composite_state
  (sub_IM (equivocator_IM IM) selection)
IM':= λ i : sub_index selection, IM (`i): sub_index selection → VLSM message
Htr: finite_valid_trace SeededXE is []
initial_descriptors: equivocator_descriptors IM'
Hpr_trX: Some ([], initial_descriptors) =
Some ([], initial_descriptors)
Hproper: proper_equivocator_descriptors IM'
  initial_descriptors
  (finite_trace_last is [])
finite_valid_trace SeededX
  (equivocators_state_project IM' initial_descriptors
     is) []
    cut (initial_state_prop SubFree (equivocators_state_project _ initial_descriptors is)).message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
is: composite_state
  (sub_IM (equivocator_IM IM) selection)
IM':= λ i : sub_index selection, IM (`i): sub_index selection → VLSM message
Htr: finite_valid_trace SeededXE is []
initial_descriptors: equivocator_descriptors IM'
Hpr_trX: Some ([], initial_descriptors) =
Some ([], initial_descriptors)
Hproper: proper_equivocator_descriptors IM'
  initial_descriptors
  (finite_trace_last is [])
initial_state_prop
  (equivocators_state_project IM' initial_descriptors
     is)
→ finite_valid_trace SeededX
    (equivocators_state_project IM'
       initial_descriptors is) []
message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
is: composite_state
  (sub_IM (equivocator_IM IM) selection)
IM':= λ i : sub_index selection, IM (`i): sub_index selection → VLSM message
Htr: finite_valid_trace SeededXE is []
initial_descriptors: equivocator_descriptors IM'
Hpr_trX: Some ([], initial_descriptors) =
Some ([], initial_descriptors)
Hproper: proper_equivocator_descriptors IM'
  initial_descriptors
  (finite_trace_last is [])
initial_state_prop
  (equivocators_state_project IM' initial_descriptors
     is)
    {message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
is: composite_state
  (sub_IM (equivocator_IM IM) selection)
IM':= λ i : sub_index selection, IM (`i): sub_index selection → VLSM message
Htr: finite_valid_trace SeededXE is []
initial_descriptors: equivocator_descriptors IM'
Hpr_trX: Some ([], initial_descriptors) =
Some ([], initial_descriptors)
Hproper: proper_equivocator_descriptors IM'
  initial_descriptors
  (finite_trace_last is [])
initial_state_prop
  (equivocators_state_project IM' initial_descriptors
     is)
→ finite_valid_trace SeededX
    (equivocators_state_project IM'
       initial_descriptors is) [] intro.message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
is: composite_state
  (sub_IM (equivocator_IM IM) selection)
IM':= λ i : sub_index selection, IM (`i): sub_index selection → VLSM message
Htr: finite_valid_trace SeededXE is []
initial_descriptors: equivocator_descriptors IM'
Hpr_trX: Some ([], initial_descriptors) =
Some ([], initial_descriptors)
Hproper: proper_equivocator_descriptors IM'
  initial_descriptors
  (finite_trace_last is [])
H: initial_state_prop
  (equivocators_state_project IM'
     initial_descriptors is)
finite_valid_trace SeededX
  (equivocators_state_project IM' initial_descriptors
     is) [] split; [| done].message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
is: composite_state
  (sub_IM (equivocator_IM IM) selection)
IM':= λ i : sub_index selection, IM (`i): sub_index selection → VLSM message
Htr: finite_valid_trace SeededXE is []
initial_descriptors: equivocator_descriptors IM'
Hpr_trX: Some ([], initial_descriptors) =
Some ([], initial_descriptors)
Hproper: proper_equivocator_descriptors IM'
  initial_descriptors
  (finite_trace_last is [])
H: initial_state_prop
  (equivocators_state_project IM'
     initial_descriptors is)
finite_valid_trace_from SeededX
  (equivocators_state_project IM' initial_descriptors
     is) [] constructor.message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
is: composite_state
  (sub_IM (equivocator_IM IM) selection)
IM':= λ i : sub_index selection, IM (`i): sub_index selection → VLSM message
Htr: finite_valid_trace SeededXE is []
initial_descriptors: equivocator_descriptors IM'
Hpr_trX: Some ([], initial_descriptors) =
Some ([], initial_descriptors)
Hproper: proper_equivocator_descriptors IM'
  initial_descriptors
  (finite_trace_last is [])
H: initial_state_prop
  (equivocators_state_project IM'
     initial_descriptors is)
valid_state_prop SeededX
  (equivocators_state_project IM' initial_descriptors
     is)
      apply valid_state_prop_iff.message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
is: composite_state
  (sub_IM (equivocator_IM IM) selection)
IM':= λ i : sub_index selection, IM (`i): sub_index selection → VLSM message
Htr: finite_valid_trace SeededXE is []
initial_descriptors: equivocator_descriptors IM'
Hpr_trX: Some ([], initial_descriptors) =
Some ([], initial_descriptors)
Hproper: proper_equivocator_descriptors IM'
  initial_descriptors
  (finite_trace_last is [])
H: initial_state_prop
  (equivocators_state_project IM'
     initial_descriptors is)
(∃ is0 : initial_state SeededX,
   equivocators_state_project IM' initial_descriptors
     is = `is0)
∨ (∃ (l : label SeededX) (som : state SeededX *
                                option message) (om' : 
                                                 option
                                                 message),
     input_valid_transition SeededX l som
       (equivocators_state_project IM'
          initial_descriptors is, om')) left.message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
is: composite_state
  (sub_IM (equivocator_IM IM) selection)
IM':= λ i : sub_index selection, IM (`i): sub_index selection → VLSM message
Htr: finite_valid_trace SeededXE is []
initial_descriptors: equivocator_descriptors IM'
Hpr_trX: Some ([], initial_descriptors) =
Some ([], initial_descriptors)
Hproper: proper_equivocator_descriptors IM'
  initial_descriptors
  (finite_trace_last is [])
H: initial_state_prop
  (equivocators_state_project IM'
     initial_descriptors is)
∃ is0 : initial_state SeededX,
  equivocators_state_project IM' initial_descriptors
    is = `is0
      by exists (exist _ _ H).
    }message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
is: composite_state
  (sub_IM (equivocator_IM IM) selection)
IM':= λ i : sub_index selection, IM (`i): sub_index selection → VLSM message
Htr: finite_valid_trace SeededXE is []
initial_descriptors: equivocator_descriptors IM'
Hpr_trX: Some ([], initial_descriptors) =
Some ([], initial_descriptors)
Hproper: proper_equivocator_descriptors IM'
  initial_descriptors
  (finite_trace_last is [])
initial_state_prop
  (equivocators_state_project IM' initial_descriptors
     is)
    apply equivocators_initial_state_project; [| done].message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
is: composite_state
  (sub_IM (equivocator_IM IM) selection)
IM':= λ i : sub_index selection, IM (`i): sub_index selection → VLSM message
Htr: finite_valid_trace SeededXE is []
initial_descriptors: equivocator_descriptors IM'
Hpr_trX: Some ([], initial_descriptors) =
Some ([], initial_descriptors)
Hproper: proper_equivocator_descriptors IM'
  initial_descriptors
  (finite_trace_last is [])
initial_state_prop is
    by apply Htr.
  -message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
is: composite_state
  (sub_IM (equivocator_IM IM) selection)
IM':= λ i : sub_index selection, IM (`i): sub_index selection → VLSM message
tr, tr': list
  (composite_transition_item
     (sub_IM (equivocator_IM IM) selection))
lst: composite_transition_item
  (sub_IM (equivocator_IM IM) selection)
H: ∀ y : nat,
  y < length (tr' ++ [lst])
  → ∀ tr : list
             (composite_transition_item
                (sub_IM 
                   (equivocator_IM IM) selection)),
      finite_valid_trace SeededXE is tr
      → let final_state := 
          finite_trace_last is tr in
        y = length tr
        → ∀ final_descriptors : 
             equivocator_descriptors IM',
            proper_equivocator_descriptors IM'
              final_descriptors final_state
            → ∀ (initial_descriptors : 
                 equivocator_descriptors IM') 
                (trX : 
                 list
                   (composite_transition_item IM')),
                equivocators_trace_project IM'
                  final_descriptors tr =
                Some (trX, initial_descriptors)
                → finite_valid_trace SeededX
                    (equivocators_state_project
                      IM' initial_descriptors is)
                    trX
Htr: finite_valid_trace SeededXE is (tr' ++ [lst])
final_state:= finite_trace_last is tr: state
  (composite_type
     (sub_IM (equivocator_IM IM)
        selection))
final_descriptors: equivocator_descriptors IM'
Hproper: proper_equivocator_descriptors IM'
  final_descriptors final_state
initial_descriptors: equivocator_descriptors IM'
trX: list (composite_transition_item IM')
Hpr_trX: equivocators_trace_project IM'
  final_descriptors (tr' ++ [lst]) =
Some (trX, initial_descriptors)
Htr_lst: tr = tr' ++ [lst]
finite_valid_trace SeededX
  (equivocators_state_project IM' initial_descriptors
     is) trX specialize (H (length tr')) as H'.message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
is: composite_state
  (sub_IM (equivocator_IM IM) selection)
IM':= λ i : sub_index selection, IM (`i): sub_index selection → VLSM message
tr, tr': list
  (composite_transition_item
     (sub_IM (equivocator_IM IM) selection))
lst: composite_transition_item
  (sub_IM (equivocator_IM IM) selection)
H: ∀ y : nat,
  y < length (tr' ++ [lst])
  → ∀ tr : list
             (composite_transition_item
                (sub_IM 
                   (equivocator_IM IM) selection)),
      finite_valid_trace SeededXE is tr
      → let final_state := 
          finite_trace_last is tr in
        y = length tr
        → ∀ final_descriptors : 
             equivocator_descriptors IM',
            proper_equivocator_descriptors IM'
              final_descriptors final_state
            → ∀ (initial_descriptors : 
                 equivocator_descriptors IM') 
                (trX : 
                 list
                   (composite_transition_item IM')),
                equivocators_trace_project IM'
                  final_descriptors tr =
                Some (trX, initial_descriptors)
                → finite_valid_trace SeededX
                    (equivocators_state_project
                      IM' initial_descriptors is)
                    trX
Htr: finite_valid_trace SeededXE is (tr' ++ [lst])
final_state:= finite_trace_last is tr: state
  (composite_type
     (sub_IM (equivocator_IM IM)
        selection))
final_descriptors: equivocator_descriptors IM'
Hproper: proper_equivocator_descriptors IM'
  final_descriptors final_state
initial_descriptors: equivocator_descriptors IM'
trX: list (composite_transition_item IM')
Hpr_trX: equivocators_trace_project IM'
  final_descriptors (tr' ++ [lst]) =
Some (trX, initial_descriptors)
Htr_lst: tr = tr' ++ [lst]
H': length tr' < length (tr' ++ [lst])
→ ∀ tr : list
           (composite_transition_item
              (sub_IM (equivocator_IM IM)
                 selection)),
    finite_valid_trace SeededXE is tr
    → let final_state := 
        finite_trace_last is tr in
      length tr' = length tr
      → ∀ final_descriptors : 
           equivocator_descriptors IM',
          proper_equivocator_descriptors IM'
            final_descriptors final_state
          → ∀ (initial_descriptors : 
               equivocator_descriptors IM') 
              (trX : list
                      (composite_transition_item
                      IM')),
              equivocators_trace_project IM'
                final_descriptors tr =
              Some (trX, initial_descriptors)
              → finite_valid_trace SeededX
                  (equivocators_state_project IM'
                     initial_descriptors is) trX
finite_valid_trace SeededX
  (equivocators_state_project IM' initial_descriptors
     is) trX
    spec H'; [by rewrite app_length; cbn; lia |].message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
is: composite_state
  (sub_IM (equivocator_IM IM) selection)
IM':= λ i : sub_index selection, IM (`i): sub_index selection → VLSM message
tr, tr': list
  (composite_transition_item
     (sub_IM (equivocator_IM IM) selection))
lst: composite_transition_item
  (sub_IM (equivocator_IM IM) selection)
H: ∀ y : nat,
  y < length (tr' ++ [lst])
  → ∀ tr : list
             (composite_transition_item
                (sub_IM 
                   (equivocator_IM IM) selection)),
      finite_valid_trace SeededXE is tr
      → let final_state := 
          finite_trace_last is tr in
        y = length tr
        → ∀ final_descriptors : 
             equivocator_descriptors IM',
            proper_equivocator_descriptors IM'
              final_descriptors final_state
            → ∀ (initial_descriptors : 
                 equivocator_descriptors IM') 
                (trX : 
                 list
                   (composite_transition_item IM')),
                equivocators_trace_project IM'
                  final_descriptors tr =
                Some (trX, initial_descriptors)
                → finite_valid_trace SeededX
                    (equivocators_state_project
                      IM' initial_descriptors is)
                    trX
Htr: finite_valid_trace SeededXE is (tr' ++ [lst])
final_state:= finite_trace_last is tr: state
  (composite_type
     (sub_IM (equivocator_IM IM)
        selection))
final_descriptors: equivocator_descriptors IM'
Hproper: proper_equivocator_descriptors IM'
  final_descriptors final_state
initial_descriptors: equivocator_descriptors IM'
trX: list (composite_transition_item IM')
Hpr_trX: equivocators_trace_project IM'
  final_descriptors (tr' ++ [lst]) =
Some (trX, initial_descriptors)
Htr_lst: tr = tr' ++ [lst]
H': ∀ tr : list
         (composite_transition_item
            (sub_IM (equivocator_IM IM) selection)),
  finite_valid_trace SeededXE is tr
  → let final_state := finite_trace_last is tr in
    length tr' = length tr
    → ∀ final_descriptors : 
         equivocator_descriptors IM',
        proper_equivocator_descriptors IM'
          final_descriptors final_state
        → ∀ (initial_descriptors : 
             equivocator_descriptors IM') 
            (trX : list
                     (composite_transition_item
                      IM')),
            equivocators_trace_project IM'
              final_descriptors tr =
            Some (trX, initial_descriptors)
            → finite_valid_trace SeededX
                (equivocators_state_project IM'
                   initial_descriptors is) trX
finite_valid_trace SeededX
  (equivocators_state_project IM' initial_descriptors
     is) trX
    destruct Htr as [Htr Hinit].message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
is: composite_state
  (sub_IM (equivocator_IM IM) selection)
IM':= λ i : sub_index selection, IM (`i): sub_index selection → VLSM message
tr, tr': list
  (composite_transition_item
     (sub_IM (equivocator_IM IM) selection))
lst: composite_transition_item
  (sub_IM (equivocator_IM IM) selection)
H: ∀ y : nat,
  y < length (tr' ++ [lst])
  → ∀ tr : list
             (composite_transition_item
                (sub_IM 
                   (equivocator_IM IM) selection)),
      finite_valid_trace SeededXE is tr
      → let final_state := 
          finite_trace_last is tr in
        y = length tr
        → ∀ final_descriptors : 
             equivocator_descriptors IM',
            proper_equivocator_descriptors IM'
              final_descriptors final_state
            → ∀ (initial_descriptors : 
                 equivocator_descriptors IM') 
                (trX : 
                 list
                   (composite_transition_item IM')),
                equivocators_trace_project IM'
                  final_descriptors tr =
                Some (trX, initial_descriptors)
                → finite_valid_trace SeededX
                    (equivocators_state_project
                      IM' initial_descriptors is)
                    trX
Htr: finite_valid_trace_from SeededXE is
  (tr' ++ [lst])
Hinit: initial_state_prop is
final_state:= finite_trace_last is tr: state
  (composite_type
     (sub_IM (equivocator_IM IM)
        selection))
final_descriptors: equivocator_descriptors IM'
Hproper: proper_equivocator_descriptors IM'
  final_descriptors final_state
initial_descriptors: equivocator_descriptors IM'
trX: list (composite_transition_item IM')
Hpr_trX: equivocators_trace_project IM'
  final_descriptors (tr' ++ [lst]) =
Some (trX, initial_descriptors)
Htr_lst: tr = tr' ++ [lst]
H': ∀ tr : list
         (composite_transition_item
            (sub_IM (equivocator_IM IM) selection)),
  finite_valid_trace SeededXE is tr
  → let final_state := finite_trace_last is tr in
    length tr' = length tr
    → ∀ final_descriptors : 
         equivocator_descriptors IM',
        proper_equivocator_descriptors IM'
          final_descriptors final_state
        → ∀ (initial_descriptors : 
             equivocator_descriptors IM') 
            (trX : list
                     (composite_transition_item
                      IM')),
            equivocators_trace_project IM'
              final_descriptors tr =
            Some (trX, initial_descriptors)
            → finite_valid_trace SeededX
                (equivocators_state_project IM'
                   initial_descriptors is) trX
finite_valid_trace SeededX
  (equivocators_state_project IM' initial_descriptors
     is) trX
    apply finite_valid_trace_from_app_iff in Htr.message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
is: composite_state
  (sub_IM (equivocator_IM IM) selection)
IM':= λ i : sub_index selection, IM (`i): sub_index selection → VLSM message
tr, tr': list
  (composite_transition_item
     (sub_IM (equivocator_IM IM) selection))
lst: composite_transition_item
  (sub_IM (equivocator_IM IM) selection)
H: ∀ y : nat,
  y < length (tr' ++ [lst])
  → ∀ tr : list
             (composite_transition_item
                (sub_IM 
                   (equivocator_IM IM) selection)),
      finite_valid_trace SeededXE is tr
      → let final_state := 
          finite_trace_last is tr in
        y = length tr
        → ∀ final_descriptors : 
             equivocator_descriptors IM',
            proper_equivocator_descriptors IM'
              final_descriptors final_state
            → ∀ (initial_descriptors : 
                 equivocator_descriptors IM') 
                (trX : 
                 list
                   (composite_transition_item IM')),
                equivocators_trace_project IM'
                  final_descriptors tr =
                Some (trX, initial_descriptors)
                → finite_valid_trace SeededX
                    (equivocators_state_project
                      IM' initial_descriptors is)
                    trX
Htr: finite_valid_trace_from SeededXE is tr'
∧ finite_valid_trace_from SeededXE
    (finite_trace_last is tr') [lst]
Hinit: initial_state_prop is
final_state:= finite_trace_last is tr: state
  (composite_type
     (sub_IM (equivocator_IM IM)
        selection))
final_descriptors: equivocator_descriptors IM'
Hproper: proper_equivocator_descriptors IM'
  final_descriptors final_state
initial_descriptors: equivocator_descriptors IM'
trX: list (composite_transition_item IM')
Hpr_trX: equivocators_trace_project IM'
  final_descriptors (tr' ++ [lst]) =
Some (trX, initial_descriptors)
Htr_lst: tr = tr' ++ [lst]
H': ∀ tr : list
         (composite_transition_item
            (sub_IM (equivocator_IM IM) selection)),
  finite_valid_trace SeededXE is tr
  → let final_state := finite_trace_last is tr in
    length tr' = length tr
    → ∀ final_descriptors : 
         equivocator_descriptors IM',
        proper_equivocator_descriptors IM'
          final_descriptors final_state
        → ∀ (initial_descriptors : 
             equivocator_descriptors IM') 
            (trX : list
                     (composite_transition_item
                      IM')),
            equivocators_trace_project IM'
              final_descriptors tr =
            Some (trX, initial_descriptors)
            → finite_valid_trace SeededX
                (equivocators_state_project IM'
                   initial_descriptors is) trX
finite_valid_trace SeededX
  (equivocators_state_project IM' initial_descriptors
     is) trX
    destruct Htr as [Htr Hlst].message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
is: composite_state
  (sub_IM (equivocator_IM IM) selection)
IM':= λ i : sub_index selection, IM (`i): sub_index selection → VLSM message
tr, tr': list
  (composite_transition_item
     (sub_IM (equivocator_IM IM) selection))
lst: composite_transition_item
  (sub_IM (equivocator_IM IM) selection)
H: ∀ y : nat,
  y < length (tr' ++ [lst])
  → ∀ tr : list
             (composite_transition_item
                (sub_IM 
                   (equivocator_IM IM) selection)),
      finite_valid_trace SeededXE is tr
      → let final_state := 
          finite_trace_last is tr in
        y = length tr
        → ∀ final_descriptors : 
             equivocator_descriptors IM',
            proper_equivocator_descriptors IM'
              final_descriptors final_state
            → ∀ (initial_descriptors : 
                 equivocator_descriptors IM') 
                (trX : 
                 list
                   (composite_transition_item IM')),
                equivocators_trace_project IM'
                  final_descriptors tr =
                Some (trX, initial_descriptors)
                → finite_valid_trace SeededX
                    (equivocators_state_project
                      IM' initial_descriptors is)
                    trX
Htr: finite_valid_trace_from SeededXE is tr'
Hlst: finite_valid_trace_from SeededXE
  (finite_trace_last is tr') [lst]
Hinit: initial_state_prop is
final_state:= finite_trace_last is tr: state
  (composite_type
     (sub_IM (equivocator_IM IM)
        selection))
final_descriptors: equivocator_descriptors IM'
Hproper: proper_equivocator_descriptors IM'
  final_descriptors final_state
initial_descriptors: equivocator_descriptors IM'
trX: list (composite_transition_item IM')
Hpr_trX: equivocators_trace_project IM'
  final_descriptors (tr' ++ [lst]) =
Some (trX, initial_descriptors)
Htr_lst: tr = tr' ++ [lst]
H': ∀ tr : list
         (composite_transition_item
            (sub_IM (equivocator_IM IM) selection)),
  finite_valid_trace SeededXE is tr
  → let final_state := finite_trace_last is tr in
    length tr' = length tr
    → ∀ final_descriptors : 
         equivocator_descriptors IM',
        proper_equivocator_descriptors IM'
          final_descriptors final_state
        → ∀ (initial_descriptors : 
             equivocator_descriptors IM') 
            (trX : list
                     (composite_transition_item
                      IM')),
            equivocators_trace_project IM'
              final_descriptors tr =
            Some (trX, initial_descriptors)
            → finite_valid_trace SeededX
                (equivocators_state_project IM'
                   initial_descriptors is) trX
finite_valid_trace SeededX
  (equivocators_state_project IM' initial_descriptors
     is) trX
    specialize (H' tr' (conj Htr Hinit) eq_refl).message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
is: composite_state
  (sub_IM (equivocator_IM IM) selection)
IM':= λ i : sub_index selection, IM (`i): sub_index selection → VLSM message
tr, tr': list
  (composite_transition_item
     (sub_IM (equivocator_IM IM) selection))
lst: composite_transition_item
  (sub_IM (equivocator_IM IM) selection)
H: ∀ y : nat,
  y < length (tr' ++ [lst])
  → ∀ tr : list
             (composite_transition_item
                (sub_IM 
                   (equivocator_IM IM) selection)),
      finite_valid_trace SeededXE is tr
      → let final_state := 
          finite_trace_last is tr in
        y = length tr
        → ∀ final_descriptors : 
             equivocator_descriptors IM',
            proper_equivocator_descriptors IM'
              final_descriptors final_state
            → ∀ (initial_descriptors : 
                 equivocator_descriptors IM') 
                (trX : 
                 list
                   (composite_transition_item IM')),
                equivocators_trace_project IM'
                  final_descriptors tr =
                Some (trX, initial_descriptors)
                → finite_valid_trace SeededX
                    (equivocators_state_project
                      IM' initial_descriptors is)
                    trX
Htr: finite_valid_trace_from SeededXE is tr'
Hlst: finite_valid_trace_from SeededXE
  (finite_trace_last is tr') [lst]
Hinit: initial_state_prop is
final_state:= finite_trace_last is tr: state
  (composite_type
     (sub_IM (equivocator_IM IM)
        selection))
final_descriptors: equivocator_descriptors IM'
Hproper: proper_equivocator_descriptors IM'
  final_descriptors final_state
initial_descriptors: equivocator_descriptors IM'
trX: list (composite_transition_item IM')
Hpr_trX: equivocators_trace_project IM'
  final_descriptors (tr' ++ [lst]) =
Some (trX, initial_descriptors)
Htr_lst: tr = tr' ++ [lst]
H': ∀ final_descriptors : equivocator_descriptors IM',
  proper_equivocator_descriptors IM'
    final_descriptors (finite_trace_last is tr')
  → ∀ (initial_descriptors : 
       equivocator_descriptors IM') 
      (trX : list (composite_transition_item IM')),
      equivocators_trace_project IM'
        final_descriptors tr' =
      Some (trX, initial_descriptors)
      → finite_valid_trace SeededX
          (equivocators_state_project IM'
             initial_descriptors is) trX
finite_valid_trace SeededX
  (equivocators_state_project IM' initial_descriptors
     is) trX
    specialize (equivocators_transition_item_project_proper_characterization _
      final_descriptors lst) as Hproperx.message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
is: composite_state
  (sub_IM (equivocator_IM IM) selection)
IM':= λ i : sub_index selection, IM (`i): sub_index selection → VLSM message
tr, tr': list
  (composite_transition_item
     (sub_IM (equivocator_IM IM) selection))
lst: composite_transition_item
  (sub_IM (equivocator_IM IM) selection)
H: ∀ y : nat,
  y < length (tr' ++ [lst])
  → ∀ tr : list
             (composite_transition_item
                (sub_IM 
                   (equivocator_IM IM) selection)),
      finite_valid_trace SeededXE is tr
      → let final_state := 
          finite_trace_last is tr in
        y = length tr
        → ∀ final_descriptors : 
             equivocator_descriptors IM',
            proper_equivocator_descriptors IM'
              final_descriptors final_state
            → ∀ (initial_descriptors : 
                 equivocator_descriptors IM') 
                (trX : 
                 list
                   (composite_transition_item IM')),
                equivocators_trace_project IM'
                  final_descriptors tr =
                Some (trX, initial_descriptors)
                → finite_valid_trace SeededX
                    (equivocators_state_project
                      IM' initial_descriptors is)
                    trX
Htr: finite_valid_trace_from SeededXE is tr'
Hlst: finite_valid_trace_from SeededXE
  (finite_trace_last is tr') [lst]
Hinit: initial_state_prop is
final_state:= finite_trace_last is tr: state
  (composite_type
     (sub_IM (equivocator_IM IM)
        selection))
final_descriptors: equivocator_descriptors IM'
Hproper: proper_equivocator_descriptors IM'
  final_descriptors final_state
initial_descriptors: equivocator_descriptors IM'
trX: list (composite_transition_item IM')
Hpr_trX: equivocators_trace_project IM'
  final_descriptors (tr' ++ [lst]) =
Some (trX, initial_descriptors)
Htr_lst: tr = tr' ++ [lst]
H': ∀ final_descriptors : equivocator_descriptors IM',
  proper_equivocator_descriptors IM'
    final_descriptors (finite_trace_last is tr')
  → ∀ (initial_descriptors : 
       equivocator_descriptors IM') 
      (trX : list (composite_transition_item IM')),
      equivocators_trace_project IM'
        final_descriptors tr' =
      Some (trX, initial_descriptors)
      → finite_valid_trace SeededX
          (equivocators_state_project IM'
             initial_descriptors is) trX
Hproperx: proper_equivocator_descriptors IM'
  final_descriptors (destination lst)
→ ∃ (oitem : option
               (composite_transition_item
                  IM')) 
    (eqv_descriptors' : 
     equivocator_descriptors IM'),
    equivocators_transition_item_project
      IM' final_descriptors lst =
    Some (oitem, eqv_descriptors')
    ∧ match oitem with
      | Some itemx =>
          (∃ Hex : existing_equivocator_label
                     (IM' (projT1 (l lst)))
                     (projT2 (l lst)),
             existT (projT1 (l lst))
               (existing_equivocator_label_extract
                  (IM' (projT1 (l lst)))
                  (projT2 (l lst)) Hex) =
             l itemx)
          ∧ input lst = input itemx
            ∧ output lst = output itemx
              ∧ equivocators_state_project
                  IM' final_descriptors
                  (destination lst) =
                destination itemx
                ∧ eqv_descriptors'
                    (projT1 (l lst)) =
                  equivocator_label_descriptor
                    (projT2 (l lst))
      | None => True
      end
      ∧ (∀ s : composite_state
                 (equivocator_IM IM'),
           composite_valid
             (equivocator_IM IM') 
             (l lst) (s, input lst)
           → composite_transition
               (equivocator_IM IM') 
               (l lst) (
               s, input lst) =
             (destination lst, output lst)
             → proper_equivocator_descriptors
                 IM' eqv_descriptors' s
               ∧ eqv_descriptors' =
                 equivocator_descriptors_update
                   IM' final_descriptors
                   (projT1 (l lst))
                   (eqv_descriptors'
                      (projT1 (l lst)))
                 ∧ s =
                   state_update
                     (equivocator_IM IM')
                     (destination lst)
                     (projT1 (l lst))
                     (s (projT1 (l lst)))
                   ∧ previous_state_descriptor_prop
                      (IM' (projT1 (l lst)))
                      (final_descriptors
                      (projT1 (l lst)))
                      (s (projT1 (l lst)))
                      (eqv_descriptors'
                      (projT1 (l lst)))
                     ∧ 
                     match oitem with
                     | Some itemx =>
                      ∀ sx : 
                      composite_state IM',
                      sx =
                      equivocators_state_project
                      IM' eqv_descriptors'
                      s
                      → 
                      composite_valid IM'
                      (l itemx)
                      (sx, input itemx)
                      ∧ 
                      composite_transition
                      IM' 
                      (l itemx)
                      (sx, input itemx) =
                      (
                      destination itemx,
                      output itemx)
                     | None =>
                      equivocators_state_project
                      IM' final_descriptors
                      (destination lst) =
                      equivocators_state_project
                      IM' eqv_descriptors'
                      s
                     end)
finite_valid_trace SeededX
  (equivocators_state_project IM' initial_descriptors
     is) trX
    unfold final_state in Hproper.message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
is: composite_state
  (sub_IM (equivocator_IM IM) selection)
IM':= λ i : sub_index selection, IM (`i): sub_index selection → VLSM message
tr, tr': list
  (composite_transition_item
     (sub_IM (equivocator_IM IM) selection))
lst: composite_transition_item
  (sub_IM (equivocator_IM IM) selection)
H: ∀ y : nat,
  y < length (tr' ++ [lst])
  → ∀ tr : list
             (composite_transition_item
                (sub_IM 
                   (equivocator_IM IM) selection)),
      finite_valid_trace SeededXE is tr
      → let final_state := 
          finite_trace_last is tr in
        y = length tr
        → ∀ final_descriptors : 
             equivocator_descriptors IM',
            proper_equivocator_descriptors IM'
              final_descriptors final_state
            → ∀ (initial_descriptors : 
                 equivocator_descriptors IM') 
                (trX : 
                 list
                   (composite_transition_item IM')),
                equivocators_trace_project IM'
                  final_descriptors tr =
                Some (trX, initial_descriptors)
                → finite_valid_trace SeededX
                    (equivocators_state_project
                      IM' initial_descriptors is)
                    trX
Htr: finite_valid_trace_from SeededXE is tr'
Hlst: finite_valid_trace_from SeededXE
  (finite_trace_last is tr') [lst]
Hinit: initial_state_prop is
final_state:= finite_trace_last is tr: state
  (composite_type
     (sub_IM (equivocator_IM IM)
        selection))
final_descriptors: equivocator_descriptors IM'
Hproper: proper_equivocator_descriptors IM'
  final_descriptors
  (finite_trace_last is tr)
initial_descriptors: equivocator_descriptors IM'
trX: list (composite_transition_item IM')
Hpr_trX: equivocators_trace_project IM'
  final_descriptors (tr' ++ [lst]) =
Some (trX, initial_descriptors)
Htr_lst: tr = tr' ++ [lst]
H': ∀ final_descriptors : equivocator_descriptors IM',
  proper_equivocator_descriptors IM'
    final_descriptors (finite_trace_last is tr')
  → ∀ (initial_descriptors : 
       equivocator_descriptors IM') 
      (trX : list (composite_transition_item IM')),
      equivocators_trace_project IM'
        final_descriptors tr' =
      Some (trX, initial_descriptors)
      → finite_valid_trace SeededX
          (equivocators_state_project IM'
             initial_descriptors is) trX
Hproperx: proper_equivocator_descriptors IM'
  final_descriptors (destination lst)
→ ∃ (oitem : option
               (composite_transition_item
                  IM')) 
    (eqv_descriptors' : 
     equivocator_descriptors IM'),
    equivocators_transition_item_project
      IM' final_descriptors lst =
    Some (oitem, eqv_descriptors')
    ∧ match oitem with
      | Some itemx =>
          (∃ Hex : existing_equivocator_label
                     (IM' (projT1 (l lst)))
                     (projT2 (l lst)),
             existT (projT1 (l lst))
               (existing_equivocator_label_extract
                  (IM' (projT1 (l lst)))
                  (projT2 (l lst)) Hex) =
             l itemx)
          ∧ input lst = input itemx
            ∧ output lst = output itemx
              ∧ equivocators_state_project
                  IM' final_descriptors
                  (destination lst) =
                destination itemx
                ∧ eqv_descriptors'
                    (projT1 (l lst)) =
                  equivocator_label_descriptor
                    (projT2 (l lst))
      | None => True
      end
      ∧ (∀ s : composite_state
                 (equivocator_IM IM'),
           composite_valid
             (equivocator_IM IM') 
             (l lst) (s, input lst)
           → composite_transition
               (equivocator_IM IM') 
               (l lst) (
               s, input lst) =
             (destination lst, output lst)
             → proper_equivocator_descriptors
                 IM' eqv_descriptors' s
               ∧ eqv_descriptors' =
                 equivocator_descriptors_update
                   IM' final_descriptors
                   (projT1 (l lst))
                   (eqv_descriptors'
                      (projT1 (l lst)))
                 ∧ s =
                   state_update
                     (equivocator_IM IM')
                     (destination lst)
                     (projT1 (l lst))
                     (s (projT1 (l lst)))
                   ∧ previous_state_descriptor_prop
                      (IM' (projT1 (l lst)))
                      (final_descriptors
                      (projT1 (l lst)))
                      (s (projT1 (l lst)))
                      (eqv_descriptors'
                      (projT1 (l lst)))
                     ∧ 
                     match oitem with
                     | Some itemx =>
                      ∀ sx : 
                      composite_state IM',
                      sx =
                      equivocators_state_project
                      IM' eqv_descriptors'
                      s
                      → 
                      composite_valid IM'
                      (l itemx)
                      (sx, input itemx)
                      ∧ 
                      composite_transition
                      IM' 
                      (l itemx)
                      (sx, input itemx) =
                      (
                      destination itemx,
                      output itemx)
                     | None =>
                      equivocators_state_project
                      IM' final_descriptors
                      (destination lst) =
                      equivocators_state_project
                      IM' eqv_descriptors'
                      s
                     end)
finite_valid_trace SeededX
  (equivocators_state_project IM' initial_descriptors
     is) trX rewrite Htr_lst in Hproper.message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
is: composite_state
  (sub_IM (equivocator_IM IM) selection)
IM':= λ i : sub_index selection, IM (`i): sub_index selection → VLSM message
tr, tr': list
  (composite_transition_item
     (sub_IM (equivocator_IM IM) selection))
lst: composite_transition_item
  (sub_IM (equivocator_IM IM) selection)
H: ∀ y : nat,
  y < length (tr' ++ [lst])
  → ∀ tr : list
             (composite_transition_item
                (sub_IM 
                   (equivocator_IM IM) selection)),
      finite_valid_trace SeededXE is tr
      → let final_state := 
          finite_trace_last is tr in
        y = length tr
        → ∀ final_descriptors : 
             equivocator_descriptors IM',
            proper_equivocator_descriptors IM'
              final_descriptors final_state
            → ∀ (initial_descriptors : 
                 equivocator_descriptors IM') 
                (trX : 
                 list
                   (composite_transition_item IM')),
                equivocators_trace_project IM'
                  final_descriptors tr =
                Some (trX, initial_descriptors)
                → finite_valid_trace SeededX
                    (equivocators_state_project
                      IM' initial_descriptors is)
                    trX
Htr: finite_valid_trace_from SeededXE is tr'
Hlst: finite_valid_trace_from SeededXE
  (finite_trace_last is tr') [lst]
Hinit: initial_state_prop is
final_state:= finite_trace_last is tr: state
  (composite_type
     (sub_IM (equivocator_IM IM)
        selection))
final_descriptors: equivocator_descriptors IM'
Hproper: proper_equivocator_descriptors IM'
  final_descriptors
  (finite_trace_last is (tr' ++ [lst]))
initial_descriptors: equivocator_descriptors IM'
trX: list (composite_transition_item IM')
Hpr_trX: equivocators_trace_project IM'
  final_descriptors (tr' ++ [lst]) =
Some (trX, initial_descriptors)
Htr_lst: tr = tr' ++ [lst]
H': ∀ final_descriptors : equivocator_descriptors IM',
  proper_equivocator_descriptors IM'
    final_descriptors (finite_trace_last is tr')
  → ∀ (initial_descriptors : 
       equivocator_descriptors IM') 
      (trX : list (composite_transition_item IM')),
      equivocators_trace_project IM'
        final_descriptors tr' =
      Some (trX, initial_descriptors)
      → finite_valid_trace SeededX
          (equivocators_state_project IM'
             initial_descriptors is) trX
Hproperx: proper_equivocator_descriptors IM'
  final_descriptors (destination lst)
→ ∃ (oitem : option
               (composite_transition_item
                  IM')) 
    (eqv_descriptors' : 
     equivocator_descriptors IM'),
    equivocators_transition_item_project
      IM' final_descriptors lst =
    Some (oitem, eqv_descriptors')
    ∧ match oitem with
      | Some itemx =>
          (∃ Hex : existing_equivocator_label
                     (IM' (projT1 (l lst)))
                     (projT2 (l lst)),
             existT (projT1 (l lst))
               (existing_equivocator_label_extract
                  (IM' (projT1 (l lst)))
                  (projT2 (l lst)) Hex) =
             l itemx)
          ∧ input lst = input itemx
            ∧ output lst = output itemx
              ∧ equivocators_state_project
                  IM' final_descriptors
                  (destination lst) =
                destination itemx
                ∧ eqv_descriptors'
                    (projT1 (l lst)) =
                  equivocator_label_descriptor
                    (projT2 (l lst))
      | None => True
      end
      ∧ (∀ s : composite_state
                 (equivocator_IM IM'),
           composite_valid
             (equivocator_IM IM') 
             (l lst) (s, input lst)
           → composite_transition
               (equivocator_IM IM') 
               (l lst) (
               s, input lst) =
             (destination lst, output lst)
             → proper_equivocator_descriptors
                 IM' eqv_descriptors' s
               ∧ eqv_descriptors' =
                 equivocator_descriptors_update
                   IM' final_descriptors
                   (projT1 (l lst))
                   (eqv_descriptors'
                      (projT1 (l lst)))
                 ∧ s =
                   state_update
                     (equivocator_IM IM')
                     (destination lst)
                     (projT1 (l lst))
                     (s (projT1 (l lst)))
                   ∧ previous_state_descriptor_prop
                      (IM' (projT1 (l lst)))
                      (final_descriptors
                      (projT1 (l lst)))
                      (s (projT1 (l lst)))
                      (eqv_descriptors'
                      (projT1 (l lst)))
                     ∧ 
                     match oitem with
                     | Some itemx =>
                      ∀ sx : 
                      composite_state IM',
                      sx =
                      equivocators_state_project
                      IM' eqv_descriptors'
                      s
                      → 
                      composite_valid IM'
                      (l itemx)
                      (sx, input itemx)
                      ∧ 
                      composite_transition
                      IM' 
                      (l itemx)
                      (sx, input itemx) =
                      (
                      destination itemx,
                      output itemx)
                     | None =>
                      equivocators_state_project
                      IM' final_descriptors
                      (destination lst) =
                      equivocators_state_project
                      IM' eqv_descriptors'
                      s
                     end)
finite_valid_trace SeededX
  (equivocators_state_project IM' initial_descriptors
     is) trX
    rewrite finite_trace_last_is_last in Hproper.message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
is: composite_state
  (sub_IM (equivocator_IM IM) selection)
IM':= λ i : sub_index selection, IM (`i): sub_index selection → VLSM message
tr, tr': list
  (composite_transition_item
     (sub_IM (equivocator_IM IM) selection))
lst: composite_transition_item
  (sub_IM (equivocator_IM IM) selection)
H: ∀ y : nat,
  y < length (tr' ++ [lst])
  → ∀ tr : list
             (composite_transition_item
                (sub_IM 
                   (equivocator_IM IM) selection)),
      finite_valid_trace SeededXE is tr
      → let final_state := 
          finite_trace_last is tr in
        y = length tr
        → ∀ final_descriptors : 
             equivocator_descriptors IM',
            proper_equivocator_descriptors IM'
              final_descriptors final_state
            → ∀ (initial_descriptors : 
                 equivocator_descriptors IM') 
                (trX : 
                 list
                   (composite_transition_item IM')),
                equivocators_trace_project IM'
                  final_descriptors tr =
                Some (trX, initial_descriptors)
                → finite_valid_trace SeededX
                    (equivocators_state_project
                      IM' initial_descriptors is)
                    trX
Htr: finite_valid_trace_from SeededXE is tr'
Hlst: finite_valid_trace_from SeededXE
  (finite_trace_last is tr') [lst]
Hinit: initial_state_prop is
final_state:= finite_trace_last is tr: state
  (composite_type
     (sub_IM (equivocator_IM IM)
        selection))
final_descriptors: equivocator_descriptors IM'
Hproper: proper_equivocator_descriptors IM'
  final_descriptors (destination lst)
initial_descriptors: equivocator_descriptors IM'
trX: list (composite_transition_item IM')
Hpr_trX: equivocators_trace_project IM'
  final_descriptors (tr' ++ [lst]) =
Some (trX, initial_descriptors)
Htr_lst: tr = tr' ++ [lst]
H': ∀ final_descriptors : equivocator_descriptors IM',
  proper_equivocator_descriptors IM'
    final_descriptors (finite_trace_last is tr')
  → ∀ (initial_descriptors : 
       equivocator_descriptors IM') 
      (trX : list (composite_transition_item IM')),
      equivocators_trace_project IM'
        final_descriptors tr' =
      Some (trX, initial_descriptors)
      → finite_valid_trace SeededX
          (equivocators_state_project IM'
             initial_descriptors is) trX
Hproperx: proper_equivocator_descriptors IM'
  final_descriptors (destination lst)
→ ∃ (oitem : option
               (composite_transition_item
                  IM')) 
    (eqv_descriptors' : 
     equivocator_descriptors IM'),
    equivocators_transition_item_project
      IM' final_descriptors lst =
    Some (oitem, eqv_descriptors')
    ∧ match oitem with
      | Some itemx =>
          (∃ Hex : existing_equivocator_label
                     (IM' (projT1 (l lst)))
                     (projT2 (l lst)),
             existT (projT1 (l lst))
               (existing_equivocator_label_extract
                  (IM' (projT1 (l lst)))
                  (projT2 (l lst)) Hex) =
             l itemx)
          ∧ input lst = input itemx
            ∧ output lst = output itemx
              ∧ equivocators_state_project
                  IM' final_descriptors
                  (destination lst) =
                destination itemx
                ∧ eqv_descriptors'
                    (projT1 (l lst)) =
                  equivocator_label_descriptor
                    (projT2 (l lst))
      | None => True
      end
      ∧ (∀ s : composite_state
                 (equivocator_IM IM'),
           composite_valid
             (equivocator_IM IM') 
             (l lst) (s, input lst)
           → composite_transition
               (equivocator_IM IM') 
               (l lst) (
               s, input lst) =
             (destination lst, output lst)
             → proper_equivocator_descriptors
                 IM' eqv_descriptors' s
               ∧ eqv_descriptors' =
                 equivocator_descriptors_update
                   IM' final_descriptors
                   (projT1 (l lst))
                   (eqv_descriptors'
                      (projT1 (l lst)))
                 ∧ s =
                   state_update
                     (equivocator_IM IM')
                     (destination lst)
                     (projT1 (l lst))
                     (s (projT1 (l lst)))
                   ∧ previous_state_descriptor_prop
                      (IM' (projT1 (l lst)))
                      (final_descriptors
                      (projT1 (l lst)))
                      (s (projT1 (l lst)))
                      (eqv_descriptors'
                      (projT1 (l lst)))
                     ∧ 
                     match oitem with
                     | Some itemx =>
                      ∀ sx : 
                      composite_state IM',
                      sx =
                      equivocators_state_project
                      IM' eqv_descriptors'
                      s
                      → 
                      composite_valid IM'
                      (l itemx)
                      (sx, input itemx)
                      ∧ 
                      composite_transition
                      IM' 
                      (l itemx)
                      (sx, input itemx) =
                      (
                      destination itemx,
                      output itemx)
                     | None =>
                      equivocators_state_project
                      IM' final_descriptors
                      (destination lst) =
                      equivocators_state_project
                      IM' eqv_descriptors'
                      s
                     end)
finite_valid_trace SeededX
  (equivocators_state_project IM' initial_descriptors
     is) trX
    specialize (Hproperx Hproper).message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
is: composite_state
  (sub_IM (equivocator_IM IM) selection)
IM':= λ i : sub_index selection, IM (`i): sub_index selection → VLSM message
tr, tr': list
  (composite_transition_item
     (sub_IM (equivocator_IM IM) selection))
lst: composite_transition_item
  (sub_IM (equivocator_IM IM) selection)
H: ∀ y : nat,
  y < length (tr' ++ [lst])
  → ∀ tr : list
             (composite_transition_item
                (sub_IM 
                   (equivocator_IM IM) selection)),
      finite_valid_trace SeededXE is tr
      → let final_state := 
          finite_trace_last is tr in
        y = length tr
        → ∀ final_descriptors : 
             equivocator_descriptors IM',
            proper_equivocator_descriptors IM'
              final_descriptors final_state
            → ∀ (initial_descriptors : 
                 equivocator_descriptors IM') 
                (trX : 
                 list
                   (composite_transition_item IM')),
                equivocators_trace_project IM'
                  final_descriptors tr =
                Some (trX, initial_descriptors)
                → finite_valid_trace SeededX
                    (equivocators_state_project
                      IM' initial_descriptors is)
                    trX
Htr: finite_valid_trace_from SeededXE is tr'
Hlst: finite_valid_trace_from SeededXE
  (finite_trace_last is tr') [lst]
Hinit: initial_state_prop is
final_state:= finite_trace_last is tr: state
  (composite_type
     (sub_IM (equivocator_IM IM)
        selection))
final_descriptors: equivocator_descriptors IM'
Hproper: proper_equivocator_descriptors IM'
  final_descriptors (destination lst)
initial_descriptors: equivocator_descriptors IM'
trX: list (composite_transition_item IM')
Hpr_trX: equivocators_trace_project IM'
  final_descriptors (tr' ++ [lst]) =
Some (trX, initial_descriptors)
Htr_lst: tr = tr' ++ [lst]
H': ∀ final_descriptors : equivocator_descriptors IM',
  proper_equivocator_descriptors IM'
    final_descriptors (finite_trace_last is tr')
  → ∀ (initial_descriptors : 
       equivocator_descriptors IM') 
      (trX : list (composite_transition_item IM')),
      equivocators_trace_project IM'
        final_descriptors tr' =
      Some (trX, initial_descriptors)
      → finite_valid_trace SeededX
          (equivocators_state_project IM'
             initial_descriptors is) trX
Hproperx: ∃ (oitem : option
             (composite_transition_item IM')) 
  (eqv_descriptors' : equivocator_descriptors
                      IM'),
  equivocators_transition_item_project IM'
    final_descriptors lst =
  Some (oitem, eqv_descriptors')
  ∧ match oitem with
    | Some itemx =>
        (∃ Hex : existing_equivocator_label
                   (IM' (projT1 (l lst)))
                   (projT2 (l lst)),
           existT (projT1 (l lst))
             (existing_equivocator_label_extract
                (IM' (projT1 (l lst)))
                (projT2 (l lst)) Hex) =
           l itemx)
        ∧ input lst = input itemx
          ∧ output lst = output itemx
            ∧ equivocators_state_project
                IM' final_descriptors
                (destination lst) =
              destination itemx
              ∧ eqv_descriptors'
                  (projT1 (l lst)) =
                equivocator_label_descriptor
                  (projT2 (l lst))
    | None => True
    end
    ∧ (∀ s : composite_state
               (equivocator_IM IM'),
         composite_valid
           (equivocator_IM IM') 
           (l lst) (s, input lst)
         → composite_transition
             (equivocator_IM IM') 
             (l lst) (s, input lst) =
           (destination lst, output lst)
           → proper_equivocator_descriptors
               IM' eqv_descriptors' s
             ∧ eqv_descriptors' =
               equivocator_descriptors_update
                 IM' final_descriptors
                 (projT1 (l lst))
                 (eqv_descriptors'
                    (projT1 (l lst)))
               ∧ s =
                 state_update
                   (equivocator_IM IM')
                   (destination lst)
                   (projT1 (l lst))
                   (s (projT1 (l lst)))
                 ∧ previous_state_descriptor_prop
                     (IM' (projT1 (l lst)))
                     (final_descriptors
                      (projT1 (l lst)))
                     (s (projT1 (l lst)))
                     (eqv_descriptors'
                      (projT1 (l lst)))
                   ∧ match oitem with
                     | Some itemx =>
                      ∀ sx : 
                      composite_state IM',
                      sx =
                      equivocators_state_project
                      IM' eqv_descriptors'
                      s
                      → 
                      composite_valid IM'
                      (l itemx)
                      (sx, input itemx)
                      ∧ 
                      composite_transition
                      IM' 
                      (l itemx)
                      (sx, input itemx) =
                      (
                      destination itemx,
                      output itemx)
                     | None =>
                      equivocators_state_project
                      IM' final_descriptors
                      (destination lst) =
                      equivocators_state_project
                      IM' eqv_descriptors'
                      s
                     end)
finite_valid_trace SeededX
  (equivocators_state_project IM' initial_descriptors
     is) trX
    destruct Hproperx as [oitem [final_descriptors' [Hprojectx [Hitemx Hproperx]]]].message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
is: composite_state
  (sub_IM (equivocator_IM IM) selection)
IM':= λ i : sub_index selection, IM (`i): sub_index selection → VLSM message
tr, tr': list
  (composite_transition_item
     (sub_IM (equivocator_IM IM) selection))
lst: composite_transition_item
  (sub_IM (equivocator_IM IM) selection)
H: ∀ y : nat,
  y < length (tr' ++ [lst])
  → ∀ tr : list
             (composite_transition_item
                (sub_IM 
                   (equivocator_IM IM) selection)),
      finite_valid_trace SeededXE is tr
      → let final_state := 
          finite_trace_last is tr in
        y = length tr
        → ∀ final_descriptors : 
             equivocator_descriptors IM',
            proper_equivocator_descriptors IM'
              final_descriptors final_state
            → ∀ (initial_descriptors : 
                 equivocator_descriptors IM') 
                (trX : 
                 list
                   (composite_transition_item IM')),
                equivocators_trace_project IM'
                  final_descriptors tr =
                Some (trX, initial_descriptors)
                → finite_valid_trace SeededX
                    (equivocators_state_project
                      IM' initial_descriptors is)
                    trX
Htr: finite_valid_trace_from SeededXE is tr'
Hlst: finite_valid_trace_from SeededXE
  (finite_trace_last is tr') [lst]
Hinit: initial_state_prop is
final_state:= finite_trace_last is tr: state
  (composite_type
     (sub_IM (equivocator_IM IM)
        selection))
final_descriptors: equivocator_descriptors IM'
Hproper: proper_equivocator_descriptors IM'
  final_descriptors (destination lst)
initial_descriptors: equivocator_descriptors IM'
trX: list (composite_transition_item IM')
Hpr_trX: equivocators_trace_project IM'
  final_descriptors (tr' ++ [lst]) =
Some (trX, initial_descriptors)
Htr_lst: tr = tr' ++ [lst]
H': ∀ final_descriptors : equivocator_descriptors IM',
  proper_equivocator_descriptors IM'
    final_descriptors (finite_trace_last is tr')
  → ∀ (initial_descriptors : 
       equivocator_descriptors IM') 
      (trX : list (composite_transition_item IM')),
      equivocators_trace_project IM'
        final_descriptors tr' =
      Some (trX, initial_descriptors)
      → finite_valid_trace SeededX
          (equivocators_state_project IM'
             initial_descriptors is) trX
oitem: option (composite_transition_item IM')
final_descriptors': equivocator_descriptors IM'
Hprojectx: equivocators_transition_item_project IM'
  final_descriptors lst =
Some (oitem, final_descriptors')
Hitemx: match oitem with
| Some itemx =>
    (∃ Hex : existing_equivocator_label
               (IM' (projT1 (l lst)))
               (projT2 (l lst)),
       existT (projT1 (l lst))
         (existing_equivocator_label_extract
            (IM' (projT1 (l lst)))
            (projT2 (l lst)) Hex) = 
       l itemx)
    ∧ input lst = input itemx
      ∧ output lst = output itemx
        ∧ equivocators_state_project IM'
            final_descriptors
            (destination lst) =
          destination itemx
          ∧ final_descriptors'
              (projT1 (l lst)) =
            equivocator_label_descriptor
              (projT2 (l lst))
| None => True
end
Hproperx: ∀ s : composite_state (equivocator_IM IM'),
  composite_valid (equivocator_IM IM')
    (l lst) (s, input lst)
  → composite_transition
      (equivocator_IM IM') 
      (l lst) (s, input lst) =
    (destination lst, output lst)
    → proper_equivocator_descriptors IM'
        final_descriptors' s
      ∧ final_descriptors' =
        equivocator_descriptors_update IM'
          final_descriptors
          (projT1 (l lst))
          (final_descriptors'
             (projT1 (l lst)))
        ∧ s =
          state_update 
            (equivocator_IM IM')
            (destination lst)
            (projT1 (l lst))
            (s (projT1 (l lst)))
          ∧ previous_state_descriptor_prop
              (IM' (projT1 (l lst)))
              (final_descriptors
                 (projT1 (l lst)))
              (s (projT1 (l lst)))
              (final_descriptors'
                 (projT1 (l lst)))
            ∧ match oitem with
              | Some itemx =>
                  ∀ sx : 
                     composite_state IM',
                    sx =
                    equivocators_state_project
                      IM'
                      final_descriptors' s
                    → composite_valid IM'
                      (l itemx)
                      (sx, input itemx)
                      ∧ 
                      composite_transition
                      IM' 
                      (l itemx)
                      (sx, input itemx) =
                      (
                      destination itemx,
                      output itemx)
              | None =>
                  equivocators_state_project
                    IM' final_descriptors
                    (destination lst) =
                  equivocators_state_project
                    IM' final_descriptors'
                    s
              end
finite_valid_trace SeededX
  (equivocators_state_project IM' initial_descriptors
     is) trX
    specialize (Hproperx (finite_trace_last is tr')).message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
is: composite_state
  (sub_IM (equivocator_IM IM) selection)
IM':= λ i : sub_index selection, IM (`i): sub_index selection → VLSM message
tr, tr': list
  (composite_transition_item
     (sub_IM (equivocator_IM IM) selection))
lst: composite_transition_item
  (sub_IM (equivocator_IM IM) selection)
H: ∀ y : nat,
  y < length (tr' ++ [lst])
  → ∀ tr : list
             (composite_transition_item
                (sub_IM 
                   (equivocator_IM IM) selection)),
      finite_valid_trace SeededXE is tr
      → let final_state := 
          finite_trace_last is tr in
        y = length tr
        → ∀ final_descriptors : 
             equivocator_descriptors IM',
            proper_equivocator_descriptors IM'
              final_descriptors final_state
            → ∀ (initial_descriptors : 
                 equivocator_descriptors IM') 
                (trX : 
                 list
                   (composite_transition_item IM')),
                equivocators_trace_project IM'
                  final_descriptors tr =
                Some (trX, initial_descriptors)
                → finite_valid_trace SeededX
                    (equivocators_state_project
                      IM' initial_descriptors is)
                    trX
Htr: finite_valid_trace_from SeededXE is tr'
Hlst: finite_valid_trace_from SeededXE
  (finite_trace_last is tr') [lst]
Hinit: initial_state_prop is
final_state:= finite_trace_last is tr: state
  (composite_type
     (sub_IM (equivocator_IM IM)
        selection))
final_descriptors: equivocator_descriptors IM'
Hproper: proper_equivocator_descriptors IM'
  final_descriptors (destination lst)
initial_descriptors: equivocator_descriptors IM'
trX: list (composite_transition_item IM')
Hpr_trX: equivocators_trace_project IM'
  final_descriptors (tr' ++ [lst]) =
Some (trX, initial_descriptors)
Htr_lst: tr = tr' ++ [lst]
H': ∀ final_descriptors : equivocator_descriptors IM',
  proper_equivocator_descriptors IM'
    final_descriptors (finite_trace_last is tr')
  → ∀ (initial_descriptors : 
       equivocator_descriptors IM') 
      (trX : list (composite_transition_item IM')),
      equivocators_trace_project IM'
        final_descriptors tr' =
      Some (trX, initial_descriptors)
      → finite_valid_trace SeededX
          (equivocators_state_project IM'
             initial_descriptors is) trX
oitem: option (composite_transition_item IM')
final_descriptors': equivocator_descriptors IM'
Hprojectx: equivocators_transition_item_project IM'
  final_descriptors lst =
Some (oitem, final_descriptors')
Hitemx: match oitem with
| Some itemx =>
    (∃ Hex : existing_equivocator_label
               (IM' (projT1 (l lst)))
               (projT2 (l lst)),
       existT (projT1 (l lst))
         (existing_equivocator_label_extract
            (IM' (projT1 (l lst)))
            (projT2 (l lst)) Hex) = 
       l itemx)
    ∧ input lst = input itemx
      ∧ output lst = output itemx
        ∧ equivocators_state_project IM'
            final_descriptors
            (destination lst) =
          destination itemx
          ∧ final_descriptors'
              (projT1 (l lst)) =
            equivocator_label_descriptor
              (projT2 (l lst))
| None => True
end
Hproperx: composite_valid (equivocator_IM IM')
  (l lst)
  (finite_trace_last is tr', input lst)
→ composite_transition 
    (equivocator_IM IM') 
    (l lst)
    (finite_trace_last is tr', input lst) =
  (destination lst, output lst)
  → proper_equivocator_descriptors IM'
      final_descriptors'
      (finite_trace_last is tr')
    ∧ final_descriptors' =
      equivocator_descriptors_update IM'
        final_descriptors 
        (projT1 (l lst))
        (final_descriptors'
           (projT1 (l lst)))
      ∧ finite_trace_last is tr' =
        state_update (equivocator_IM IM')
          (destination lst)
          (projT1 (l lst))
          (finite_trace_last is tr'
             (projT1 (l lst)))
        ∧ previous_state_descriptor_prop
            (IM' (projT1 (l lst)))
            (final_descriptors
               (projT1 (l lst)))
            (finite_trace_last is tr'
               (projT1 (l lst)))
            (final_descriptors'
               (projT1 (l lst)))
          ∧ match oitem with
            | Some itemx =>
                ∀ sx : composite_state IM',
                  sx =
                  equivocators_state_project
                    IM' final_descriptors'
                    (finite_trace_last is
                      tr')
                  → composite_valid IM'
                      (l itemx)
                      (
                      sx, input itemx)
                    ∧ composite_transition
                      IM' 
                      (l itemx)
                      (sx, input itemx) =
                      (
                      destination itemx,
                      output itemx)
            | None =>
                equivocators_state_project
                  IM' final_descriptors
                  (destination lst) =
                equivocators_state_project
                  IM' final_descriptors'
                  (finite_trace_last is tr')
            end
finite_valid_trace SeededX
  (equivocators_state_project IM' initial_descriptors
     is) trX
    apply equivocators_trace_project_app_iff in Hpr_trX.message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
is: composite_state
  (sub_IM (equivocator_IM IM) selection)
IM':= λ i : sub_index selection, IM (`i): sub_index selection → VLSM message
tr, tr': list
  (composite_transition_item
     (sub_IM (equivocator_IM IM) selection))
lst: composite_transition_item
  (sub_IM (equivocator_IM IM) selection)
H: ∀ y : nat,
  y < length (tr' ++ [lst])
  → ∀ tr : list
             (composite_transition_item
                (sub_IM 
                   (equivocator_IM IM) selection)),
      finite_valid_trace SeededXE is tr
      → let final_state := 
          finite_trace_last is tr in
        y = length tr
        → ∀ final_descriptors : 
             equivocator_descriptors IM',
            proper_equivocator_descriptors IM'
              final_descriptors final_state
            → ∀ (initial_descriptors : 
                 equivocator_descriptors IM') 
                (trX : 
                 list
                   (composite_transition_item IM')),
                equivocators_trace_project IM'
                  final_descriptors tr =
                Some (trX, initial_descriptors)
                → finite_valid_trace SeededX
                    (equivocators_state_project
                      IM' initial_descriptors is)
                    trX
Htr: finite_valid_trace_from SeededXE is tr'
Hlst: finite_valid_trace_from SeededXE
  (finite_trace_last is tr') [lst]
Hinit: initial_state_prop is
final_state:= finite_trace_last is tr: state
  (composite_type
     (sub_IM (equivocator_IM IM)
        selection))
final_descriptors: equivocator_descriptors IM'
Hproper: proper_equivocator_descriptors IM'
  final_descriptors (destination lst)
initial_descriptors: equivocator_descriptors IM'
trX: list (composite_transition_item IM')
Hpr_trX: ∃ (preX
   sufX : list
            (composite_transition_item
               (λ i : sub_index selection,
                  IM (`i)))) 
  (eqv_descriptors' : equivocator_descriptors
                      (λ i : 
                      sub_index selection,
                      IM (`i))),
  equivocators_trace_project
    (λ i : sub_index selection, IM (`i))
    final_descriptors [lst] =
  Some (sufX, eqv_descriptors')
  ∧ equivocators_trace_project
      (λ i : sub_index selection, IM (`i))
      eqv_descriptors' tr' =
    Some (preX, initial_descriptors)
    ∧ trX = preX ++ sufX
Htr_lst: tr = tr' ++ [lst]
H': ∀ final_descriptors : equivocator_descriptors IM',
  proper_equivocator_descriptors IM'
    final_descriptors (finite_trace_last is tr')
  → ∀ (initial_descriptors : 
       equivocator_descriptors IM') 
      (trX : list (composite_transition_item IM')),
      equivocators_trace_project IM'
        final_descriptors tr' =
      Some (trX, initial_descriptors)
      → finite_valid_trace SeededX
          (equivocators_state_project IM'
             initial_descriptors is) trX
oitem: option (composite_transition_item IM')
final_descriptors': equivocator_descriptors IM'
Hprojectx: equivocators_transition_item_project IM'
  final_descriptors lst =
Some (oitem, final_descriptors')
Hitemx: match oitem with
| Some itemx =>
    (∃ Hex : existing_equivocator_label
               (IM' (projT1 (l lst)))
               (projT2 (l lst)),
       existT (projT1 (l lst))
         (existing_equivocator_label_extract
            (IM' (projT1 (l lst)))
            (projT2 (l lst)) Hex) = 
       l itemx)
    ∧ input lst = input itemx
      ∧ output lst = output itemx
        ∧ equivocators_state_project IM'
            final_descriptors
            (destination lst) =
          destination itemx
          ∧ final_descriptors'
              (projT1 (l lst)) =
            equivocator_label_descriptor
              (projT2 (l lst))
| None => True
end
Hproperx: composite_valid (equivocator_IM IM')
  (l lst)
  (finite_trace_last is tr', input lst)
→ composite_transition 
    (equivocator_IM IM') 
    (l lst)
    (finite_trace_last is tr', input lst) =
  (destination lst, output lst)
  → proper_equivocator_descriptors IM'
      final_descriptors'
      (finite_trace_last is tr')
    ∧ final_descriptors' =
      equivocator_descriptors_update IM'
        final_descriptors 
        (projT1 (l lst))
        (final_descriptors'
           (projT1 (l lst)))
      ∧ finite_trace_last is tr' =
        state_update (equivocator_IM IM')
          (destination lst)
          (projT1 (l lst))
          (finite_trace_last is tr'
             (projT1 (l lst)))
        ∧ previous_state_descriptor_prop
            (IM' (projT1 (l lst)))
            (final_descriptors
               (projT1 (l lst)))
            (finite_trace_last is tr'
               (projT1 (l lst)))
            (final_descriptors'
               (projT1 (l lst)))
          ∧ match oitem with
            | Some itemx =>
                ∀ sx : composite_state IM',
                  sx =
                  equivocators_state_project
                    IM' final_descriptors'
                    (finite_trace_last is
                      tr')
                  → composite_valid IM'
                      (l itemx)
                      (
                      sx, input itemx)
                    ∧ composite_transition
                      IM' 
                      (l itemx)
                      (sx, input itemx) =
                      (
                      destination itemx,
                      output itemx)
            | None =>
                equivocators_state_project
                  IM' final_descriptors
                  (destination lst) =
                equivocators_state_project
                  IM' final_descriptors'
                  (finite_trace_last is tr')
            end
finite_valid_trace SeededX
  (equivocators_state_project IM' initial_descriptors
     is) trX
    destruct Hpr_trX as [trX' [lstX [_final_descriptors' [_Hprojectx [Hpr_trX' Heq_trX]]]]].message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
is: composite_state
  (sub_IM (equivocator_IM IM) selection)
IM':= λ i : sub_index selection, IM (`i): sub_index selection → VLSM message
tr, tr': list
  (composite_transition_item
     (sub_IM (equivocator_IM IM) selection))
lst: composite_transition_item
  (sub_IM (equivocator_IM IM) selection)
H: ∀ y : nat,
  y < length (tr' ++ [lst])
  → ∀ tr : list
             (composite_transition_item
                (sub_IM 
                   (equivocator_IM IM) selection)),
      finite_valid_trace SeededXE is tr
      → let final_state := 
          finite_trace_last is tr in
        y = length tr
        → ∀ final_descriptors : 
             equivocator_descriptors IM',
            proper_equivocator_descriptors IM'
              final_descriptors final_state
            → ∀ (initial_descriptors : 
                 equivocator_descriptors IM') 
                (trX : 
                 list
                   (composite_transition_item IM')),
                equivocators_trace_project IM'
                  final_descriptors tr =
                Some (trX, initial_descriptors)
                → finite_valid_trace SeededX
                    (equivocators_state_project
                      IM' initial_descriptors is)
                    trX
Htr: finite_valid_trace_from SeededXE is tr'
Hlst: finite_valid_trace_from SeededXE
  (finite_trace_last is tr') [lst]
Hinit: initial_state_prop is
final_state:= finite_trace_last is tr: state
  (composite_type
     (sub_IM (equivocator_IM IM)
        selection))
final_descriptors: equivocator_descriptors IM'
Hproper: proper_equivocator_descriptors IM'
  final_descriptors (destination lst)
initial_descriptors: equivocator_descriptors IM'
trX: list (composite_transition_item IM')
trX', lstX: list
  (composite_transition_item
     (λ i : sub_index selection, IM (`i)))
_final_descriptors': equivocator_descriptors
  (λ i : sub_index selection,
     IM (`i))
_Hprojectx: equivocators_trace_project
  (λ i : sub_index selection, IM (`i))
  final_descriptors [lst] =
Some (lstX, _final_descriptors')
Hpr_trX': equivocators_trace_project
  (λ i : sub_index selection, IM (`i))
  _final_descriptors' tr' =
Some (trX', initial_descriptors)
Heq_trX: trX = trX' ++ lstX
Htr_lst: tr = tr' ++ [lst]
H': ∀ final_descriptors : equivocator_descriptors IM',
  proper_equivocator_descriptors IM'
    final_descriptors (finite_trace_last is tr')
  → ∀ (initial_descriptors : 
       equivocator_descriptors IM') 
      (trX : list (composite_transition_item IM')),
      equivocators_trace_project IM'
        final_descriptors tr' =
      Some (trX, initial_descriptors)
      → finite_valid_trace SeededX
          (equivocators_state_project IM'
             initial_descriptors is) trX
oitem: option (composite_transition_item IM')
final_descriptors': equivocator_descriptors IM'
Hprojectx: equivocators_transition_item_project IM'
  final_descriptors lst =
Some (oitem, final_descriptors')
Hitemx: match oitem with
| Some itemx =>
    (∃ Hex : existing_equivocator_label
               (IM' (projT1 (l lst)))
               (projT2 (l lst)),
       existT (projT1 (l lst))
         (existing_equivocator_label_extract
            (IM' (projT1 (l lst)))
            (projT2 (l lst)) Hex) = 
       l itemx)
    ∧ input lst = input itemx
      ∧ output lst = output itemx
        ∧ equivocators_state_project IM'
            final_descriptors
            (destination lst) =
          destination itemx
          ∧ final_descriptors'
              (projT1 (l lst)) =
            equivocator_label_descriptor
              (projT2 (l lst))
| None => True
end
Hproperx: composite_valid (equivocator_IM IM')
  (l lst)
  (finite_trace_last is tr', input lst)
→ composite_transition 
    (equivocator_IM IM') 
    (l lst)
    (finite_trace_last is tr', input lst) =
  (destination lst, output lst)
  → proper_equivocator_descriptors IM'
      final_descriptors'
      (finite_trace_last is tr')
    ∧ final_descriptors' =
      equivocator_descriptors_update IM'
        final_descriptors 
        (projT1 (l lst))
        (final_descriptors'
           (projT1 (l lst)))
      ∧ finite_trace_last is tr' =
        state_update (equivocator_IM IM')
          (destination lst)
          (projT1 (l lst))
          (finite_trace_last is tr'
             (projT1 (l lst)))
        ∧ previous_state_descriptor_prop
            (IM' (projT1 (l lst)))
            (final_descriptors
               (projT1 (l lst)))
            (finite_trace_last is tr'
               (projT1 (l lst)))
            (final_descriptors'
               (projT1 (l lst)))
          ∧ match oitem with
            | Some itemx =>
                ∀ sx : composite_state IM',
                  sx =
                  equivocators_state_project
                    IM' final_descriptors'
                    (finite_trace_last is
                      tr')
                  → composite_valid IM'
                      (l itemx)
                      (
                      sx, input itemx)
                    ∧ composite_transition
                      IM' 
                      (l itemx)
                      (sx, input itemx) =
                      (
                      destination itemx,
                      output itemx)
            | None =>
                equivocators_state_project
                  IM' final_descriptors
                  (destination lst) =
                equivocators_state_project
                  IM' final_descriptors'
                  (finite_trace_last is tr')
            end
finite_valid_trace SeededX
  (equivocators_state_project IM' initial_descriptors
     is) trX
    subst trX tr.message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
is: composite_state
  (sub_IM (equivocator_IM IM) selection)
IM':= λ i : sub_index selection, IM (`i): sub_index selection → VLSM message
tr': list
  (composite_transition_item
     (sub_IM (equivocator_IM IM) selection))
lst: composite_transition_item
  (sub_IM (equivocator_IM IM) selection)
H: ∀ y : nat,
  y < length (tr' ++ [lst])
  → ∀ tr : list
             (composite_transition_item
                (sub_IM 
                   (equivocator_IM IM) selection)),
      finite_valid_trace SeededXE is tr
      → let final_state := 
          finite_trace_last is tr in
        y = length tr
        → ∀ final_descriptors : 
             equivocator_descriptors IM',
            proper_equivocator_descriptors IM'
              final_descriptors final_state
            → ∀ (initial_descriptors : 
                 equivocator_descriptors IM') 
                (trX : 
                 list
                   (composite_transition_item IM')),
                equivocators_trace_project IM'
                  final_descriptors tr =
                Some (trX, initial_descriptors)
                → finite_valid_trace SeededX
                    (equivocators_state_project
                      IM' initial_descriptors is)
                    trX
Htr: finite_valid_trace_from SeededXE is tr'
Hlst: finite_valid_trace_from SeededXE
  (finite_trace_last is tr') [lst]
Hinit: initial_state_prop is
final_state:= finite_trace_last is (tr' ++ [lst]): state
  (composite_type
     (sub_IM (equivocator_IM IM)
        selection))
final_descriptors: equivocator_descriptors IM'
Hproper: proper_equivocator_descriptors IM'
  final_descriptors (destination lst)
initial_descriptors: equivocator_descriptors IM'
trX', lstX: list
  (composite_transition_item
     (λ i : sub_index selection, IM (`i)))
_final_descriptors': equivocator_descriptors
  (λ i : sub_index selection,
     IM (`i))
_Hprojectx: equivocators_trace_project
  (λ i : sub_index selection, IM (`i))
  final_descriptors [lst] =
Some (lstX, _final_descriptors')
Hpr_trX': equivocators_trace_project
  (λ i : sub_index selection, IM (`i))
  _final_descriptors' tr' =
Some (trX', initial_descriptors)
H': ∀ final_descriptors : equivocator_descriptors IM',
  proper_equivocator_descriptors IM'
    final_descriptors (finite_trace_last is tr')
  → ∀ (initial_descriptors : 
       equivocator_descriptors IM') 
      (trX : list (composite_transition_item IM')),
      equivocators_trace_project IM'
        final_descriptors tr' =
      Some (trX, initial_descriptors)
      → finite_valid_trace SeededX
          (equivocators_state_project IM'
             initial_descriptors is) trX
oitem: option (composite_transition_item IM')
final_descriptors': equivocator_descriptors IM'
Hprojectx: equivocators_transition_item_project IM'
  final_descriptors lst =
Some (oitem, final_descriptors')
Hitemx: match oitem with
| Some itemx =>
    (∃ Hex : existing_equivocator_label
               (IM' (projT1 (l lst)))
               (projT2 (l lst)),
       existT (projT1 (l lst))
         (existing_equivocator_label_extract
            (IM' (projT1 (l lst)))
            (projT2 (l lst)) Hex) = 
       l itemx)
    ∧ input lst = input itemx
      ∧ output lst = output itemx
        ∧ equivocators_state_project IM'
            final_descriptors
            (destination lst) =
          destination itemx
          ∧ final_descriptors'
              (projT1 (l lst)) =
            equivocator_label_descriptor
              (projT2 (l lst))
| None => True
end
Hproperx: composite_valid (equivocator_IM IM')
  (l lst)
  (finite_trace_last is tr', input lst)
→ composite_transition 
    (equivocator_IM IM') 
    (l lst)
    (finite_trace_last is tr', input lst) =
  (destination lst, output lst)
  → proper_equivocator_descriptors IM'
      final_descriptors'
      (finite_trace_last is tr')
    ∧ final_descriptors' =
      equivocator_descriptors_update IM'
        final_descriptors 
        (projT1 (l lst))
        (final_descriptors'
           (projT1 (l lst)))
      ∧ finite_trace_last is tr' =
        state_update (equivocator_IM IM')
          (destination lst)
          (projT1 (l lst))
          (finite_trace_last is tr'
             (projT1 (l lst)))
        ∧ previous_state_descriptor_prop
            (IM' (projT1 (l lst)))
            (final_descriptors
               (projT1 (l lst)))
            (finite_trace_last is tr'
               (projT1 (l lst)))
            (final_descriptors'
               (projT1 (l lst)))
          ∧ match oitem with
            | Some itemx =>
                ∀ sx : composite_state IM',
                  sx =
                  equivocators_state_project
                    IM' final_descriptors'
                    (finite_trace_last is
                      tr')
                  → composite_valid IM'
                      (l itemx)
                      (
                      sx, input itemx)
                    ∧ composite_transition
                      IM' 
                      (l itemx)
                      (sx, input itemx) =
                      (
                      destination itemx,
                      output itemx)
            | None =>
                equivocators_state_project
                  IM' final_descriptors
                  (destination lst) =
                equivocators_state_project
                  IM' final_descriptors'
                  (finite_trace_last is tr')
            end
finite_valid_trace SeededX
  (equivocators_state_project IM' initial_descriptors
     is) (trX' ++ lstX)
    simpl in _Hprojectx.message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
is: composite_state
  (sub_IM (equivocator_IM IM) selection)
IM':= λ i : sub_index selection, IM (`i): sub_index selection → VLSM message
tr': list
  (composite_transition_item
     (sub_IM (equivocator_IM IM) selection))
lst: composite_transition_item
  (sub_IM (equivocator_IM IM) selection)
H: ∀ y : nat,
  y < length (tr' ++ [lst])
  → ∀ tr : list
             (composite_transition_item
                (sub_IM 
                   (equivocator_IM IM) selection)),
      finite_valid_trace SeededXE is tr
      → let final_state := 
          finite_trace_last is tr in
        y = length tr
        → ∀ final_descriptors : 
             equivocator_descriptors IM',
            proper_equivocator_descriptors IM'
              final_descriptors final_state
            → ∀ (initial_descriptors : 
                 equivocator_descriptors IM') 
                (trX : 
                 list
                   (composite_transition_item IM')),
                equivocators_trace_project IM'
                  final_descriptors tr =
                Some (trX, initial_descriptors)
                → finite_valid_trace SeededX
                    (equivocators_state_project
                      IM' initial_descriptors is)
                    trX
Htr: finite_valid_trace_from SeededXE is tr'
Hlst: finite_valid_trace_from SeededXE
  (finite_trace_last is tr') [lst]
Hinit: initial_state_prop is
final_state:= finite_trace_last is (tr' ++ [lst]): state
  (composite_type
     (sub_IM (equivocator_IM IM)
        selection))
final_descriptors: equivocator_descriptors IM'
Hproper: proper_equivocator_descriptors IM'
  final_descriptors (destination lst)
initial_descriptors: equivocator_descriptors IM'
trX', lstX: list
  (composite_transition_item
     (λ i : sub_index selection, IM (`i)))
_final_descriptors': equivocator_descriptors
  (λ i : sub_index selection,
     IM (`i))
_Hprojectx: match
  equivocators_transition_item_project
    (λ i : sub_index selection, IM (`i))
    final_descriptors lst
with
| Some (Some item', odescriptor) =>
    Some ([item'], odescriptor)
| Some (None, odescriptor) =>
    Some ([], odescriptor)
| None => None
end = Some (lstX, _final_descriptors')
Hpr_trX': equivocators_trace_project
  (λ i : sub_index selection, IM (`i))
  _final_descriptors' tr' =
Some (trX', initial_descriptors)
H': ∀ final_descriptors : equivocator_descriptors IM',
  proper_equivocator_descriptors IM'
    final_descriptors (finite_trace_last is tr')
  → ∀ (initial_descriptors : 
       equivocator_descriptors IM') 
      (trX : list (composite_transition_item IM')),
      equivocators_trace_project IM'
        final_descriptors tr' =
      Some (trX, initial_descriptors)
      → finite_valid_trace SeededX
          (equivocators_state_project IM'
             initial_descriptors is) trX
oitem: option (composite_transition_item IM')
final_descriptors': equivocator_descriptors IM'
Hprojectx: equivocators_transition_item_project IM'
  final_descriptors lst =
Some (oitem, final_descriptors')
Hitemx: match oitem with
| Some itemx =>
    (∃ Hex : existing_equivocator_label
               (IM' (projT1 (l lst)))
               (projT2 (l lst)),
       existT (projT1 (l lst))
         (existing_equivocator_label_extract
            (IM' (projT1 (l lst)))
            (projT2 (l lst)) Hex) = 
       l itemx)
    ∧ input lst = input itemx
      ∧ output lst = output itemx
        ∧ equivocators_state_project IM'
            final_descriptors
            (destination lst) =
          destination itemx
          ∧ final_descriptors'
              (projT1 (l lst)) =
            equivocator_label_descriptor
              (projT2 (l lst))
| None => True
end
Hproperx: composite_valid (equivocator_IM IM')
  (l lst)
  (finite_trace_last is tr', input lst)
→ composite_transition 
    (equivocator_IM IM') 
    (l lst)
    (finite_trace_last is tr', input lst) =
  (destination lst, output lst)
  → proper_equivocator_descriptors IM'
      final_descriptors'
      (finite_trace_last is tr')
    ∧ final_descriptors' =
      equivocator_descriptors_update IM'
        final_descriptors 
        (projT1 (l lst))
        (final_descriptors'
           (projT1 (l lst)))
      ∧ finite_trace_last is tr' =
        state_update (equivocator_IM IM')
          (destination lst)
          (projT1 (l lst))
          (finite_trace_last is tr'
             (projT1 (l lst)))
        ∧ previous_state_descriptor_prop
            (IM' (projT1 (l lst)))
            (final_descriptors
               (projT1 (l lst)))
            (finite_trace_last is tr'
               (projT1 (l lst)))
            (final_descriptors'
               (projT1 (l lst)))
          ∧ match oitem with
            | Some itemx =>
                ∀ sx : composite_state IM',
                  sx =
                  equivocators_state_project
                    IM' final_descriptors'
                    (finite_trace_last is
                      tr')
                  → composite_valid IM'
                      (l itemx)
                      (
                      sx, input itemx)
                    ∧ composite_transition
                      IM' 
                      (l itemx)
                      (sx, input itemx) =
                      (
                      destination itemx,
                      output itemx)
            | None =>
                equivocators_state_project
                  IM' final_descriptors
                  (destination lst) =
                equivocators_state_project
                  IM' final_descriptors'
                  (finite_trace_last is tr')
            end
finite_valid_trace SeededX
  (equivocators_state_project IM' initial_descriptors
     is) (trX' ++ lstX)
    replace (equivocators_transition_item_project _ _ _) with (Some (oitem, final_descriptors'))
      in _Hprojectx.message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
is: composite_state
  (sub_IM (equivocator_IM IM) selection)
IM':= λ i : sub_index selection, IM (`i): sub_index selection → VLSM message
tr': list
  (composite_transition_item
     (sub_IM (equivocator_IM IM) selection))
lst: composite_transition_item
  (sub_IM (equivocator_IM IM) selection)
H: ∀ y : nat,
  y < length (tr' ++ [lst])
  → ∀ tr : list
             (composite_transition_item
                (sub_IM 
                   (equivocator_IM IM) selection)),
      finite_valid_trace SeededXE is tr
      → let final_state := 
          finite_trace_last is tr in
        y = length tr
        → ∀ final_descriptors : 
             equivocator_descriptors IM',
            proper_equivocator_descriptors IM'
              final_descriptors final_state
            → ∀ (initial_descriptors : 
                 equivocator_descriptors IM') 
                (trX : 
                 list
                   (composite_transition_item IM')),
                equivocators_trace_project IM'
                  final_descriptors tr =
                Some (trX, initial_descriptors)
                → finite_valid_trace SeededX
                    (equivocators_state_project
                      IM' initial_descriptors is)
                    trX
Htr: finite_valid_trace_from SeededXE is tr'
Hlst: finite_valid_trace_from SeededXE
  (finite_trace_last is tr') [lst]
Hinit: initial_state_prop is
final_state:= finite_trace_last is (tr' ++ [lst]): state
  (composite_type
     (sub_IM (equivocator_IM IM)
        selection))
final_descriptors: equivocator_descriptors IM'
Hproper: proper_equivocator_descriptors IM'
  final_descriptors (destination lst)
initial_descriptors: equivocator_descriptors IM'
trX', lstX: list
  (composite_transition_item
     (λ i : sub_index selection, IM (`i)))
_final_descriptors': equivocator_descriptors
  (λ i : sub_index selection,
     IM (`i))
oitem: option (composite_transition_item IM')
final_descriptors': equivocator_descriptors IM'
_Hprojectx: match oitem with
| Some item' =>
    Some ([item'], final_descriptors')
| None => Some ([], final_descriptors')
end = Some (lstX, _final_descriptors')
Hpr_trX': equivocators_trace_project
  (λ i : sub_index selection, IM (`i))
  _final_descriptors' tr' =
Some (trX', initial_descriptors)
H': ∀ final_descriptors : equivocator_descriptors IM',
  proper_equivocator_descriptors IM'
    final_descriptors (finite_trace_last is tr')
  → ∀ (initial_descriptors : 
       equivocator_descriptors IM') 
      (trX : list (composite_transition_item IM')),
      equivocators_trace_project IM'
        final_descriptors tr' =
      Some (trX, initial_descriptors)
      → finite_valid_trace SeededX
          (equivocators_state_project IM'
             initial_descriptors is) trX
Hprojectx: equivocators_transition_item_project IM'
  final_descriptors lst =
Some (oitem, final_descriptors')
Hitemx: match oitem with
| Some itemx =>
    (∃ Hex : existing_equivocator_label
               (IM' (projT1 (l lst)))
               (projT2 (l lst)),
       existT (projT1 (l lst))
         (existing_equivocator_label_extract
            (IM' (projT1 (l lst)))
            (projT2 (l lst)) Hex) = 
       l itemx)
    ∧ input lst = input itemx
      ∧ output lst = output itemx
        ∧ equivocators_state_project IM'
            final_descriptors
            (destination lst) =
          destination itemx
          ∧ final_descriptors'
              (projT1 (l lst)) =
            equivocator_label_descriptor
              (projT2 (l lst))
| None => True
end
Hproperx: composite_valid (equivocator_IM IM')
  (l lst)
  (finite_trace_last is tr', input lst)
→ composite_transition 
    (equivocator_IM IM') 
    (l lst)
    (finite_trace_last is tr', input lst) =
  (destination lst, output lst)
  → proper_equivocator_descriptors IM'
      final_descriptors'
      (finite_trace_last is tr')
    ∧ final_descriptors' =
      equivocator_descriptors_update IM'
        final_descriptors 
        (projT1 (l lst))
        (final_descriptors'
           (projT1 (l lst)))
      ∧ finite_trace_last is tr' =
        state_update (equivocator_IM IM')
          (destination lst)
          (projT1 (l lst))
          (finite_trace_last is tr'
             (projT1 (l lst)))
        ∧ previous_state_descriptor_prop
            (IM' (projT1 (l lst)))
            (final_descriptors
               (projT1 (l lst)))
            (finite_trace_last is tr'
               (projT1 (l lst)))
            (final_descriptors'
               (projT1 (l lst)))
          ∧ match oitem with
            | Some itemx =>
                ∀ sx : composite_state IM',
                  sx =
                  equivocators_state_project
                    IM' final_descriptors'
                    (finite_trace_last is
                      tr')
                  → composite_valid IM'
                      (l itemx)
                      (
                      sx, input itemx)
                    ∧ composite_transition
                      IM' 
                      (l itemx)
                      (sx, input itemx) =
                      (
                      destination itemx,
                      output itemx)
            | None =>
                equivocators_state_project
                  IM' final_descriptors
                  (destination lst) =
                equivocators_state_project
                  IM' final_descriptors'
                  (finite_trace_last is tr')
            end
finite_valid_trace SeededX
  (equivocators_state_project IM' initial_descriptors
     is) (trX' ++ lstX)
    assert (Heq_final_descriptors' : final_descriptors' = _final_descriptors')
      by (destruct oitem; inversion _Hprojectx; done).message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
is: composite_state
  (sub_IM (equivocator_IM IM) selection)
IM':= λ i : sub_index selection, IM (`i): sub_index selection → VLSM message
tr': list
  (composite_transition_item
     (sub_IM (equivocator_IM IM) selection))
lst: composite_transition_item
  (sub_IM (equivocator_IM IM) selection)
H: ∀ y : nat,
  y < length (tr' ++ [lst])
  → ∀ tr : list
             (composite_transition_item
                (sub_IM 
                   (equivocator_IM IM) selection)),
      finite_valid_trace SeededXE is tr
      → let final_state := 
          finite_trace_last is tr in
        y = length tr
        → ∀ final_descriptors : 
             equivocator_descriptors IM',
            proper_equivocator_descriptors IM'
              final_descriptors final_state
            → ∀ (initial_descriptors : 
                 equivocator_descriptors IM') 
                (trX : 
                 list
                   (composite_transition_item IM')),
                equivocators_trace_project IM'
                  final_descriptors tr =
                Some (trX, initial_descriptors)
                → finite_valid_trace SeededX
                    (equivocators_state_project
                      IM' initial_descriptors is)
                    trX
Htr: finite_valid_trace_from SeededXE is tr'
Hlst: finite_valid_trace_from SeededXE
  (finite_trace_last is tr') [lst]
Hinit: initial_state_prop is
final_state:= finite_trace_last is (tr' ++ [lst]): state
  (composite_type
     (sub_IM (equivocator_IM IM)
        selection))
final_descriptors: equivocator_descriptors IM'
Hproper: proper_equivocator_descriptors IM'
  final_descriptors (destination lst)
initial_descriptors: equivocator_descriptors IM'
trX', lstX: list
  (composite_transition_item
     (λ i : sub_index selection, IM (`i)))
_final_descriptors': equivocator_descriptors
  (λ i : sub_index selection,
     IM (`i))
oitem: option (composite_transition_item IM')
final_descriptors': equivocator_descriptors IM'
_Hprojectx: match oitem with
| Some item' =>
    Some ([item'], final_descriptors')
| None => Some ([], final_descriptors')
end = Some (lstX, _final_descriptors')
Hpr_trX': equivocators_trace_project
  (λ i : sub_index selection, IM (`i))
  _final_descriptors' tr' =
Some (trX', initial_descriptors)
H': ∀ final_descriptors : equivocator_descriptors IM',
  proper_equivocator_descriptors IM'
    final_descriptors (finite_trace_last is tr')
  → ∀ (initial_descriptors : 
       equivocator_descriptors IM') 
      (trX : list (composite_transition_item IM')),
      equivocators_trace_project IM'
        final_descriptors tr' =
      Some (trX, initial_descriptors)
      → finite_valid_trace SeededX
          (equivocators_state_project IM'
             initial_descriptors is) trX
Hprojectx: equivocators_transition_item_project IM'
  final_descriptors lst =
Some (oitem, final_descriptors')
Hitemx: match oitem with
| Some itemx =>
    (∃ Hex : existing_equivocator_label
               (IM' (projT1 (l lst)))
               (projT2 (l lst)),
       existT (projT1 (l lst))
         (existing_equivocator_label_extract
            (IM' (projT1 (l lst)))
            (projT2 (l lst)) Hex) = 
       l itemx)
    ∧ input lst = input itemx
      ∧ output lst = output itemx
        ∧ equivocators_state_project IM'
            final_descriptors
            (destination lst) =
          destination itemx
          ∧ final_descriptors'
              (projT1 (l lst)) =
            equivocator_label_descriptor
              (projT2 (l lst))
| None => True
end
Hproperx: composite_valid (equivocator_IM IM')
  (l lst)
  (finite_trace_last is tr', input lst)
→ composite_transition 
    (equivocator_IM IM') 
    (l lst)
    (finite_trace_last is tr', input lst) =
  (destination lst, output lst)
  → proper_equivocator_descriptors IM'
      final_descriptors'
      (finite_trace_last is tr')
    ∧ final_descriptors' =
      equivocator_descriptors_update IM'
        final_descriptors 
        (projT1 (l lst))
        (final_descriptors'
           (projT1 (l lst)))
      ∧ finite_trace_last is tr' =
        state_update (equivocator_IM IM')
          (destination lst)
          (projT1 (l lst))
          (finite_trace_last is tr'
             (projT1 (l lst)))
        ∧ previous_state_descriptor_prop
            (IM' (projT1 (l lst)))
            (final_descriptors
               (projT1 (l lst)))
            (finite_trace_last is tr'
               (projT1 (l lst)))
            (final_descriptors'
               (projT1 (l lst)))
          ∧ match oitem with
            | Some itemx =>
                ∀ sx : composite_state IM',
                  sx =
                  equivocators_state_project
                    IM' final_descriptors'
                    (finite_trace_last is
                      tr')
                  → composite_valid IM'
                      (l itemx)
                      (
                      sx, input itemx)
                    ∧ composite_transition
                      IM' 
                      (l itemx)
                      (sx, input itemx) =
                      (
                      destination itemx,
                      output itemx)
            | None =>
                equivocators_state_project
                  IM' final_descriptors
                  (destination lst) =
                equivocators_state_project
                  IM' final_descriptors'
                  (finite_trace_last is tr')
            end
Heq_final_descriptors': final_descriptors' =
_final_descriptors'
finite_valid_trace SeededX
  (equivocators_state_project IM' initial_descriptors
     is) (trX' ++ lstX)
    subst _final_descriptors'.message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
is: composite_state
  (sub_IM (equivocator_IM IM) selection)
IM':= λ i : sub_index selection, IM (`i): sub_index selection → VLSM message
tr': list
  (composite_transition_item
     (sub_IM (equivocator_IM IM) selection))
lst: composite_transition_item
  (sub_IM (equivocator_IM IM) selection)
H: ∀ y : nat,
  y < length (tr' ++ [lst])
  → ∀ tr : list
             (composite_transition_item
                (sub_IM 
                   (equivocator_IM IM) selection)),
      finite_valid_trace SeededXE is tr
      → let final_state := 
          finite_trace_last is tr in
        y = length tr
        → ∀ final_descriptors : 
             equivocator_descriptors IM',
            proper_equivocator_descriptors IM'
              final_descriptors final_state
            → ∀ (initial_descriptors : 
                 equivocator_descriptors IM') 
                (trX : 
                 list
                   (composite_transition_item IM')),
                equivocators_trace_project IM'
                  final_descriptors tr =
                Some (trX, initial_descriptors)
                → finite_valid_trace SeededX
                    (equivocators_state_project
                      IM' initial_descriptors is)
                    trX
Htr: finite_valid_trace_from SeededXE is tr'
Hlst: finite_valid_trace_from SeededXE
  (finite_trace_last is tr') [lst]
Hinit: initial_state_prop is
final_state:= finite_trace_last is (tr' ++ [lst]): state
  (composite_type
     (sub_IM (equivocator_IM IM)
        selection))
final_descriptors: equivocator_descriptors IM'
Hproper: proper_equivocator_descriptors IM'
  final_descriptors (destination lst)
initial_descriptors: equivocator_descriptors IM'
trX', lstX: list
  (composite_transition_item
     (λ i : sub_index selection, IM (`i)))
oitem: option (composite_transition_item IM')
final_descriptors': equivocator_descriptors IM'
Hpr_trX': equivocators_trace_project
  (λ i : sub_index selection, IM (`i))
  final_descriptors' tr' =
Some (trX', initial_descriptors)
_Hprojectx: match oitem with
| Some item' =>
    Some ([item'], final_descriptors')
| None => Some ([], final_descriptors')
end = Some (lstX, final_descriptors')
H': ∀ final_descriptors : equivocator_descriptors IM',
  proper_equivocator_descriptors IM'
    final_descriptors (finite_trace_last is tr')
  → ∀ (initial_descriptors : 
       equivocator_descriptors IM') 
      (trX : list (composite_transition_item IM')),
      equivocators_trace_project IM'
        final_descriptors tr' =
      Some (trX, initial_descriptors)
      → finite_valid_trace SeededX
          (equivocators_state_project IM'
             initial_descriptors is) trX
Hprojectx: equivocators_transition_item_project IM'
  final_descriptors lst =
Some (oitem, final_descriptors')
Hitemx: match oitem with
| Some itemx =>
    (∃ Hex : existing_equivocator_label
               (IM' (projT1 (l lst)))
               (projT2 (l lst)),
       existT (projT1 (l lst))
         (existing_equivocator_label_extract
            (IM' (projT1 (l lst)))
            (projT2 (l lst)) Hex) = 
       l itemx)
    ∧ input lst = input itemx
      ∧ output lst = output itemx
        ∧ equivocators_state_project IM'
            final_descriptors
            (destination lst) =
          destination itemx
          ∧ final_descriptors'
              (projT1 (l lst)) =
            equivocator_label_descriptor
              (projT2 (l lst))
| None => True
end
Hproperx: composite_valid (equivocator_IM IM')
  (l lst)
  (finite_trace_last is tr', input lst)
→ composite_transition 
    (equivocator_IM IM') 
    (l lst)
    (finite_trace_last is tr', input lst) =
  (destination lst, output lst)
  → proper_equivocator_descriptors IM'
      final_descriptors'
      (finite_trace_last is tr')
    ∧ final_descriptors' =
      equivocator_descriptors_update IM'
        final_descriptors 
        (projT1 (l lst))
        (final_descriptors'
           (projT1 (l lst)))
      ∧ finite_trace_last is tr' =
        state_update (equivocator_IM IM')
          (destination lst)
          (projT1 (l lst))
          (finite_trace_last is tr'
             (projT1 (l lst)))
        ∧ previous_state_descriptor_prop
            (IM' (projT1 (l lst)))
            (final_descriptors
               (projT1 (l lst)))
            (finite_trace_last is tr'
               (projT1 (l lst)))
            (final_descriptors'
               (projT1 (l lst)))
          ∧ match oitem with
            | Some itemx =>
                ∀ sx : composite_state IM',
                  sx =
                  equivocators_state_project
                    IM' final_descriptors'
                    (finite_trace_last is
                      tr')
                  → composite_valid IM'
                      (l itemx)
                      (
                      sx, input itemx)
                    ∧ composite_transition
                      IM' 
                      (l itemx)
                      (sx, input itemx) =
                      (
                      destination itemx,
                      output itemx)
            | None =>
                equivocators_state_project
                  IM' final_descriptors
                  (destination lst) =
                equivocators_state_project
                  IM' final_descriptors'
                  (finite_trace_last is tr')
            end
finite_valid_trace SeededX
  (equivocators_state_project IM' initial_descriptors
     is) (trX' ++ lstX)
    inversion Hlst.message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
is: composite_state
  (sub_IM (equivocator_IM IM) selection)
IM':= λ i : sub_index selection, IM (`i): sub_index selection → VLSM message
tr': list
  (composite_transition_item
     (sub_IM (equivocator_IM IM) selection))
lst: composite_transition_item
  (sub_IM (equivocator_IM IM) selection)
H: ∀ y : nat,
  y < length (tr' ++ [lst])
  → ∀ tr : list
             (composite_transition_item
                (sub_IM 
                   (equivocator_IM IM) selection)),
      finite_valid_trace SeededXE is tr
      → let final_state := 
          finite_trace_last is tr in
        y = length tr
        → ∀ final_descriptors : 
             equivocator_descriptors IM',
            proper_equivocator_descriptors IM'
              final_descriptors final_state
            → ∀ (initial_descriptors : 
                 equivocator_descriptors IM') 
                (trX : 
                 list
                   (composite_transition_item IM')),
                equivocators_trace_project IM'
                  final_descriptors tr =
                Some (trX, initial_descriptors)
                → finite_valid_trace SeededX
                    (equivocators_state_project
                      IM' initial_descriptors is)
                    trX
Htr: finite_valid_trace_from SeededXE is tr'
Hlst: finite_valid_trace_from SeededXE
  (finite_trace_last is tr') [lst]
Hinit: initial_state_prop is
final_state:= finite_trace_last is (tr' ++ [lst]): state
  (composite_type
     (sub_IM (equivocator_IM IM)
        selection))
final_descriptors: equivocator_descriptors IM'
Hproper: proper_equivocator_descriptors IM'
  final_descriptors (destination lst)
initial_descriptors: equivocator_descriptors IM'
trX', lstX: list
  (composite_transition_item
     (λ i : sub_index selection, IM (`i)))
oitem: option (composite_transition_item IM')
final_descriptors': equivocator_descriptors IM'
Hpr_trX': equivocators_trace_project
  (λ i : sub_index selection, IM (`i))
  final_descriptors' tr' =
Some (trX', initial_descriptors)
_Hprojectx: match oitem with
| Some item' =>
    Some ([item'], final_descriptors')
| None => Some ([], final_descriptors')
end = Some (lstX, final_descriptors')
H': ∀ final_descriptors : equivocator_descriptors IM',
  proper_equivocator_descriptors IM'
    final_descriptors (finite_trace_last is tr')
  → ∀ (initial_descriptors : 
       equivocator_descriptors IM') 
      (trX : list (composite_transition_item IM')),
      equivocators_trace_project IM'
        final_descriptors tr' =
      Some (trX, initial_descriptors)
      → finite_valid_trace SeededX
          (equivocators_state_project IM'
             initial_descriptors is) trX
Hprojectx: equivocators_transition_item_project IM'
  final_descriptors lst =
Some (oitem, final_descriptors')
Hitemx: match oitem with
| Some itemx =>
    (∃ Hex : existing_equivocator_label
               (IM' (projT1 (VLSM.l lst)))
               (projT2 (VLSM.l lst)),
       existT (projT1 (VLSM.l lst))
         (existing_equivocator_label_extract
            (IM' (projT1 (VLSM.l lst)))
            (projT2 (VLSM.l lst)) Hex) =
       VLSM.l itemx)
    ∧ input lst = input itemx
      ∧ output lst = output itemx
        ∧ equivocators_state_project IM'
            final_descriptors
            (destination lst) =
          destination itemx
          ∧ final_descriptors'
              (projT1 (VLSM.l lst)) =
            equivocator_label_descriptor
              (projT2 (VLSM.l lst))
| None => True
end
Hproperx: composite_valid (equivocator_IM IM')
  (VLSM.l lst)
  (finite_trace_last is tr', input lst)
→ composite_transition 
    (equivocator_IM IM') 
    (VLSM.l lst)
    (finite_trace_last is tr', input lst) =
  (destination lst, output lst)
  → proper_equivocator_descriptors IM'
      final_descriptors'
      (finite_trace_last is tr')
    ∧ final_descriptors' =
      equivocator_descriptors_update IM'
        final_descriptors
        (projT1 (VLSM.l lst))
        (final_descriptors'
           (projT1 (VLSM.l lst)))
      ∧ finite_trace_last is tr' =
        state_update (equivocator_IM IM')
          (destination lst)
          (projT1 (VLSM.l lst))
          (finite_trace_last is tr'
             (projT1 (VLSM.l lst)))
        ∧ previous_state_descriptor_prop
            (IM' (projT1 (VLSM.l lst)))
            (final_descriptors
               (projT1 (VLSM.l lst)))
            (finite_trace_last is tr'
               (projT1 (VLSM.l lst)))
            (final_descriptors'
               (projT1 (VLSM.l lst)))
          ∧ match oitem with
            | Some itemx =>
                ∀ sx : composite_state IM',
                  sx =
                  equivocators_state_project
                    IM' final_descriptors'
                    (finite_trace_last is
                      tr')
                  → composite_valid IM'
                      (VLSM.l itemx)
                      (
                      sx, input itemx)
                    ∧ composite_transition
                      IM' 
                      (VLSM.l itemx)
                      (sx, input itemx) =
                      (
                      destination itemx,
                      output itemx)
            | None =>
                equivocators_state_project
                  IM' final_descriptors
                  (destination lst) =
                equivocators_state_project
                  IM' final_descriptors'
                  (finite_trace_last is tr')
            end
s: state SeededXE
tl: list transition_item
s': state SeededXE
iom, oom: option message
l: label SeededXE
Htl: finite_valid_trace_from SeededXE s []
Ht: input_valid_transition SeededXE l
  (finite_trace_last is tr', iom) (
  s, oom)
H1: s' = finite_trace_last is tr'
H3: {|
  l := l;
  input := iom;
  destination := s;
  output := oom
|} = lst
H4: tl = []
finite_valid_trace SeededX
  (equivocators_state_project IM' initial_descriptors
     is) (trX' ++ lstX) subst tl s' lst.message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
is: composite_state
  (sub_IM (equivocator_IM IM) selection)
IM':= λ i : sub_index selection, IM (`i): sub_index selection → VLSM message
tr': list
  (composite_transition_item
     (sub_IM (equivocator_IM IM) selection))
s: state SeededXE
iom, oom: option message
l: label SeededXE
H: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}])
  → ∀ tr : list
             (composite_transition_item
                (sub_IM 
                   (equivocator_IM IM) selection)),
      finite_valid_trace SeededXE is tr
      → let final_state := 
          finite_trace_last is tr in
        y = length tr
        → ∀ final_descriptors : 
             equivocator_descriptors IM',
            proper_equivocator_descriptors IM'
              final_descriptors final_state
            → ∀ (initial_descriptors : 
                 equivocator_descriptors IM') 
                (trX : 
                 list
                   (composite_transition_item IM')),
                equivocators_trace_project IM'
                  final_descriptors tr =
                Some (trX, initial_descriptors)
                → finite_valid_trace SeededX
                    (equivocators_state_project
                      IM' initial_descriptors is)
                    trX
Htr: finite_valid_trace_from SeededXE is tr'
Hlst: finite_valid_trace_from SeededXE
  (finite_trace_last is tr')
  [{|
     l := l;
     input := iom;
     destination := s;
     output := oom
   |}]
Hinit: initial_state_prop is
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input := iom;
      destination := s;
      output := oom
    |}]): state
  (composite_type
     (sub_IM (equivocator_IM IM)
        selection))
final_descriptors: equivocator_descriptors IM'
Hproper: proper_equivocator_descriptors IM'
  final_descriptors
  (destination
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |})
initial_descriptors: equivocator_descriptors IM'
trX', lstX: list
  (composite_transition_item
     (λ i : sub_index selection, IM (`i)))
oitem: option (composite_transition_item IM')
final_descriptors': equivocator_descriptors IM'
Hpr_trX': equivocators_trace_project
  (λ i : sub_index selection, IM (`i))
  final_descriptors' tr' =
Some (trX', initial_descriptors)
_Hprojectx: match oitem with
| Some item' =>
    Some ([item'], final_descriptors')
| None => Some ([], final_descriptors')
end = Some (lstX, final_descriptors')
H': ∀ final_descriptors : equivocator_descriptors IM',
  proper_equivocator_descriptors IM'
    final_descriptors (finite_trace_last is tr')
  → ∀ (initial_descriptors : 
       equivocator_descriptors IM') 
      (trX : list (composite_transition_item IM')),
      equivocators_trace_project IM'
        final_descriptors tr' =
      Some (trX, initial_descriptors)
      → finite_valid_trace SeededX
          (equivocators_state_project IM'
             initial_descriptors is) trX
Hproperx: composite_valid (equivocator_IM IM')
  (VLSM.l
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |})
  (finite_trace_last is tr',
   input
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |})
→ composite_transition 
    (equivocator_IM IM')
    (VLSM.l
       {|
         l := l;
         input := iom;
         destination := s;
         output := oom
       |})
    (finite_trace_last is tr',
     input
       {|
         l := l;
         input := iom;
         destination := s;
         output := oom
       |}) =
  (destination
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |},
   output
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |})
  → proper_equivocator_descriptors IM'
      final_descriptors'
      (finite_trace_last is tr')
    ∧ final_descriptors' =
      equivocator_descriptors_update IM'
        final_descriptors
        (projT1
           (VLSM.l
              {|
                l := l;
                input := iom;
                destination := s;
                output := oom
              |}))
        (final_descriptors'
           (projT1
              (VLSM.l
                 {|
                   l := l;
                   input := iom;
                   destination := s;
                   output := oom
                 |})))
      ∧ finite_trace_last is tr' =
        state_update (equivocator_IM IM')
          (destination
             {|
               l := l;
               input := iom;
               destination := s;
               output := oom
             |})
          (projT1
             (VLSM.l
                {|
                  l := l;
                  input := iom;
                  destination := s;
                  output := oom
                |}))
          (finite_trace_last is tr'
             (projT1
                (VLSM.l
                   {|
                     l := l;
                     input := iom;
                     destination := s;
                     output := oom
                   |})))
        ∧ previous_state_descriptor_prop
            (IM'
               (projT1
                  (VLSM.l
                     {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                     |})))
            (final_descriptors
               (projT1
                  (VLSM.l
                     {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                     |})))
            (finite_trace_last is tr'
               (projT1
                  (VLSM.l
                     {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                     |})))
            (final_descriptors'
               (projT1
                  (VLSM.l
                     {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                     |})))
          ∧ match oitem with
            | Some itemx =>
                ∀ sx : composite_state IM',
                  sx =
                  equivocators_state_project
                    IM' final_descriptors'
                    (finite_trace_last is
                      tr')
                  → composite_valid IM'
                      (VLSM.l itemx)
                      (
                      sx, input itemx)
                    ∧ composite_transition
                      IM' 
                      (VLSM.l itemx)
                      (sx, input itemx) =
                      (
                      destination itemx,
                      output itemx)
            | None =>
                equivocators_state_project
                  IM' final_descriptors
                  (destination
                     {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                     |}) =
                equivocators_state_project
                  IM' final_descriptors'
                  (finite_trace_last is tr')
            end
Hitemx: match oitem with
| Some itemx =>
    (∃ Hex : existing_equivocator_label
               (IM'
                  (projT1
                     (VLSM.l
                      {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                      |})))
               (projT2
                  (VLSM.l
                     {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                     |})),
       existT
         (projT1
            (VLSM.l
               {|
                 l := l;
                 input := iom;
                 destination := s;
                 output := oom
               |}))
         (existing_equivocator_label_extract
            (IM'
               (projT1
                  (VLSM.l
                     {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                     |})))
            (projT2
               (VLSM.l
                  {|
                    l := l;
                    input := iom;
                    destination := s;
                    output := oom
                  |})) Hex) = 
       VLSM.l itemx)
    ∧ input
        {|
          l := l;
          input := iom;
          destination := s;
          output := oom
        |} = input itemx
      ∧ output
          {|
            l := l;
            input := iom;
            destination := s;
            output := oom
          |} = output itemx
        ∧ equivocators_state_project IM'
            final_descriptors
            (destination
               {|
                 l := l;
                 input := iom;
                 destination := s;
                 output := oom
               |}) = destination itemx
          ∧ final_descriptors'
              (projT1
                 (VLSM.l
                    {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                    |})) =
            equivocator_label_descriptor
              (projT2
                 (VLSM.l
                    {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                    |}))
| None => True
end
Hprojectx: equivocators_transition_item_project IM'
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some (oitem, final_descriptors')
Htl: finite_valid_trace_from SeededXE s []
Ht: input_valid_transition SeededXE l
  (finite_trace_last is tr', iom) (
  s, oom)
finite_valid_trace SeededX
  (equivocators_state_project IM' initial_descriptors
     is) (trX' ++ lstX)
    destruct Ht as [[Hs [Hiom [Hv Hc]]] Ht].message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
is: composite_state
  (sub_IM (equivocator_IM IM) selection)
IM':= λ i : sub_index selection, IM (`i): sub_index selection → VLSM message
tr': list
  (composite_transition_item
     (sub_IM (equivocator_IM IM) selection))
s: state SeededXE
iom, oom: option message
l: label SeededXE
H: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}])
  → ∀ tr : list
             (composite_transition_item
                (sub_IM 
                   (equivocator_IM IM) selection)),
      finite_valid_trace SeededXE is tr
      → let final_state := 
          finite_trace_last is tr in
        y = length tr
        → ∀ final_descriptors : 
             equivocator_descriptors IM',
            proper_equivocator_descriptors IM'
              final_descriptors final_state
            → ∀ (initial_descriptors : 
                 equivocator_descriptors IM') 
                (trX : 
                 list
                   (composite_transition_item IM')),
                equivocators_trace_project IM'
                  final_descriptors tr =
                Some (trX, initial_descriptors)
                → finite_valid_trace SeededX
                    (equivocators_state_project
                      IM' initial_descriptors is)
                    trX
Htr: finite_valid_trace_from SeededXE is tr'
Hlst: finite_valid_trace_from SeededXE
  (finite_trace_last is tr')
  [{|
     l := l;
     input := iom;
     destination := s;
     output := oom
   |}]
Hinit: initial_state_prop is
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input := iom;
      destination := s;
      output := oom
    |}]): state
  (composite_type
     (sub_IM (equivocator_IM IM)
        selection))
final_descriptors: equivocator_descriptors IM'
Hproper: proper_equivocator_descriptors IM'
  final_descriptors
  (destination
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |})
initial_descriptors: equivocator_descriptors IM'
trX', lstX: list
  (composite_transition_item
     (λ i : sub_index selection, IM (`i)))
oitem: option (composite_transition_item IM')
final_descriptors': equivocator_descriptors IM'
Hpr_trX': equivocators_trace_project
  (λ i : sub_index selection, IM (`i))
  final_descriptors' tr' =
Some (trX', initial_descriptors)
_Hprojectx: match oitem with
| Some item' =>
    Some ([item'], final_descriptors')
| None => Some ([], final_descriptors')
end = Some (lstX, final_descriptors')
H': ∀ final_descriptors : equivocator_descriptors IM',
  proper_equivocator_descriptors IM'
    final_descriptors (finite_trace_last is tr')
  → ∀ (initial_descriptors : 
       equivocator_descriptors IM') 
      (trX : list (composite_transition_item IM')),
      equivocators_trace_project IM'
        final_descriptors tr' =
      Some (trX, initial_descriptors)
      → finite_valid_trace SeededX
          (equivocators_state_project IM'
             initial_descriptors is) trX
Hproperx: composite_valid (equivocator_IM IM')
  (VLSM.l
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |})
  (finite_trace_last is tr',
   input
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |})
→ composite_transition 
    (equivocator_IM IM')
    (VLSM.l
       {|
         l := l;
         input := iom;
         destination := s;
         output := oom
       |})
    (finite_trace_last is tr',
     input
       {|
         l := l;
         input := iom;
         destination := s;
         output := oom
       |}) =
  (destination
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |},
   output
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |})
  → proper_equivocator_descriptors IM'
      final_descriptors'
      (finite_trace_last is tr')
    ∧ final_descriptors' =
      equivocator_descriptors_update IM'
        final_descriptors
        (projT1
           (VLSM.l
              {|
                l := l;
                input := iom;
                destination := s;
                output := oom
              |}))
        (final_descriptors'
           (projT1
              (VLSM.l
                 {|
                   l := l;
                   input := iom;
                   destination := s;
                   output := oom
                 |})))
      ∧ finite_trace_last is tr' =
        state_update (equivocator_IM IM')
          (destination
             {|
               l := l;
               input := iom;
               destination := s;
               output := oom
             |})
          (projT1
             (VLSM.l
                {|
                  l := l;
                  input := iom;
                  destination := s;
                  output := oom
                |}))
          (finite_trace_last is tr'
             (projT1
                (VLSM.l
                   {|
                     l := l;
                     input := iom;
                     destination := s;
                     output := oom
                   |})))
        ∧ previous_state_descriptor_prop
            (IM'
               (projT1
                  (VLSM.l
                     {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                     |})))
            (final_descriptors
               (projT1
                  (VLSM.l
                     {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                     |})))
            (finite_trace_last is tr'
               (projT1
                  (VLSM.l
                     {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                     |})))
            (final_descriptors'
               (projT1
                  (VLSM.l
                     {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                     |})))
          ∧ match oitem with
            | Some itemx =>
                ∀ sx : composite_state IM',
                  sx =
                  equivocators_state_project
                    IM' final_descriptors'
                    (finite_trace_last is
                      tr')
                  → composite_valid IM'
                      (VLSM.l itemx)
                      (
                      sx, input itemx)
                    ∧ composite_transition
                      IM' 
                      (VLSM.l itemx)
                      (sx, input itemx) =
                      (
                      destination itemx,
                      output itemx)
            | None =>
                equivocators_state_project
                  IM' final_descriptors
                  (destination
                     {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                     |}) =
                equivocators_state_project
                  IM' final_descriptors'
                  (finite_trace_last is tr')
            end
Hitemx: match oitem with
| Some itemx =>
    (∃ Hex : existing_equivocator_label
               (IM'
                  (projT1
                     (VLSM.l
                      {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                      |})))
               (projT2
                  (VLSM.l
                     {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                     |})),
       existT
         (projT1
            (VLSM.l
               {|
                 l := l;
                 input := iom;
                 destination := s;
                 output := oom
               |}))
         (existing_equivocator_label_extract
            (IM'
               (projT1
                  (VLSM.l
                     {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                     |})))
            (projT2
               (VLSM.l
                  {|
                    l := l;
                    input := iom;
                    destination := s;
                    output := oom
                  |})) Hex) = 
       VLSM.l itemx)
    ∧ input
        {|
          l := l;
          input := iom;
          destination := s;
          output := oom
        |} = input itemx
      ∧ output
          {|
            l := l;
            input := iom;
            destination := s;
            output := oom
          |} = output itemx
        ∧ equivocators_state_project IM'
            final_descriptors
            (destination
               {|
                 l := l;
                 input := iom;
                 destination := s;
                 output := oom
               |}) = destination itemx
          ∧ final_descriptors'
              (projT1
                 (VLSM.l
                    {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                    |})) =
            equivocator_label_descriptor
              (projT2
                 (VLSM.l
                    {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                    |}))
| None => True
end
Hprojectx: equivocators_transition_item_project IM'
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some (oitem, final_descriptors')
Htl: finite_valid_trace_from SeededXE s []
Hs: valid_state_prop SeededXE
  (finite_trace_last is tr')
Hiom: option_valid_message_prop SeededXE iom
Hv: valid l (finite_trace_last is tr', iom)
Hc: no_equivocations_additional_constraint_with_preloaded
  (sub_IM (equivocator_IM IM) selection)
  (free_constraint
     (sub_IM (equivocator_IM IM) selection)) seed
  l (finite_trace_last is tr', iom)
Ht: transition l (finite_trace_last is tr', iom) =
(s, oom)
finite_valid_trace SeededX
  (equivocators_state_project IM' initial_descriptors
     is) (trX' ++ lstX)
    specialize (Hproperx Hv Ht).message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
is: composite_state
  (sub_IM (equivocator_IM IM) selection)
IM':= λ i : sub_index selection, IM (`i): sub_index selection → VLSM message
tr': list
  (composite_transition_item
     (sub_IM (equivocator_IM IM) selection))
s: state SeededXE
iom, oom: option message
l: label SeededXE
H: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}])
  → ∀ tr : list
             (composite_transition_item
                (sub_IM 
                   (equivocator_IM IM) selection)),
      finite_valid_trace SeededXE is tr
      → let final_state := 
          finite_trace_last is tr in
        y = length tr
        → ∀ final_descriptors : 
             equivocator_descriptors IM',
            proper_equivocator_descriptors IM'
              final_descriptors final_state
            → ∀ (initial_descriptors : 
                 equivocator_descriptors IM') 
                (trX : 
                 list
                   (composite_transition_item IM')),
                equivocators_trace_project IM'
                  final_descriptors tr =
                Some (trX, initial_descriptors)
                → finite_valid_trace SeededX
                    (equivocators_state_project
                      IM' initial_descriptors is)
                    trX
Htr: finite_valid_trace_from SeededXE is tr'
Hlst: finite_valid_trace_from SeededXE
  (finite_trace_last is tr')
  [{|
     l := l;
     input := iom;
     destination := s;
     output := oom
   |}]
Hinit: initial_state_prop is
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input := iom;
      destination := s;
      output := oom
    |}]): state
  (composite_type
     (sub_IM (equivocator_IM IM)
        selection))
final_descriptors: equivocator_descriptors IM'
Hproper: proper_equivocator_descriptors IM'
  final_descriptors
  (destination
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |})
initial_descriptors: equivocator_descriptors IM'
trX', lstX: list
  (composite_transition_item
     (λ i : sub_index selection, IM (`i)))
oitem: option (composite_transition_item IM')
final_descriptors': equivocator_descriptors IM'
Hpr_trX': equivocators_trace_project
  (λ i : sub_index selection, IM (`i))
  final_descriptors' tr' =
Some (trX', initial_descriptors)
_Hprojectx: match oitem with
| Some item' =>
    Some ([item'], final_descriptors')
| None => Some ([], final_descriptors')
end = Some (lstX, final_descriptors')
H': ∀ final_descriptors : equivocator_descriptors IM',
  proper_equivocator_descriptors IM'
    final_descriptors (finite_trace_last is tr')
  → ∀ (initial_descriptors : 
       equivocator_descriptors IM') 
      (trX : list (composite_transition_item IM')),
      equivocators_trace_project IM'
        final_descriptors tr' =
      Some (trX, initial_descriptors)
      → finite_valid_trace SeededX
          (equivocators_state_project IM'
             initial_descriptors is) trX
Hproperx: proper_equivocator_descriptors IM'
  final_descriptors'
  (finite_trace_last is tr')
∧ final_descriptors' =
  equivocator_descriptors_update IM'
    final_descriptors
    (projT1
       (VLSM.l
          {|
            l := l;
            input := iom;
            destination := s;
            output := oom
          |}))
    (final_descriptors'
       (projT1
          (VLSM.l
             {|
               l := l;
               input := iom;
               destination := s;
               output := oom
             |})))
  ∧ finite_trace_last is tr' =
    state_update (equivocator_IM IM')
      (destination
         {|
           l := l;
           input := iom;
           destination := s;
           output := oom
         |})
      (projT1
         (VLSM.l
            {|
              l := l;
              input := iom;
              destination := s;
              output := oom
            |}))
      (finite_trace_last is tr'
         (projT1
            (VLSM.l
               {|
                 l := l;
                 input := iom;
                 destination := s;
                 output := oom
               |})))
    ∧ previous_state_descriptor_prop
        (IM'
           (projT1
              (VLSM.l
                 {|
                   l := l;
                   input := iom;
                   destination := s;
                   output := oom
                 |})))
        (final_descriptors
           (projT1
              (VLSM.l
                 {|
                   l := l;
                   input := iom;
                   destination := s;
                   output := oom
                 |})))
        (finite_trace_last is tr'
           (projT1
              (VLSM.l
                 {|
                   l := l;
                   input := iom;
                   destination := s;
                   output := oom
                 |})))
        (final_descriptors'
           (projT1
              (VLSM.l
                 {|
                   l := l;
                   input := iom;
                   destination := s;
                   output := oom
                 |})))
      ∧ match oitem with
        | Some itemx =>
            ∀ sx : composite_state IM',
              sx =
              equivocators_state_project
                IM' final_descriptors'
                (finite_trace_last is tr')
              → composite_valid IM'
                  (VLSM.l itemx)
                  (sx, input itemx)
                ∧ composite_transition IM'
                    (VLSM.l itemx)
                    (sx, input itemx) =
                  (destination itemx,
                   output itemx)
        | None =>
            equivocators_state_project IM'
              final_descriptors
              (destination
                 {|
                   l := l;
                   input := iom;
                   destination := s;
                   output := oom
                 |}) =
            equivocators_state_project IM'
              final_descriptors'
              (finite_trace_last is tr')
        end
Hitemx: match oitem with
| Some itemx =>
    (∃ Hex : existing_equivocator_label
               (IM'
                  (projT1
                     (VLSM.l
                      {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                      |})))
               (projT2
                  (VLSM.l
                     {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                     |})),
       existT
         (projT1
            (VLSM.l
               {|
                 l := l;
                 input := iom;
                 destination := s;
                 output := oom
               |}))
         (existing_equivocator_label_extract
            (IM'
               (projT1
                  (VLSM.l
                     {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                     |})))
            (projT2
               (VLSM.l
                  {|
                    l := l;
                    input := iom;
                    destination := s;
                    output := oom
                  |})) Hex) = 
       VLSM.l itemx)
    ∧ input
        {|
          l := l;
          input := iom;
          destination := s;
          output := oom
        |} = input itemx
      ∧ output
          {|
            l := l;
            input := iom;
            destination := s;
            output := oom
          |} = output itemx
        ∧ equivocators_state_project IM'
            final_descriptors
            (destination
               {|
                 l := l;
                 input := iom;
                 destination := s;
                 output := oom
               |}) = destination itemx
          ∧ final_descriptors'
              (projT1
                 (VLSM.l
                    {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                    |})) =
            equivocator_label_descriptor
              (projT2
                 (VLSM.l
                    {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                    |}))
| None => True
end
Hprojectx: equivocators_transition_item_project IM'
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some (oitem, final_descriptors')
Htl: finite_valid_trace_from SeededXE s []
Hs: valid_state_prop SeededXE
  (finite_trace_last is tr')
Hiom: option_valid_message_prop SeededXE iom
Hv: valid l (finite_trace_last is tr', iom)
Hc: no_equivocations_additional_constraint_with_preloaded
  (sub_IM (equivocator_IM IM) selection)
  (free_constraint
     (sub_IM (equivocator_IM IM) selection)) seed
  l (finite_trace_last is tr', iom)
Ht: transition l (finite_trace_last is tr', iom) =
(s, oom)
finite_valid_trace SeededX
  (equivocators_state_project IM' initial_descriptors
     is) (trX' ++ lstX) clear Hv Ht.message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
is: composite_state
  (sub_IM (equivocator_IM IM) selection)
IM':= λ i : sub_index selection, IM (`i): sub_index selection → VLSM message
tr': list
  (composite_transition_item
     (sub_IM (equivocator_IM IM) selection))
s: state SeededXE
iom, oom: option message
l: label SeededXE
H: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}])
  → ∀ tr : list
             (composite_transition_item
                (sub_IM 
                   (equivocator_IM IM) selection)),
      finite_valid_trace SeededXE is tr
      → let final_state := 
          finite_trace_last is tr in
        y = length tr
        → ∀ final_descriptors : 
             equivocator_descriptors IM',
            proper_equivocator_descriptors IM'
              final_descriptors final_state
            → ∀ (initial_descriptors : 
                 equivocator_descriptors IM') 
                (trX : 
                 list
                   (composite_transition_item IM')),
                equivocators_trace_project IM'
                  final_descriptors tr =
                Some (trX, initial_descriptors)
                → finite_valid_trace SeededX
                    (equivocators_state_project
                      IM' initial_descriptors is)
                    trX
Htr: finite_valid_trace_from SeededXE is tr'
Hlst: finite_valid_trace_from SeededXE
  (finite_trace_last is tr')
  [{|
     l := l;
     input := iom;
     destination := s;
     output := oom
   |}]
Hinit: initial_state_prop is
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input := iom;
      destination := s;
      output := oom
    |}]): state
  (composite_type
     (sub_IM (equivocator_IM IM)
        selection))
final_descriptors: equivocator_descriptors IM'
Hproper: proper_equivocator_descriptors IM'
  final_descriptors
  (destination
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |})
initial_descriptors: equivocator_descriptors IM'
trX', lstX: list
  (composite_transition_item
     (λ i : sub_index selection, IM (`i)))
oitem: option (composite_transition_item IM')
final_descriptors': equivocator_descriptors IM'
Hpr_trX': equivocators_trace_project
  (λ i : sub_index selection, IM (`i))
  final_descriptors' tr' =
Some (trX', initial_descriptors)
_Hprojectx: match oitem with
| Some item' =>
    Some ([item'], final_descriptors')
| None => Some ([], final_descriptors')
end = Some (lstX, final_descriptors')
H': ∀ final_descriptors : equivocator_descriptors IM',
  proper_equivocator_descriptors IM'
    final_descriptors (finite_trace_last is tr')
  → ∀ (initial_descriptors : 
       equivocator_descriptors IM') 
      (trX : list (composite_transition_item IM')),
      equivocators_trace_project IM'
        final_descriptors tr' =
      Some (trX, initial_descriptors)
      → finite_valid_trace SeededX
          (equivocators_state_project IM'
             initial_descriptors is) trX
Hproperx: proper_equivocator_descriptors IM'
  final_descriptors'
  (finite_trace_last is tr')
∧ final_descriptors' =
  equivocator_descriptors_update IM'
    final_descriptors
    (projT1
       (VLSM.l
          {|
            l := l;
            input := iom;
            destination := s;
            output := oom
          |}))
    (final_descriptors'
       (projT1
          (VLSM.l
             {|
               l := l;
               input := iom;
               destination := s;
               output := oom
             |})))
  ∧ finite_trace_last is tr' =
    state_update (equivocator_IM IM')
      (destination
         {|
           l := l;
           input := iom;
           destination := s;
           output := oom
         |})
      (projT1
         (VLSM.l
            {|
              l := l;
              input := iom;
              destination := s;
              output := oom
            |}))
      (finite_trace_last is tr'
         (projT1
            (VLSM.l
               {|
                 l := l;
                 input := iom;
                 destination := s;
                 output := oom
               |})))
    ∧ previous_state_descriptor_prop
        (IM'
           (projT1
              (VLSM.l
                 {|
                   l := l;
                   input := iom;
                   destination := s;
                   output := oom
                 |})))
        (final_descriptors
           (projT1
              (VLSM.l
                 {|
                   l := l;
                   input := iom;
                   destination := s;
                   output := oom
                 |})))
        (finite_trace_last is tr'
           (projT1
              (VLSM.l
                 {|
                   l := l;
                   input := iom;
                   destination := s;
                   output := oom
                 |})))
        (final_descriptors'
           (projT1
              (VLSM.l
                 {|
                   l := l;
                   input := iom;
                   destination := s;
                   output := oom
                 |})))
      ∧ match oitem with
        | Some itemx =>
            ∀ sx : composite_state IM',
              sx =
              equivocators_state_project
                IM' final_descriptors'
                (finite_trace_last is tr')
              → composite_valid IM'
                  (VLSM.l itemx)
                  (sx, input itemx)
                ∧ composite_transition IM'
                    (VLSM.l itemx)
                    (sx, input itemx) =
                  (destination itemx,
                   output itemx)
        | None =>
            equivocators_state_project IM'
              final_descriptors
              (destination
                 {|
                   l := l;
                   input := iom;
                   destination := s;
                   output := oom
                 |}) =
            equivocators_state_project IM'
              final_descriptors'
              (finite_trace_last is tr')
        end
Hitemx: match oitem with
| Some itemx =>
    (∃ Hex : existing_equivocator_label
               (IM'
                  (projT1
                     (VLSM.l
                      {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                      |})))
               (projT2
                  (VLSM.l
                     {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                     |})),
       existT
         (projT1
            (VLSM.l
               {|
                 l := l;
                 input := iom;
                 destination := s;
                 output := oom
               |}))
         (existing_equivocator_label_extract
            (IM'
               (projT1
                  (VLSM.l
                     {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                     |})))
            (projT2
               (VLSM.l
                  {|
                    l := l;
                    input := iom;
                    destination := s;
                    output := oom
                  |})) Hex) = 
       VLSM.l itemx)
    ∧ input
        {|
          l := l;
          input := iom;
          destination := s;
          output := oom
        |} = input itemx
      ∧ output
          {|
            l := l;
            input := iom;
            destination := s;
            output := oom
          |} = output itemx
        ∧ equivocators_state_project IM'
            final_descriptors
            (destination
               {|
                 l := l;
                 input := iom;
                 destination := s;
                 output := oom
               |}) = destination itemx
          ∧ final_descriptors'
              (projT1
                 (VLSM.l
                    {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                    |})) =
            equivocator_label_descriptor
              (projT2
                 (VLSM.l
                    {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                    |}))
| None => True
end
Hprojectx: equivocators_transition_item_project IM'
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some (oitem, final_descriptors')
Htl: finite_valid_trace_from SeededXE s []
Hs: valid_state_prop SeededXE
  (finite_trace_last is tr')
Hiom: option_valid_message_prop SeededXE iom
Hc: no_equivocations_additional_constraint_with_preloaded
  (sub_IM (equivocator_IM IM) selection)
  (free_constraint
     (sub_IM (equivocator_IM IM) selection)) seed
  l (finite_trace_last is tr', iom)
finite_valid_trace SeededX
  (equivocators_state_project IM' initial_descriptors
     is) (trX' ++ lstX)
    destruct Hproperx as [Hproper' [Heq_final_descriptors' [_ [_ Hx]]]].message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
is: composite_state
  (sub_IM (equivocator_IM IM) selection)
IM':= λ i : sub_index selection, IM (`i): sub_index selection → VLSM message
tr': list
  (composite_transition_item
     (sub_IM (equivocator_IM IM) selection))
s: state SeededXE
iom, oom: option message
l: label SeededXE
H: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}])
  → ∀ tr : list
             (composite_transition_item
                (sub_IM 
                   (equivocator_IM IM) selection)),
      finite_valid_trace SeededXE is tr
      → let final_state := 
          finite_trace_last is tr in
        y = length tr
        → ∀ final_descriptors : 
             equivocator_descriptors IM',
            proper_equivocator_descriptors IM'
              final_descriptors final_state
            → ∀ (initial_descriptors : 
                 equivocator_descriptors IM') 
                (trX : 
                 list
                   (composite_transition_item IM')),
                equivocators_trace_project IM'
                  final_descriptors tr =
                Some (trX, initial_descriptors)
                → finite_valid_trace SeededX
                    (equivocators_state_project
                      IM' initial_descriptors is)
                    trX
Htr: finite_valid_trace_from SeededXE is tr'
Hlst: finite_valid_trace_from SeededXE
  (finite_trace_last is tr')
  [{|
     l := l;
     input := iom;
     destination := s;
     output := oom
   |}]
Hinit: initial_state_prop is
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input := iom;
      destination := s;
      output := oom
    |}]): state
  (composite_type
     (sub_IM (equivocator_IM IM)
        selection))
final_descriptors: equivocator_descriptors IM'
Hproper: proper_equivocator_descriptors IM'
  final_descriptors
  (destination
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |})
initial_descriptors: equivocator_descriptors IM'
trX', lstX: list
  (composite_transition_item
     (λ i : sub_index selection, IM (`i)))
oitem: option (composite_transition_item IM')
final_descriptors': equivocator_descriptors IM'
Hpr_trX': equivocators_trace_project
  (λ i : sub_index selection, IM (`i))
  final_descriptors' tr' =
Some (trX', initial_descriptors)
_Hprojectx: match oitem with
| Some item' =>
    Some ([item'], final_descriptors')
| None => Some ([], final_descriptors')
end = Some (lstX, final_descriptors')
H': ∀ final_descriptors : equivocator_descriptors IM',
  proper_equivocator_descriptors IM'
    final_descriptors (finite_trace_last is tr')
  → ∀ (initial_descriptors : 
       equivocator_descriptors IM') 
      (trX : list (composite_transition_item IM')),
      equivocators_trace_project IM'
        final_descriptors tr' =
      Some (trX, initial_descriptors)
      → finite_valid_trace SeededX
          (equivocators_state_project IM'
             initial_descriptors is) trX
Hproper': proper_equivocator_descriptors IM'
  final_descriptors'
  (finite_trace_last is tr')
Heq_final_descriptors': final_descriptors' =
equivocator_descriptors_update
  IM' final_descriptors
  (projT1
     (VLSM.l
        {|
          l := l;
          input := iom;
          destination := s;
          output := oom
        |}))
  (final_descriptors'
     (projT1
        (VLSM.l
           {|
             l := l;
             input := iom;
             destination := s;
             output := oom
           |})))
Hx: match oitem with
| Some itemx =>
    ∀ sx : composite_state IM',
      sx =
      equivocators_state_project IM'
        final_descriptors'
        (finite_trace_last is tr')
      → composite_valid IM' 
          (VLSM.l itemx) (
          sx, input itemx)
        ∧ composite_transition IM' 
            (VLSM.l itemx) (
            sx, input itemx) =
          (destination itemx, output itemx)
| None =>
    equivocators_state_project IM'
      final_descriptors
      (destination
         {|
           l := l;
           input := iom;
           destination := s;
           output := oom
         |}) =
    equivocators_state_project IM'
      final_descriptors'
      (finite_trace_last is tr')
end
Hitemx: match oitem with
| Some itemx =>
    (∃ Hex : existing_equivocator_label
               (IM'
                  (projT1
                     (VLSM.l
                      {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                      |})))
               (projT2
                  (VLSM.l
                     {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                     |})),
       existT
         (projT1
            (VLSM.l
               {|
                 l := l;
                 input := iom;
                 destination := s;
                 output := oom
               |}))
         (existing_equivocator_label_extract
            (IM'
               (projT1
                  (VLSM.l
                     {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                     |})))
            (projT2
               (VLSM.l
                  {|
                    l := l;
                    input := iom;
                    destination := s;
                    output := oom
                  |})) Hex) = 
       VLSM.l itemx)
    ∧ input
        {|
          l := l;
          input := iom;
          destination := s;
          output := oom
        |} = input itemx
      ∧ output
          {|
            l := l;
            input := iom;
            destination := s;
            output := oom
          |} = output itemx
        ∧ equivocators_state_project IM'
            final_descriptors
            (destination
               {|
                 l := l;
                 input := iom;
                 destination := s;
                 output := oom
               |}) = destination itemx
          ∧ final_descriptors'
              (projT1
                 (VLSM.l
                    {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                    |})) =
            equivocator_label_descriptor
              (projT2
                 (VLSM.l
                    {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                    |}))
| None => True
end
Hprojectx: equivocators_transition_item_project IM'
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some (oitem, final_descriptors')
Htl: finite_valid_trace_from SeededXE s []
Hs: valid_state_prop SeededXE
  (finite_trace_last is tr')
Hiom: option_valid_message_prop SeededXE iom
Hc: no_equivocations_additional_constraint_with_preloaded
  (sub_IM (equivocator_IM IM) selection)
  (free_constraint
     (sub_IM (equivocator_IM IM) selection)) seed
  l (finite_trace_last is tr', iom)
finite_valid_trace SeededX
  (equivocators_state_project IM' initial_descriptors
     is) (trX' ++ lstX)
    specialize (H' _ Hproper' _ _ Hpr_trX').message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
is: composite_state
  (sub_IM (equivocator_IM IM) selection)
IM':= λ i : sub_index selection, IM (`i): sub_index selection → VLSM message
tr': list
  (composite_transition_item
     (sub_IM (equivocator_IM IM) selection))
s: state SeededXE
iom, oom: option message
l: label SeededXE
H: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}])
  → ∀ tr : list
             (composite_transition_item
                (sub_IM 
                   (equivocator_IM IM) selection)),
      finite_valid_trace SeededXE is tr
      → let final_state := 
          finite_trace_last is tr in
        y = length tr
        → ∀ final_descriptors : 
             equivocator_descriptors IM',
            proper_equivocator_descriptors IM'
              final_descriptors final_state
            → ∀ (initial_descriptors : 
                 equivocator_descriptors IM') 
                (trX : 
                 list
                   (composite_transition_item IM')),
                equivocators_trace_project IM'
                  final_descriptors tr =
                Some (trX, initial_descriptors)
                → finite_valid_trace SeededX
                    (equivocators_state_project
                      IM' initial_descriptors is)
                    trX
Htr: finite_valid_trace_from SeededXE is tr'
Hlst: finite_valid_trace_from SeededXE
  (finite_trace_last is tr')
  [{|
     l := l;
     input := iom;
     destination := s;
     output := oom
   |}]
Hinit: initial_state_prop is
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input := iom;
      destination := s;
      output := oom
    |}]): state
  (composite_type
     (sub_IM (equivocator_IM IM)
        selection))
final_descriptors: equivocator_descriptors IM'
Hproper: proper_equivocator_descriptors IM'
  final_descriptors
  (destination
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |})
initial_descriptors: equivocator_descriptors IM'
trX', lstX: list
  (composite_transition_item
     (λ i : sub_index selection, IM (`i)))
oitem: option (composite_transition_item IM')
final_descriptors': equivocator_descriptors IM'
Hpr_trX': equivocators_trace_project
  (λ i : sub_index selection, IM (`i))
  final_descriptors' tr' =
Some (trX', initial_descriptors)
_Hprojectx: match oitem with
| Some item' =>
    Some ([item'], final_descriptors')
| None => Some ([], final_descriptors')
end = Some (lstX, final_descriptors')
H': finite_valid_trace SeededX
  (equivocators_state_project IM'
     initial_descriptors is) trX'
Hproper': proper_equivocator_descriptors IM'
  final_descriptors'
  (finite_trace_last is tr')
Heq_final_descriptors': final_descriptors' =
equivocator_descriptors_update
  IM' final_descriptors
  (projT1
     (VLSM.l
        {|
          l := l;
          input := iom;
          destination := s;
          output := oom
        |}))
  (final_descriptors'
     (projT1
        (VLSM.l
           {|
             l := l;
             input := iom;
             destination := s;
             output := oom
           |})))
Hx: match oitem with
| Some itemx =>
    ∀ sx : composite_state IM',
      sx =
      equivocators_state_project IM'
        final_descriptors'
        (finite_trace_last is tr')
      → composite_valid IM' 
          (VLSM.l itemx) (
          sx, input itemx)
        ∧ composite_transition IM' 
            (VLSM.l itemx) (
            sx, input itemx) =
          (destination itemx, output itemx)
| None =>
    equivocators_state_project IM'
      final_descriptors
      (destination
         {|
           l := l;
           input := iom;
           destination := s;
           output := oom
         |}) =
    equivocators_state_project IM'
      final_descriptors'
      (finite_trace_last is tr')
end
Hitemx: match oitem with
| Some itemx =>
    (∃ Hex : existing_equivocator_label
               (IM'
                  (projT1
                     (VLSM.l
                      {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                      |})))
               (projT2
                  (VLSM.l
                     {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                     |})),
       existT
         (projT1
            (VLSM.l
               {|
                 l := l;
                 input := iom;
                 destination := s;
                 output := oom
               |}))
         (existing_equivocator_label_extract
            (IM'
               (projT1
                  (VLSM.l
                     {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                     |})))
            (projT2
               (VLSM.l
                  {|
                    l := l;
                    input := iom;
                    destination := s;
                    output := oom
                  |})) Hex) = 
       VLSM.l itemx)
    ∧ input
        {|
          l := l;
          input := iom;
          destination := s;
          output := oom
        |} = input itemx
      ∧ output
          {|
            l := l;
            input := iom;
            destination := s;
            output := oom
          |} = output itemx
        ∧ equivocators_state_project IM'
            final_descriptors
            (destination
               {|
                 l := l;
                 input := iom;
                 destination := s;
                 output := oom
               |}) = destination itemx
          ∧ final_descriptors'
              (projT1
                 (VLSM.l
                    {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                    |})) =
            equivocator_label_descriptor
              (projT2
                 (VLSM.l
                    {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                    |}))
| None => True
end
Hprojectx: equivocators_transition_item_project IM'
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some (oitem, final_descriptors')
Htl: finite_valid_trace_from SeededXE s []
Hs: valid_state_prop SeededXE
  (finite_trace_last is tr')
Hiom: option_valid_message_prop SeededXE iom
Hc: no_equivocations_additional_constraint_with_preloaded
  (sub_IM (equivocator_IM IM) selection)
  (free_constraint
     (sub_IM (equivocator_IM IM) selection)) seed
  l (finite_trace_last is tr', iom)
finite_valid_trace SeededX
  (equivocators_state_project IM' initial_descriptors
     is) (trX' ++ lstX)
    destruct H' as [HtrX' HinitX].message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
is: composite_state
  (sub_IM (equivocator_IM IM) selection)
IM':= λ i : sub_index selection, IM (`i): sub_index selection → VLSM message
tr': list
  (composite_transition_item
     (sub_IM (equivocator_IM IM) selection))
s: state SeededXE
iom, oom: option message
l: label SeededXE
H: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}])
  → ∀ tr : list
             (composite_transition_item
                (sub_IM 
                   (equivocator_IM IM) selection)),
      finite_valid_trace SeededXE is tr
      → let final_state := 
          finite_trace_last is tr in
        y = length tr
        → ∀ final_descriptors : 
             equivocator_descriptors IM',
            proper_equivocator_descriptors IM'
              final_descriptors final_state
            → ∀ (initial_descriptors : 
                 equivocator_descriptors IM') 
                (trX : 
                 list
                   (composite_transition_item IM')),
                equivocators_trace_project IM'
                  final_descriptors tr =
                Some (trX, initial_descriptors)
                → finite_valid_trace SeededX
                    (equivocators_state_project
                      IM' initial_descriptors is)
                    trX
Htr: finite_valid_trace_from SeededXE is tr'
Hlst: finite_valid_trace_from SeededXE
  (finite_trace_last is tr')
  [{|
     l := l;
     input := iom;
     destination := s;
     output := oom
   |}]
Hinit: initial_state_prop is
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input := iom;
      destination := s;
      output := oom
    |}]): state
  (composite_type
     (sub_IM (equivocator_IM IM)
        selection))
final_descriptors: equivocator_descriptors IM'
Hproper: proper_equivocator_descriptors IM'
  final_descriptors
  (destination
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |})
initial_descriptors: equivocator_descriptors IM'
trX', lstX: list
  (composite_transition_item
     (λ i : sub_index selection, IM (`i)))
oitem: option (composite_transition_item IM')
final_descriptors': equivocator_descriptors IM'
Hpr_trX': equivocators_trace_project
  (λ i : sub_index selection, IM (`i))
  final_descriptors' tr' =
Some (trX', initial_descriptors)
_Hprojectx: match oitem with
| Some item' =>
    Some ([item'], final_descriptors')
| None => Some ([], final_descriptors')
end = Some (lstX, final_descriptors')
HtrX': finite_valid_trace_from SeededX
  (equivocators_state_project IM'
     initial_descriptors is) trX'
HinitX: initial_state_prop
  (equivocators_state_project IM'
     initial_descriptors is)
Hproper': proper_equivocator_descriptors IM'
  final_descriptors'
  (finite_trace_last is tr')
Heq_final_descriptors': final_descriptors' =
equivocator_descriptors_update
  IM' final_descriptors
  (projT1
     (VLSM.l
        {|
          l := l;
          input := iom;
          destination := s;
          output := oom
        |}))
  (final_descriptors'
     (projT1
        (VLSM.l
           {|
             l := l;
             input := iom;
             destination := s;
             output := oom
           |})))
Hx: match oitem with
| Some itemx =>
    ∀ sx : composite_state IM',
      sx =
      equivocators_state_project IM'
        final_descriptors'
        (finite_trace_last is tr')
      → composite_valid IM' 
          (VLSM.l itemx) (
          sx, input itemx)
        ∧ composite_transition IM' 
            (VLSM.l itemx) (
            sx, input itemx) =
          (destination itemx, output itemx)
| None =>
    equivocators_state_project IM'
      final_descriptors
      (destination
         {|
           l := l;
           input := iom;
           destination := s;
           output := oom
         |}) =
    equivocators_state_project IM'
      final_descriptors'
      (finite_trace_last is tr')
end
Hitemx: match oitem with
| Some itemx =>
    (∃ Hex : existing_equivocator_label
               (IM'
                  (projT1
                     (VLSM.l
                      {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                      |})))
               (projT2
                  (VLSM.l
                     {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                     |})),
       existT
         (projT1
            (VLSM.l
               {|
                 l := l;
                 input := iom;
                 destination := s;
                 output := oom
               |}))
         (existing_equivocator_label_extract
            (IM'
               (projT1
                  (VLSM.l
                     {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                     |})))
            (projT2
               (VLSM.l
                  {|
                    l := l;
                    input := iom;
                    destination := s;
                    output := oom
                  |})) Hex) = 
       VLSM.l itemx)
    ∧ input
        {|
          l := l;
          input := iom;
          destination := s;
          output := oom
        |} = input itemx
      ∧ output
          {|
            l := l;
            input := iom;
            destination := s;
            output := oom
          |} = output itemx
        ∧ equivocators_state_project IM'
            final_descriptors
            (destination
               {|
                 l := l;
                 input := iom;
                 destination := s;
                 output := oom
               |}) = destination itemx
          ∧ final_descriptors'
              (projT1
                 (VLSM.l
                    {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                    |})) =
            equivocator_label_descriptor
              (projT2
                 (VLSM.l
                    {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                    |}))
| None => True
end
Hprojectx: equivocators_transition_item_project IM'
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some (oitem, final_descriptors')
Htl: finite_valid_trace_from SeededXE s []
Hs: valid_state_prop SeededXE
  (finite_trace_last is tr')
Hiom: option_valid_message_prop SeededXE iom
Hc: no_equivocations_additional_constraint_with_preloaded
  (sub_IM (equivocator_IM IM) selection)
  (free_constraint
     (sub_IM (equivocator_IM IM) selection)) seed
  l (finite_trace_last is tr', iom)
finite_valid_trace SeededX
  (equivocators_state_project IM' initial_descriptors
     is) (trX' ++ lstX)
    split; [| done].message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
is: composite_state
  (sub_IM (equivocator_IM IM) selection)
IM':= λ i : sub_index selection, IM (`i): sub_index selection → VLSM message
tr': list
  (composite_transition_item
     (sub_IM (equivocator_IM IM) selection))
s: state SeededXE
iom, oom: option message
l: label SeededXE
H: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}])
  → ∀ tr : list
             (composite_transition_item
                (sub_IM 
                   (equivocator_IM IM) selection)),
      finite_valid_trace SeededXE is tr
      → let final_state := 
          finite_trace_last is tr in
        y = length tr
        → ∀ final_descriptors : 
             equivocator_descriptors IM',
            proper_equivocator_descriptors IM'
              final_descriptors final_state
            → ∀ (initial_descriptors : 
                 equivocator_descriptors IM') 
                (trX : 
                 list
                   (composite_transition_item IM')),
                equivocators_trace_project IM'
                  final_descriptors tr =
                Some (trX, initial_descriptors)
                → finite_valid_trace SeededX
                    (equivocators_state_project
                      IM' initial_descriptors is)
                    trX
Htr: finite_valid_trace_from SeededXE is tr'
Hlst: finite_valid_trace_from SeededXE
  (finite_trace_last is tr')
  [{|
     l := l;
     input := iom;
     destination := s;
     output := oom
   |}]
Hinit: initial_state_prop is
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input := iom;
      destination := s;
      output := oom
    |}]): state
  (composite_type
     (sub_IM (equivocator_IM IM)
        selection))
final_descriptors: equivocator_descriptors IM'
Hproper: proper_equivocator_descriptors IM'
  final_descriptors
  (destination
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |})
initial_descriptors: equivocator_descriptors IM'
trX', lstX: list
  (composite_transition_item
     (λ i : sub_index selection, IM (`i)))
oitem: option (composite_transition_item IM')
final_descriptors': equivocator_descriptors IM'
Hpr_trX': equivocators_trace_project
  (λ i : sub_index selection, IM (`i))
  final_descriptors' tr' =
Some (trX', initial_descriptors)
_Hprojectx: match oitem with
| Some item' =>
    Some ([item'], final_descriptors')
| None => Some ([], final_descriptors')
end = Some (lstX, final_descriptors')
HtrX': finite_valid_trace_from SeededX
  (equivocators_state_project IM'
     initial_descriptors is) trX'
HinitX: initial_state_prop
  (equivocators_state_project IM'
     initial_descriptors is)
Hproper': proper_equivocator_descriptors IM'
  final_descriptors'
  (finite_trace_last is tr')
Heq_final_descriptors': final_descriptors' =
equivocator_descriptors_update
  IM' final_descriptors
  (projT1
     (VLSM.l
        {|
          l := l;
          input := iom;
          destination := s;
          output := oom
        |}))
  (final_descriptors'
     (projT1
        (VLSM.l
           {|
             l := l;
             input := iom;
             destination := s;
             output := oom
           |})))
Hx: match oitem with
| Some itemx =>
    ∀ sx : composite_state IM',
      sx =
      equivocators_state_project IM'
        final_descriptors'
        (finite_trace_last is tr')
      → composite_valid IM' 
          (VLSM.l itemx) (
          sx, input itemx)
        ∧ composite_transition IM' 
            (VLSM.l itemx) (
            sx, input itemx) =
          (destination itemx, output itemx)
| None =>
    equivocators_state_project IM'
      final_descriptors
      (destination
         {|
           l := l;
           input := iom;
           destination := s;
           output := oom
         |}) =
    equivocators_state_project IM'
      final_descriptors'
      (finite_trace_last is tr')
end
Hitemx: match oitem with
| Some itemx =>
    (∃ Hex : existing_equivocator_label
               (IM'
                  (projT1
                     (VLSM.l
                      {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                      |})))
               (projT2
                  (VLSM.l
                     {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                     |})),
       existT
         (projT1
            (VLSM.l
               {|
                 l := l;
                 input := iom;
                 destination := s;
                 output := oom
               |}))
         (existing_equivocator_label_extract
            (IM'
               (projT1
                  (VLSM.l
                     {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                     |})))
            (projT2
               (VLSM.l
                  {|
                    l := l;
                    input := iom;
                    destination := s;
                    output := oom
                  |})) Hex) = 
       VLSM.l itemx)
    ∧ input
        {|
          l := l;
          input := iom;
          destination := s;
          output := oom
        |} = input itemx
      ∧ output
          {|
            l := l;
            input := iom;
            destination := s;
            output := oom
          |} = output itemx
        ∧ equivocators_state_project IM'
            final_descriptors
            (destination
               {|
                 l := l;
                 input := iom;
                 destination := s;
                 output := oom
               |}) = destination itemx
          ∧ final_descriptors'
              (projT1
                 (VLSM.l
                    {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                    |})) =
            equivocator_label_descriptor
              (projT2
                 (VLSM.l
                    {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                    |}))
| None => True
end
Hprojectx: equivocators_transition_item_project IM'
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some (oitem, final_descriptors')
Htl: finite_valid_trace_from SeededXE s []
Hs: valid_state_prop SeededXE
  (finite_trace_last is tr')
Hiom: option_valid_message_prop SeededXE iom
Hc: no_equivocations_additional_constraint_with_preloaded
  (sub_IM (equivocator_IM IM) selection)
  (free_constraint
     (sub_IM (equivocator_IM IM) selection)) seed
  l (finite_trace_last is tr', iom)
finite_valid_trace_from SeededX
  (equivocators_state_project IM' initial_descriptors
     is) (trX' ++ lstX) apply finite_valid_trace_from_app_iff.message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
is: composite_state
  (sub_IM (equivocator_IM IM) selection)
IM':= λ i : sub_index selection, IM (`i): sub_index selection → VLSM message
tr': list
  (composite_transition_item
     (sub_IM (equivocator_IM IM) selection))
s: state SeededXE
iom, oom: option message
l: label SeededXE
H: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}])
  → ∀ tr : list
             (composite_transition_item
                (sub_IM 
                   (equivocator_IM IM) selection)),
      finite_valid_trace SeededXE is tr
      → let final_state := 
          finite_trace_last is tr in
        y = length tr
        → ∀ final_descriptors : 
             equivocator_descriptors IM',
            proper_equivocator_descriptors IM'
              final_descriptors final_state
            → ∀ (initial_descriptors : 
                 equivocator_descriptors IM') 
                (trX : 
                 list
                   (composite_transition_item IM')),
                equivocators_trace_project IM'
                  final_descriptors tr =
                Some (trX, initial_descriptors)
                → finite_valid_trace SeededX
                    (equivocators_state_project
                      IM' initial_descriptors is)
                    trX
Htr: finite_valid_trace_from SeededXE is tr'
Hlst: finite_valid_trace_from SeededXE
  (finite_trace_last is tr')
  [{|
     l := l;
     input := iom;
     destination := s;
     output := oom
   |}]
Hinit: initial_state_prop is
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input := iom;
      destination := s;
      output := oom
    |}]): state
  (composite_type
     (sub_IM (equivocator_IM IM)
        selection))
final_descriptors: equivocator_descriptors IM'
Hproper: proper_equivocator_descriptors IM'
  final_descriptors
  (destination
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |})
initial_descriptors: equivocator_descriptors IM'
trX', lstX: list
  (composite_transition_item
     (λ i : sub_index selection, IM (`i)))
oitem: option (composite_transition_item IM')
final_descriptors': equivocator_descriptors IM'
Hpr_trX': equivocators_trace_project
  (λ i : sub_index selection, IM (`i))
  final_descriptors' tr' =
Some (trX', initial_descriptors)
_Hprojectx: match oitem with
| Some item' =>
    Some ([item'], final_descriptors')
| None => Some ([], final_descriptors')
end = Some (lstX, final_descriptors')
HtrX': finite_valid_trace_from SeededX
  (equivocators_state_project IM'
     initial_descriptors is) trX'
HinitX: initial_state_prop
  (equivocators_state_project IM'
     initial_descriptors is)
Hproper': proper_equivocator_descriptors IM'
  final_descriptors'
  (finite_trace_last is tr')
Heq_final_descriptors': final_descriptors' =
equivocator_descriptors_update
  IM' final_descriptors
  (projT1
     (VLSM.l
        {|
          l := l;
          input := iom;
          destination := s;
          output := oom
        |}))
  (final_descriptors'
     (projT1
        (VLSM.l
           {|
             l := l;
             input := iom;
             destination := s;
             output := oom
           |})))
Hx: match oitem with
| Some itemx =>
    ∀ sx : composite_state IM',
      sx =
      equivocators_state_project IM'
        final_descriptors'
        (finite_trace_last is tr')
      → composite_valid IM' 
          (VLSM.l itemx) (
          sx, input itemx)
        ∧ composite_transition IM' 
            (VLSM.l itemx) (
            sx, input itemx) =
          (destination itemx, output itemx)
| None =>
    equivocators_state_project IM'
      final_descriptors
      (destination
         {|
           l := l;
           input := iom;
           destination := s;
           output := oom
         |}) =
    equivocators_state_project IM'
      final_descriptors'
      (finite_trace_last is tr')
end
Hitemx: match oitem with
| Some itemx =>
    (∃ Hex : existing_equivocator_label
               (IM'
                  (projT1
                     (VLSM.l
                      {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                      |})))
               (projT2
                  (VLSM.l
                     {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                     |})),
       existT
         (projT1
            (VLSM.l
               {|
                 l := l;
                 input := iom;
                 destination := s;
                 output := oom
               |}))
         (existing_equivocator_label_extract
            (IM'
               (projT1
                  (VLSM.l
                     {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                     |})))
            (projT2
               (VLSM.l
                  {|
                    l := l;
                    input := iom;
                    destination := s;
                    output := oom
                  |})) Hex) = 
       VLSM.l itemx)
    ∧ input
        {|
          l := l;
          input := iom;
          destination := s;
          output := oom
        |} = input itemx
      ∧ output
          {|
            l := l;
            input := iom;
            destination := s;
            output := oom
          |} = output itemx
        ∧ equivocators_state_project IM'
            final_descriptors
            (destination
               {|
                 l := l;
                 input := iom;
                 destination := s;
                 output := oom
               |}) = destination itemx
          ∧ final_descriptors'
              (projT1
                 (VLSM.l
                    {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                    |})) =
            equivocator_label_descriptor
              (projT2
                 (VLSM.l
                    {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                    |}))
| None => True
end
Hprojectx: equivocators_transition_item_project IM'
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some (oitem, final_descriptors')
Htl: finite_valid_trace_from SeededXE s []
Hs: valid_state_prop SeededXE
  (finite_trace_last is tr')
Hiom: option_valid_message_prop SeededXE iom
Hc: no_equivocations_additional_constraint_with_preloaded
  (sub_IM (equivocator_IM IM) selection)
  (free_constraint
     (sub_IM (equivocator_IM IM) selection)) seed
  l (finite_trace_last is tr', iom)
finite_valid_trace_from SeededX
  (equivocators_state_project IM' initial_descriptors
     is) trX'
∧ finite_valid_trace_from SeededX
    (finite_trace_last
       (equivocators_state_project IM'
          initial_descriptors is) trX') lstX
    split; [done |].message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
is: composite_state
  (sub_IM (equivocator_IM IM) selection)
IM':= λ i : sub_index selection, IM (`i): sub_index selection → VLSM message
tr': list
  (composite_transition_item
     (sub_IM (equivocator_IM IM) selection))
s: state SeededXE
iom, oom: option message
l: label SeededXE
H: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}])
  → ∀ tr : list
             (composite_transition_item
                (sub_IM 
                   (equivocator_IM IM) selection)),
      finite_valid_trace SeededXE is tr
      → let final_state := 
          finite_trace_last is tr in
        y = length tr
        → ∀ final_descriptors : 
             equivocator_descriptors IM',
            proper_equivocator_descriptors IM'
              final_descriptors final_state
            → ∀ (initial_descriptors : 
                 equivocator_descriptors IM') 
                (trX : 
                 list
                   (composite_transition_item IM')),
                equivocators_trace_project IM'
                  final_descriptors tr =
                Some (trX, initial_descriptors)
                → finite_valid_trace SeededX
                    (equivocators_state_project
                      IM' initial_descriptors is)
                    trX
Htr: finite_valid_trace_from SeededXE is tr'
Hlst: finite_valid_trace_from SeededXE
  (finite_trace_last is tr')
  [{|
     l := l;
     input := iom;
     destination := s;
     output := oom
   |}]
Hinit: initial_state_prop is
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input := iom;
      destination := s;
      output := oom
    |}]): state
  (composite_type
     (sub_IM (equivocator_IM IM)
        selection))
final_descriptors: equivocator_descriptors IM'
Hproper: proper_equivocator_descriptors IM'
  final_descriptors
  (destination
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |})
initial_descriptors: equivocator_descriptors IM'
trX', lstX: list
  (composite_transition_item
     (λ i : sub_index selection, IM (`i)))
oitem: option (composite_transition_item IM')
final_descriptors': equivocator_descriptors IM'
Hpr_trX': equivocators_trace_project
  (λ i : sub_index selection, IM (`i))
  final_descriptors' tr' =
Some (trX', initial_descriptors)
_Hprojectx: match oitem with
| Some item' =>
    Some ([item'], final_descriptors')
| None => Some ([], final_descriptors')
end = Some (lstX, final_descriptors')
HtrX': finite_valid_trace_from SeededX
  (equivocators_state_project IM'
     initial_descriptors is) trX'
HinitX: initial_state_prop
  (equivocators_state_project IM'
     initial_descriptors is)
Hproper': proper_equivocator_descriptors IM'
  final_descriptors'
  (finite_trace_last is tr')
Heq_final_descriptors': final_descriptors' =
equivocator_descriptors_update
  IM' final_descriptors
  (projT1
     (VLSM.l
        {|
          l := l;
          input := iom;
          destination := s;
          output := oom
        |}))
  (final_descriptors'
     (projT1
        (VLSM.l
           {|
             l := l;
             input := iom;
             destination := s;
             output := oom
           |})))
Hx: match oitem with
| Some itemx =>
    ∀ sx : composite_state IM',
      sx =
      equivocators_state_project IM'
        final_descriptors'
        (finite_trace_last is tr')
      → composite_valid IM' 
          (VLSM.l itemx) (
          sx, input itemx)
        ∧ composite_transition IM' 
            (VLSM.l itemx) (
            sx, input itemx) =
          (destination itemx, output itemx)
| None =>
    equivocators_state_project IM'
      final_descriptors
      (destination
         {|
           l := l;
           input := iom;
           destination := s;
           output := oom
         |}) =
    equivocators_state_project IM'
      final_descriptors'
      (finite_trace_last is tr')
end
Hitemx: match oitem with
| Some itemx =>
    (∃ Hex : existing_equivocator_label
               (IM'
                  (projT1
                     (VLSM.l
                      {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                      |})))
               (projT2
                  (VLSM.l
                     {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                     |})),
       existT
         (projT1
            (VLSM.l
               {|
                 l := l;
                 input := iom;
                 destination := s;
                 output := oom
               |}))
         (existing_equivocator_label_extract
            (IM'
               (projT1
                  (VLSM.l
                     {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                     |})))
            (projT2
               (VLSM.l
                  {|
                    l := l;
                    input := iom;
                    destination := s;
                    output := oom
                  |})) Hex) = 
       VLSM.l itemx)
    ∧ input
        {|
          l := l;
          input := iom;
          destination := s;
          output := oom
        |} = input itemx
      ∧ output
          {|
            l := l;
            input := iom;
            destination := s;
            output := oom
          |} = output itemx
        ∧ equivocators_state_project IM'
            final_descriptors
            (destination
               {|
                 l := l;
                 input := iom;
                 destination := s;
                 output := oom
               |}) = destination itemx
          ∧ final_descriptors'
              (projT1
                 (VLSM.l
                    {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                    |})) =
            equivocator_label_descriptor
              (projT2
                 (VLSM.l
                    {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                    |}))
| None => True
end
Hprojectx: equivocators_transition_item_project IM'
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some (oitem, final_descriptors')
Htl: finite_valid_trace_from SeededXE s []
Hs: valid_state_prop SeededXE
  (finite_trace_last is tr')
Hiom: option_valid_message_prop SeededXE iom
Hc: no_equivocations_additional_constraint_with_preloaded
  (sub_IM (equivocator_IM IM) selection)
  (free_constraint
     (sub_IM (equivocator_IM IM) selection)) seed
  l (finite_trace_last is tr', iom)
finite_valid_trace_from SeededX
  (finite_trace_last
     (equivocators_state_project IM'
        initial_descriptors is) trX') lstX
    assert (Hlst_trX' : valid_state_prop SeededX (finite_trace_last
      (equivocators_state_project _ initial_descriptors is) trX')).message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
is: composite_state
  (sub_IM (equivocator_IM IM) selection)
IM':= λ i : sub_index selection, IM (`i): sub_index selection → VLSM message
tr': list
  (composite_transition_item
     (sub_IM (equivocator_IM IM) selection))
s: state SeededXE
iom, oom: option message
l: label SeededXE
H: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}])
  → ∀ tr : list
             (composite_transition_item
                (sub_IM 
                   (equivocator_IM IM) selection)),
      finite_valid_trace SeededXE is tr
      → let final_state := 
          finite_trace_last is tr in
        y = length tr
        → ∀ final_descriptors : 
             equivocator_descriptors IM',
            proper_equivocator_descriptors IM'
              final_descriptors final_state
            → ∀ (initial_descriptors : 
                 equivocator_descriptors IM') 
                (trX : 
                 list
                   (composite_transition_item IM')),
                equivocators_trace_project IM'
                  final_descriptors tr =
                Some (trX, initial_descriptors)
                → finite_valid_trace SeededX
                    (equivocators_state_project
                      IM' initial_descriptors is)
                    trX
Htr: finite_valid_trace_from SeededXE is tr'
Hlst: finite_valid_trace_from SeededXE
  (finite_trace_last is tr')
  [{|
     l := l;
     input := iom;
     destination := s;
     output := oom
   |}]
Hinit: initial_state_prop is
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input := iom;
      destination := s;
      output := oom
    |}]): state
  (composite_type
     (sub_IM (equivocator_IM IM)
        selection))
final_descriptors: equivocator_descriptors IM'
Hproper: proper_equivocator_descriptors IM'
  final_descriptors
  (destination
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |})
initial_descriptors: equivocator_descriptors IM'
trX', lstX: list
  (composite_transition_item
     (λ i : sub_index selection, IM (`i)))
oitem: option (composite_transition_item IM')
final_descriptors': equivocator_descriptors IM'
Hpr_trX': equivocators_trace_project
  (λ i : sub_index selection, IM (`i))
  final_descriptors' tr' =
Some (trX', initial_descriptors)
_Hprojectx: match oitem with
| Some item' =>
    Some ([item'], final_descriptors')
| None => Some ([], final_descriptors')
end = Some (lstX, final_descriptors')
HtrX': finite_valid_trace_from SeededX
  (equivocators_state_project IM'
     initial_descriptors is) trX'
HinitX: initial_state_prop
  (equivocators_state_project IM'
     initial_descriptors is)
Hproper': proper_equivocator_descriptors IM'
  final_descriptors'
  (finite_trace_last is tr')
Heq_final_descriptors': final_descriptors' =
equivocator_descriptors_update
  IM' final_descriptors
  (projT1
     (VLSM.l
        {|
          l := l;
          input := iom;
          destination := s;
          output := oom
        |}))
  (final_descriptors'
     (projT1
        (VLSM.l
           {|
             l := l;
             input := iom;
             destination := s;
             output := oom
           |})))
Hx: match oitem with
| Some itemx =>
    ∀ sx : composite_state IM',
      sx =
      equivocators_state_project IM'
        final_descriptors'
        (finite_trace_last is tr')
      → composite_valid IM' 
          (VLSM.l itemx) (
          sx, input itemx)
        ∧ composite_transition IM' 
            (VLSM.l itemx) (
            sx, input itemx) =
          (destination itemx, output itemx)
| None =>
    equivocators_state_project IM'
      final_descriptors
      (destination
         {|
           l := l;
           input := iom;
           destination := s;
           output := oom
         |}) =
    equivocators_state_project IM'
      final_descriptors'
      (finite_trace_last is tr')
end
Hitemx: match oitem with
| Some itemx =>
    (∃ Hex : existing_equivocator_label
               (IM'
                  (projT1
                     (VLSM.l
                      {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                      |})))
               (projT2
                  (VLSM.l
                     {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                     |})),
       existT
         (projT1
            (VLSM.l
               {|
                 l := l;
                 input := iom;
                 destination := s;
                 output := oom
               |}))
         (existing_equivocator_label_extract
            (IM'
               (projT1
                  (VLSM.l
                     {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                     |})))
            (projT2
               (VLSM.l
                  {|
                    l := l;
                    input := iom;
                    destination := s;
                    output := oom
                  |})) Hex) = 
       VLSM.l itemx)
    ∧ input
        {|
          l := l;
          input := iom;
          destination := s;
          output := oom
        |} = input itemx
      ∧ output
          {|
            l := l;
            input := iom;
            destination := s;
            output := oom
          |} = output itemx
        ∧ equivocators_state_project IM'
            final_descriptors
            (destination
               {|
                 l := l;
                 input := iom;
                 destination := s;
                 output := oom
               |}) = destination itemx
          ∧ final_descriptors'
              (projT1
                 (VLSM.l
                    {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                    |})) =
            equivocator_label_descriptor
              (projT2
                 (VLSM.l
                    {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                    |}))
| None => True
end
Hprojectx: equivocators_transition_item_project IM'
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some (oitem, final_descriptors')
Htl: finite_valid_trace_from SeededXE s []
Hs: valid_state_prop SeededXE
  (finite_trace_last is tr')
Hiom: option_valid_message_prop SeededXE iom
Hc: no_equivocations_additional_constraint_with_preloaded
  (sub_IM (equivocator_IM IM) selection)
  (free_constraint
     (sub_IM (equivocator_IM IM) selection)) seed
  l (finite_trace_last is tr', iom)
valid_state_prop SeededX
  (finite_trace_last
     (equivocators_state_project IM'
        initial_descriptors is) trX')
message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
is: composite_state
  (sub_IM (equivocator_IM IM) selection)
IM':= λ i : sub_index selection, IM (`i): sub_index selection → VLSM message
tr': list
  (composite_transition_item
     (sub_IM (equivocator_IM IM) selection))
s: state SeededXE
iom, oom: option message
l: label SeededXE
H: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}])
  → ∀ tr : list
             (composite_transition_item
                (sub_IM 
                   (equivocator_IM IM) selection)),
      finite_valid_trace SeededXE is tr
      → let final_state := 
          finite_trace_last is tr in
        y = length tr
        → ∀ final_descriptors : 
             equivocator_descriptors IM',
            proper_equivocator_descriptors IM'
              final_descriptors final_state
            → ∀ (initial_descriptors : 
                 equivocator_descriptors IM') 
                (trX : 
                 list
                   (composite_transition_item IM')),
                equivocators_trace_project IM'
                  final_descriptors tr =
                Some (trX, initial_descriptors)
                → finite_valid_trace SeededX
                    (equivocators_state_project
                      IM' initial_descriptors is)
                    trX
Htr: finite_valid_trace_from SeededXE is tr'
Hlst: finite_valid_trace_from SeededXE
  (finite_trace_last is tr')
  [{|
     l := l;
     input := iom;
     destination := s;
     output := oom
   |}]
Hinit: initial_state_prop is
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input := iom;
      destination := s;
      output := oom
    |}]): state
  (composite_type
     (sub_IM (equivocator_IM IM)
        selection))
final_descriptors: equivocator_descriptors IM'
Hproper: proper_equivocator_descriptors IM'
  final_descriptors
  (destination
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |})
initial_descriptors: equivocator_descriptors IM'
trX', lstX: list
  (composite_transition_item
     (λ i : sub_index selection, IM (`i)))
oitem: option (composite_transition_item IM')
final_descriptors': equivocator_descriptors IM'
Hpr_trX': equivocators_trace_project
  (λ i : sub_index selection, IM (`i))
  final_descriptors' tr' =
Some (trX', initial_descriptors)
_Hprojectx: match oitem with
| Some item' =>
    Some ([item'], final_descriptors')
| None => Some ([], final_descriptors')
end = Some (lstX, final_descriptors')
HtrX': finite_valid_trace_from SeededX
  (equivocators_state_project IM'
     initial_descriptors is) trX'
HinitX: initial_state_prop
  (equivocators_state_project IM'
     initial_descriptors is)
Hproper': proper_equivocator_descriptors IM'
  final_descriptors'
  (finite_trace_last is tr')
Heq_final_descriptors': final_descriptors' =
equivocator_descriptors_update
  IM' final_descriptors
  (projT1
     (VLSM.l
        {|
          l := l;
          input := iom;
          destination := s;
          output := oom
        |}))
  (final_descriptors'
     (projT1
        (VLSM.l
           {|
             l := l;
             input := iom;
             destination := s;
             output := oom
           |})))
Hx: match oitem with
| Some itemx =>
    ∀ sx : composite_state IM',
      sx =
      equivocators_state_project IM'
        final_descriptors'
        (finite_trace_last is tr')
      → composite_valid IM' 
          (VLSM.l itemx) (
          sx, input itemx)
        ∧ composite_transition IM' 
            (VLSM.l itemx) (
            sx, input itemx) =
          (destination itemx, output itemx)
| None =>
    equivocators_state_project IM'
      final_descriptors
      (destination
         {|
           l := l;
           input := iom;
           destination := s;
           output := oom
         |}) =
    equivocators_state_project IM'
      final_descriptors'
      (finite_trace_last is tr')
end
Hitemx: match oitem with
| Some itemx =>
    (∃ Hex : existing_equivocator_label
               (IM'
                  (projT1
                     (VLSM.l
                      {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                      |})))
               (projT2
                  (VLSM.l
                     {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                     |})),
       existT
         (projT1
            (VLSM.l
               {|
                 l := l;
                 input := iom;
                 destination := s;
                 output := oom
               |}))
         (existing_equivocator_label_extract
            (IM'
               (projT1
                  (VLSM.l
                     {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                     |})))
            (projT2
               (VLSM.l
                  {|
                    l := l;
                    input := iom;
                    destination := s;
                    output := oom
                  |})) Hex) = 
       VLSM.l itemx)
    ∧ input
        {|
          l := l;
          input := iom;
          destination := s;
          output := oom
        |} = input itemx
      ∧ output
          {|
            l := l;
            input := iom;
            destination := s;
            output := oom
          |} = output itemx
        ∧ equivocators_state_project IM'
            final_descriptors
            (destination
               {|
                 l := l;
                 input := iom;
                 destination := s;
                 output := oom
               |}) = destination itemx
          ∧ final_descriptors'
              (projT1
                 (VLSM.l
                    {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                    |})) =
            equivocator_label_descriptor
              (projT2
                 (VLSM.l
                    {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                    |}))
| None => True
end
Hprojectx: equivocators_transition_item_project IM'
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some (oitem, final_descriptors')
Htl: finite_valid_trace_from SeededXE s []
Hs: valid_state_prop SeededXE
  (finite_trace_last is tr')
Hiom: option_valid_message_prop SeededXE iom
Hc: no_equivocations_additional_constraint_with_preloaded
  (sub_IM (equivocator_IM IM) selection)
  (free_constraint
     (sub_IM (equivocator_IM IM) selection)) seed
  l (finite_trace_last is tr', iom)
Hlst_trX': valid_state_prop SeededX
  (finite_trace_last
     (equivocators_state_project IM'
        initial_descriptors is) trX')
finite_valid_trace_from SeededX
  (finite_trace_last
     (equivocators_state_project IM'
        initial_descriptors is) trX') lstX
    {message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
is: composite_state
  (sub_IM (equivocator_IM IM) selection)
IM':= λ i : sub_index selection, IM (`i): sub_index selection → VLSM message
tr': list
  (composite_transition_item
     (sub_IM (equivocator_IM IM) selection))
s: state SeededXE
iom, oom: option message
l: label SeededXE
H: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}])
  → ∀ tr : list
             (composite_transition_item
                (sub_IM 
                   (equivocator_IM IM) selection)),
      finite_valid_trace SeededXE is tr
      → let final_state := 
          finite_trace_last is tr in
        y = length tr
        → ∀ final_descriptors : 
             equivocator_descriptors IM',
            proper_equivocator_descriptors IM'
              final_descriptors final_state
            → ∀ (initial_descriptors : 
                 equivocator_descriptors IM') 
                (trX : 
                 list
                   (composite_transition_item IM')),
                equivocators_trace_project IM'
                  final_descriptors tr =
                Some (trX, initial_descriptors)
                → finite_valid_trace SeededX
                    (equivocators_state_project
                      IM' initial_descriptors is)
                    trX
Htr: finite_valid_trace_from SeededXE is tr'
Hlst: finite_valid_trace_from SeededXE
  (finite_trace_last is tr')
  [{|
     l := l;
     input := iom;
     destination := s;
     output := oom
   |}]
Hinit: initial_state_prop is
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input := iom;
      destination := s;
      output := oom
    |}]): state
  (composite_type
     (sub_IM (equivocator_IM IM)
        selection))
final_descriptors: equivocator_descriptors IM'
Hproper: proper_equivocator_descriptors IM'
  final_descriptors
  (destination
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |})
initial_descriptors: equivocator_descriptors IM'
trX', lstX: list
  (composite_transition_item
     (λ i : sub_index selection, IM (`i)))
oitem: option (composite_transition_item IM')
final_descriptors': equivocator_descriptors IM'
Hpr_trX': equivocators_trace_project
  (λ i : sub_index selection, IM (`i))
  final_descriptors' tr' =
Some (trX', initial_descriptors)
_Hprojectx: match oitem with
| Some item' =>
    Some ([item'], final_descriptors')
| None => Some ([], final_descriptors')
end = Some (lstX, final_descriptors')
HtrX': finite_valid_trace_from SeededX
  (equivocators_state_project IM'
     initial_descriptors is) trX'
HinitX: initial_state_prop
  (equivocators_state_project IM'
     initial_descriptors is)
Hproper': proper_equivocator_descriptors IM'
  final_descriptors'
  (finite_trace_last is tr')
Heq_final_descriptors': final_descriptors' =
equivocator_descriptors_update
  IM' final_descriptors
  (projT1
     (VLSM.l
        {|
          l := l;
          input := iom;
          destination := s;
          output := oom
        |}))
  (final_descriptors'
     (projT1
        (VLSM.l
           {|
             l := l;
             input := iom;
             destination := s;
             output := oom
           |})))
Hx: match oitem with
| Some itemx =>
    ∀ sx : composite_state IM',
      sx =
      equivocators_state_project IM'
        final_descriptors'
        (finite_trace_last is tr')
      → composite_valid IM' 
          (VLSM.l itemx) (
          sx, input itemx)
        ∧ composite_transition IM' 
            (VLSM.l itemx) (
            sx, input itemx) =
          (destination itemx, output itemx)
| None =>
    equivocators_state_project IM'
      final_descriptors
      (destination
         {|
           l := l;
           input := iom;
           destination := s;
           output := oom
         |}) =
    equivocators_state_project IM'
      final_descriptors'
      (finite_trace_last is tr')
end
Hitemx: match oitem with
| Some itemx =>
    (∃ Hex : existing_equivocator_label
               (IM'
                  (projT1
                     (VLSM.l
                      {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                      |})))
               (projT2
                  (VLSM.l
                     {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                     |})),
       existT
         (projT1
            (VLSM.l
               {|
                 l := l;
                 input := iom;
                 destination := s;
                 output := oom
               |}))
         (existing_equivocator_label_extract
            (IM'
               (projT1
                  (VLSM.l
                     {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                     |})))
            (projT2
               (VLSM.l
                  {|
                    l := l;
                    input := iom;
                    destination := s;
                    output := oom
                  |})) Hex) = 
       VLSM.l itemx)
    ∧ input
        {|
          l := l;
          input := iom;
          destination := s;
          output := oom
        |} = input itemx
      ∧ output
          {|
            l := l;
            input := iom;
            destination := s;
            output := oom
          |} = output itemx
        ∧ equivocators_state_project IM'
            final_descriptors
            (destination
               {|
                 l := l;
                 input := iom;
                 destination := s;
                 output := oom
               |}) = destination itemx
          ∧ final_descriptors'
              (projT1
                 (VLSM.l
                    {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                    |})) =
            equivocator_label_descriptor
              (projT2
                 (VLSM.l
                    {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                    |}))
| None => True
end
Hprojectx: equivocators_transition_item_project IM'
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some (oitem, final_descriptors')
Htl: finite_valid_trace_from SeededXE s []
Hs: valid_state_prop SeededXE
  (finite_trace_last is tr')
Hiom: option_valid_message_prop SeededXE iom
Hc: no_equivocations_additional_constraint_with_preloaded
  (sub_IM (equivocator_IM IM) selection)
  (free_constraint
     (sub_IM (equivocator_IM IM) selection)) seed
  l (finite_trace_last is tr', iom)
valid_state_prop SeededX
  (finite_trace_last
     (equivocators_state_project IM'
        initial_descriptors is) trX') by apply (finite_valid_trace_last_pstate SeededX) in HtrX'. }message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
is: composite_state
  (sub_IM (equivocator_IM IM) selection)
IM':= λ i : sub_index selection, IM (`i): sub_index selection → VLSM message
tr': list
  (composite_transition_item
     (sub_IM (equivocator_IM IM) selection))
s: state SeededXE
iom, oom: option message
l: label SeededXE
H: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}])
  → ∀ tr : list
             (composite_transition_item
                (sub_IM 
                   (equivocator_IM IM) selection)),
      finite_valid_trace SeededXE is tr
      → let final_state := 
          finite_trace_last is tr in
        y = length tr
        → ∀ final_descriptors : 
             equivocator_descriptors IM',
            proper_equivocator_descriptors IM'
              final_descriptors final_state
            → ∀ (initial_descriptors : 
                 equivocator_descriptors IM') 
                (trX : 
                 list
                   (composite_transition_item IM')),
                equivocators_trace_project IM'
                  final_descriptors tr =
                Some (trX, initial_descriptors)
                → finite_valid_trace SeededX
                    (equivocators_state_project
                      IM' initial_descriptors is)
                    trX
Htr: finite_valid_trace_from SeededXE is tr'
Hlst: finite_valid_trace_from SeededXE
  (finite_trace_last is tr')
  [{|
     l := l;
     input := iom;
     destination := s;
     output := oom
   |}]
Hinit: initial_state_prop is
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input := iom;
      destination := s;
      output := oom
    |}]): state
  (composite_type
     (sub_IM (equivocator_IM IM)
        selection))
final_descriptors: equivocator_descriptors IM'
Hproper: proper_equivocator_descriptors IM'
  final_descriptors
  (destination
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |})
initial_descriptors: equivocator_descriptors IM'
trX', lstX: list
  (composite_transition_item
     (λ i : sub_index selection, IM (`i)))
oitem: option (composite_transition_item IM')
final_descriptors': equivocator_descriptors IM'
Hpr_trX': equivocators_trace_project
  (λ i : sub_index selection, IM (`i))
  final_descriptors' tr' =
Some (trX', initial_descriptors)
_Hprojectx: match oitem with
| Some item' =>
    Some ([item'], final_descriptors')
| None => Some ([], final_descriptors')
end = Some (lstX, final_descriptors')
HtrX': finite_valid_trace_from SeededX
  (equivocators_state_project IM'
     initial_descriptors is) trX'
HinitX: initial_state_prop
  (equivocators_state_project IM'
     initial_descriptors is)
Hproper': proper_equivocator_descriptors IM'
  final_descriptors'
  (finite_trace_last is tr')
Heq_final_descriptors': final_descriptors' =
equivocator_descriptors_update
  IM' final_descriptors
  (projT1
     (VLSM.l
        {|
          l := l;
          input := iom;
          destination := s;
          output := oom
        |}))
  (final_descriptors'
     (projT1
        (VLSM.l
           {|
             l := l;
             input := iom;
             destination := s;
             output := oom
           |})))
Hx: match oitem with
| Some itemx =>
    ∀ sx : composite_state IM',
      sx =
      equivocators_state_project IM'
        final_descriptors'
        (finite_trace_last is tr')
      → composite_valid IM' 
          (VLSM.l itemx) (
          sx, input itemx)
        ∧ composite_transition IM' 
            (VLSM.l itemx) (
            sx, input itemx) =
          (destination itemx, output itemx)
| None =>
    equivocators_state_project IM'
      final_descriptors
      (destination
         {|
           l := l;
           input := iom;
           destination := s;
           output := oom
         |}) =
    equivocators_state_project IM'
      final_descriptors'
      (finite_trace_last is tr')
end
Hitemx: match oitem with
| Some itemx =>
    (∃ Hex : existing_equivocator_label
               (IM'
                  (projT1
                     (VLSM.l
                      {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                      |})))
               (projT2
                  (VLSM.l
                     {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                     |})),
       existT
         (projT1
            (VLSM.l
               {|
                 l := l;
                 input := iom;
                 destination := s;
                 output := oom
               |}))
         (existing_equivocator_label_extract
            (IM'
               (projT1
                  (VLSM.l
                     {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                     |})))
            (projT2
               (VLSM.l
                  {|
                    l := l;
                    input := iom;
                    destination := s;
                    output := oom
                  |})) Hex) = 
       VLSM.l itemx)
    ∧ input
        {|
          l := l;
          input := iom;
          destination := s;
          output := oom
        |} = input itemx
      ∧ output
          {|
            l := l;
            input := iom;
            destination := s;
            output := oom
          |} = output itemx
        ∧ equivocators_state_project IM'
            final_descriptors
            (destination
               {|
                 l := l;
                 input := iom;
                 destination := s;
                 output := oom
               |}) = destination itemx
          ∧ final_descriptors'
              (projT1
                 (VLSM.l
                    {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                    |})) =
            equivocator_label_descriptor
              (projT2
                 (VLSM.l
                    {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                    |}))
| None => True
end
Hprojectx: equivocators_transition_item_project IM'
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some (oitem, final_descriptors')
Htl: finite_valid_trace_from SeededXE s []
Hs: valid_state_prop SeededXE
  (finite_trace_last is tr')
Hiom: option_valid_message_prop SeededXE iom
Hc: no_equivocations_additional_constraint_with_preloaded
  (sub_IM (equivocator_IM IM) selection)
  (free_constraint
     (sub_IM (equivocator_IM IM) selection)) seed
  l (finite_trace_last is tr', iom)
Hlst_trX': valid_state_prop SeededX
  (finite_trace_last
     (equivocators_state_project IM'
        initial_descriptors is) trX')
finite_valid_trace_from SeededX
  (finite_trace_last
     (equivocators_state_project IM'
        initial_descriptors is) trX') lstX
    destruct oitem as [item |]; inversion _Hprojectx; subst lstX; clear _Hprojectx
    ; [| by constructor].message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
is: composite_state
  (sub_IM (equivocator_IM IM) selection)
IM':= λ i : sub_index selection, IM (`i): sub_index selection → VLSM message
tr': list
  (composite_transition_item
     (sub_IM (equivocator_IM IM) selection))
s: state SeededXE
iom, oom: option message
l: label SeededXE
H: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}])
  → ∀ tr : list
             (composite_transition_item
                (sub_IM 
                   (equivocator_IM IM) selection)),
      finite_valid_trace SeededXE is tr
      → let final_state := 
          finite_trace_last is tr in
        y = length tr
        → ∀ final_descriptors : 
             equivocator_descriptors IM',
            proper_equivocator_descriptors IM'
              final_descriptors final_state
            → ∀ (initial_descriptors : 
                 equivocator_descriptors IM') 
                (trX : 
                 list
                   (composite_transition_item IM')),
                equivocators_trace_project IM'
                  final_descriptors tr =
                Some (trX, initial_descriptors)
                → finite_valid_trace SeededX
                    (equivocators_state_project
                      IM' initial_descriptors is)
                    trX
Htr: finite_valid_trace_from SeededXE is tr'
Hlst: finite_valid_trace_from SeededXE
  (finite_trace_last is tr')
  [{|
     l := l;
     input := iom;
     destination := s;
     output := oom
   |}]
Hinit: initial_state_prop is
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input := iom;
      destination := s;
      output := oom
    |}]): state
  (composite_type
     (sub_IM (equivocator_IM IM)
        selection))
final_descriptors: equivocator_descriptors IM'
Hproper: proper_equivocator_descriptors IM'
  final_descriptors
  (destination
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |})
initial_descriptors: equivocator_descriptors IM'
trX': list
  (composite_transition_item
     (λ i : sub_index selection, IM (`i)))
item: composite_transition_item IM'
final_descriptors': equivocator_descriptors IM'
Hpr_trX': equivocators_trace_project
  (λ i : sub_index selection, IM (`i))
  final_descriptors' tr' =
Some (trX', initial_descriptors)
HtrX': finite_valid_trace_from SeededX
  (equivocators_state_project IM'
     initial_descriptors is) trX'
HinitX: initial_state_prop
  (equivocators_state_project IM'
     initial_descriptors is)
Hproper': proper_equivocator_descriptors IM'
  final_descriptors'
  (finite_trace_last is tr')
Heq_final_descriptors': final_descriptors' =
equivocator_descriptors_update
  IM' final_descriptors
  (projT1
     (VLSM.l
        {|
          l := l;
          input := iom;
          destination := s;
          output := oom
        |}))
  (final_descriptors'
     (projT1
        (VLSM.l
           {|
             l := l;
             input := iom;
             destination := s;
             output := oom
           |})))
Hx: ∀ sx : composite_state IM',
  sx =
  equivocators_state_project IM'
    final_descriptors' 
    (finite_trace_last is tr')
  → composite_valid IM' 
      (VLSM.l item) (sx, input item)
    ∧ composite_transition IM' 
        (VLSM.l item) (
        sx, input item) =
      (destination item, output item)
Hitemx: (∃ Hex : existing_equivocator_label
           (IM'
              (projT1
                 (VLSM.l
                    {|
                      l := l;
                      input := iom;
                      destination := s;
                      output := oom
                    |})))
           (projT2
              (VLSM.l
                 {|
                   l := l;
                   input := iom;
                   destination := s;
                   output := oom
                 |})),
   existT
     (projT1
        (VLSM.l
           {|
             l := l;
             input := iom;
             destination := s;
             output := oom
           |}))
     (existing_equivocator_label_extract
        (IM'
           (projT1
              (VLSM.l
                 {|
                   l := l;
                   input := iom;
                   destination := s;
                   output := oom
                 |})))
        (projT2
           (VLSM.l
              {|
                l := l;
                input := iom;
                destination := s;
                output := oom
              |})) Hex) = 
   VLSM.l item)
∧ input
    {|
      l := l;
      input := iom;
      destination := s;
      output := oom
    |} = input item
  ∧ output
      {|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |} = output item
    ∧ equivocators_state_project IM'
        final_descriptors
        (destination
           {|
             l := l;
             input := iom;
             destination := s;
             output := oom
           |}) = destination item
      ∧ final_descriptors'
          (projT1
             (VLSM.l
                {|
                  l := l;
                  input := iom;
                  destination := s;
                  output := oom
                |})) =
        equivocator_label_descriptor
          (projT2
             (VLSM.l
                {|
                  l := l;
                  input := iom;
                  destination := s;
                  output := oom
                |}))
Hprojectx: equivocators_transition_item_project IM'
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} =
Some (Some item, final_descriptors')
Htl: finite_valid_trace_from SeededXE s []
Hs: valid_state_prop SeededXE
  (finite_trace_last is tr')
Hiom: option_valid_message_prop SeededXE iom
Hc: no_equivocations_additional_constraint_with_preloaded
  (sub_IM (equivocator_IM IM) selection)
  (free_constraint
     (sub_IM (equivocator_IM IM) selection)) seed
  l (finite_trace_last is tr', iom)
Hlst_trX': valid_state_prop SeededX
  (finite_trace_last
     (equivocators_state_project IM'
        initial_descriptors is) trX')
finite_valid_trace_from SeededX
  (finite_trace_last
     (equivocators_state_project IM'
        initial_descriptors is) trX') [item]
    simpl in Hitemx.message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
is: composite_state
  (sub_IM (equivocator_IM IM) selection)
IM':= λ i : sub_index selection, IM (`i): sub_index selection → VLSM message
tr': list
  (composite_transition_item
     (sub_IM (equivocator_IM IM) selection))
s: state SeededXE
iom, oom: option message
l: label SeededXE
H: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}])
  → ∀ tr : list
             (composite_transition_item
                (sub_IM 
                   (equivocator_IM IM) selection)),
      finite_valid_trace SeededXE is tr
      → let final_state := 
          finite_trace_last is tr in
        y = length tr
        → ∀ final_descriptors : 
             equivocator_descriptors IM',
            proper_equivocator_descriptors IM'
              final_descriptors final_state
            → ∀ (initial_descriptors : 
                 equivocator_descriptors IM') 
                (trX : 
                 list
                   (composite_transition_item IM')),
                equivocators_trace_project IM'
                  final_descriptors tr =
                Some (trX, initial_descriptors)
                → finite_valid_trace SeededX
                    (equivocators_state_project
                      IM' initial_descriptors is)
                    trX
Htr: finite_valid_trace_from SeededXE is tr'
Hlst: finite_valid_trace_from SeededXE
  (finite_trace_last is tr')
  [{|
     l := l;
     input := iom;
     destination := s;
     output := oom
   |}]
Hinit: initial_state_prop is
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input := iom;
      destination := s;
      output := oom
    |}]): state
  (composite_type
     (sub_IM (equivocator_IM IM)
        selection))
final_descriptors: equivocator_descriptors IM'
Hproper: proper_equivocator_descriptors IM'
  final_descriptors
  (destination
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |})
initial_descriptors: equivocator_descriptors IM'
trX': list
  (composite_transition_item
     (λ i : sub_index selection, IM (`i)))
item: composite_transition_item IM'
final_descriptors': equivocator_descriptors IM'
Hpr_trX': equivocators_trace_project
  (λ i : sub_index selection, IM (`i))
  final_descriptors' tr' =
Some (trX', initial_descriptors)
HtrX': finite_valid_trace_from SeededX
  (equivocators_state_project IM'
     initial_descriptors is) trX'
HinitX: initial_state_prop
  (equivocators_state_project IM'
     initial_descriptors is)
Hproper': proper_equivocator_descriptors IM'
  final_descriptors'
  (finite_trace_last is tr')
Heq_final_descriptors': final_descriptors' =
equivocator_descriptors_update
  IM' final_descriptors
  (projT1
     (VLSM.l
        {|
          l := l;
          input := iom;
          destination := s;
          output := oom
        |}))
  (final_descriptors'
     (projT1
        (VLSM.l
           {|
             l := l;
             input := iom;
             destination := s;
             output := oom
           |})))
Hx: ∀ sx : composite_state IM',
  sx =
  equivocators_state_project IM'
    final_descriptors' 
    (finite_trace_last is tr')
  → composite_valid IM' 
      (VLSM.l item) (sx, input item)
    ∧ composite_transition IM' 
        (VLSM.l item) (
        sx, input item) =
      (destination item, output item)
Hitemx: (∃ Hex : existing_equivocator_label
           (IM' (projT1 l)) 
           (projT2 l),
   existT (projT1 l)
     (existing_equivocator_label_extract
        (IM' (projT1 l)) 
        (projT2 l) Hex) = 
   VLSM.l item)
∧ iom = input item
  ∧ oom = output item
    ∧ equivocators_state_project IM'
        final_descriptors s =
      destination item
      ∧ final_descriptors' (projT1 l) =
        equivocator_label_descriptor
          (projT2 l)
Hprojectx: equivocators_transition_item_project IM'
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} =
Some (Some item, final_descriptors')
Htl: finite_valid_trace_from SeededXE s []
Hs: valid_state_prop SeededXE
  (finite_trace_last is tr')
Hiom: option_valid_message_prop SeededXE iom
Hc: no_equivocations_additional_constraint_with_preloaded
  (sub_IM (equivocator_IM IM) selection)
  (free_constraint
     (sub_IM (equivocator_IM IM) selection)) seed
  l (finite_trace_last is tr', iom)
Hlst_trX': valid_state_prop SeededX
  (finite_trace_last
     (equivocators_state_project IM'
        initial_descriptors is) trX')
finite_valid_trace_from SeededX
  (finite_trace_last
     (equivocators_state_project IM'
        initial_descriptors is) trX') [item] destruct Hitemx as [Hl [Hinput [Houtput [Hdestination _]]]].message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
is: composite_state
  (sub_IM (equivocator_IM IM) selection)
IM':= λ i : sub_index selection, IM (`i): sub_index selection → VLSM message
tr': list
  (composite_transition_item
     (sub_IM (equivocator_IM IM) selection))
s: state SeededXE
iom, oom: option message
l: label SeededXE
H: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}])
  → ∀ tr : list
             (composite_transition_item
                (sub_IM 
                   (equivocator_IM IM) selection)),
      finite_valid_trace SeededXE is tr
      → let final_state := 
          finite_trace_last is tr in
        y = length tr
        → ∀ final_descriptors : 
             equivocator_descriptors IM',
            proper_equivocator_descriptors IM'
              final_descriptors final_state
            → ∀ (initial_descriptors : 
                 equivocator_descriptors IM') 
                (trX : 
                 list
                   (composite_transition_item IM')),
                equivocators_trace_project IM'
                  final_descriptors tr =
                Some (trX, initial_descriptors)
                → finite_valid_trace SeededX
                    (equivocators_state_project
                      IM' initial_descriptors is)
                    trX
Htr: finite_valid_trace_from SeededXE is tr'
Hlst: finite_valid_trace_from SeededXE
  (finite_trace_last is tr')
  [{|
     l := l;
     input := iom;
     destination := s;
     output := oom
   |}]
Hinit: initial_state_prop is
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input := iom;
      destination := s;
      output := oom
    |}]): state
  (composite_type
     (sub_IM (equivocator_IM IM)
        selection))
final_descriptors: equivocator_descriptors IM'
Hproper: proper_equivocator_descriptors IM'
  final_descriptors
  (destination
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |})
initial_descriptors: equivocator_descriptors IM'
trX': list
  (composite_transition_item
     (λ i : sub_index selection, IM (`i)))
item: composite_transition_item IM'
final_descriptors': equivocator_descriptors IM'
Hpr_trX': equivocators_trace_project
  (λ i : sub_index selection, IM (`i))
  final_descriptors' tr' =
Some (trX', initial_descriptors)
HtrX': finite_valid_trace_from SeededX
  (equivocators_state_project IM'
     initial_descriptors is) trX'
HinitX: initial_state_prop
  (equivocators_state_project IM'
     initial_descriptors is)
Hproper': proper_equivocator_descriptors IM'
  final_descriptors'
  (finite_trace_last is tr')
Heq_final_descriptors': final_descriptors' =
equivocator_descriptors_update
  IM' final_descriptors
  (projT1
     (VLSM.l
        {|
          l := l;
          input := iom;
          destination := s;
          output := oom
        |}))
  (final_descriptors'
     (projT1
        (VLSM.l
           {|
             l := l;
             input := iom;
             destination := s;
             output := oom
           |})))
Hx: ∀ sx : composite_state IM',
  sx =
  equivocators_state_project IM'
    final_descriptors' 
    (finite_trace_last is tr')
  → composite_valid IM' 
      (VLSM.l item) (sx, input item)
    ∧ composite_transition IM' 
        (VLSM.l item) (
        sx, input item) =
      (destination item, output item)
Hl: ∃ Hex : existing_equivocator_label
          (IM' (projT1 l)) 
          (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM' (projT1 l)) 
       (projT2 l) Hex) = 
  VLSM.l item
Hinput: iom = input item
Houtput: oom = output item
Hdestination: equivocators_state_project IM'
  final_descriptors s =
destination item
Hprojectx: equivocators_transition_item_project IM'
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} =
Some (Some item, final_descriptors')
Htl: finite_valid_trace_from SeededXE s []
Hs: valid_state_prop SeededXE
  (finite_trace_last is tr')
Hiom: option_valid_message_prop SeededXE iom
Hc: no_equivocations_additional_constraint_with_preloaded
  (sub_IM (equivocator_IM IM) selection)
  (free_constraint
     (sub_IM (equivocator_IM IM) selection)) seed
  l (finite_trace_last is tr', iom)
Hlst_trX': valid_state_prop SeededX
  (finite_trace_last
     (equivocators_state_project IM'
        initial_descriptors is) trX')
finite_valid_trace_from SeededX
  (finite_trace_last
     (equivocators_state_project IM'
        initial_descriptors is) trX') [item]
    specialize (Hx _ eq_refl).message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
is: composite_state
  (sub_IM (equivocator_IM IM) selection)
IM':= λ i : sub_index selection, IM (`i): sub_index selection → VLSM message
tr': list
  (composite_transition_item
     (sub_IM (equivocator_IM IM) selection))
s: state SeededXE
iom, oom: option message
l: label SeededXE
H: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}])
  → ∀ tr : list
             (composite_transition_item
                (sub_IM 
                   (equivocator_IM IM) selection)),
      finite_valid_trace SeededXE is tr
      → let final_state := 
          finite_trace_last is tr in
        y = length tr
        → ∀ final_descriptors : 
             equivocator_descriptors IM',
            proper_equivocator_descriptors IM'
              final_descriptors final_state
            → ∀ (initial_descriptors : 
                 equivocator_descriptors IM') 
                (trX : 
                 list
                   (composite_transition_item IM')),
                equivocators_trace_project IM'
                  final_descriptors tr =
                Some (trX, initial_descriptors)
                → finite_valid_trace SeededX
                    (equivocators_state_project
                      IM' initial_descriptors is)
                    trX
Htr: finite_valid_trace_from SeededXE is tr'
Hlst: finite_valid_trace_from SeededXE
  (finite_trace_last is tr')
  [{|
     l := l;
     input := iom;
     destination := s;
     output := oom
   |}]
Hinit: initial_state_prop is
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input := iom;
      destination := s;
      output := oom
    |}]): state
  (composite_type
     (sub_IM (equivocator_IM IM)
        selection))
final_descriptors: equivocator_descriptors IM'
Hproper: proper_equivocator_descriptors IM'
  final_descriptors
  (destination
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |})
initial_descriptors: equivocator_descriptors IM'
trX': list
  (composite_transition_item
     (λ i : sub_index selection, IM (`i)))
item: composite_transition_item IM'
final_descriptors': equivocator_descriptors IM'
Hpr_trX': equivocators_trace_project
  (λ i : sub_index selection, IM (`i))
  final_descriptors' tr' =
Some (trX', initial_descriptors)
HtrX': finite_valid_trace_from SeededX
  (equivocators_state_project IM'
     initial_descriptors is) trX'
HinitX: initial_state_prop
  (equivocators_state_project IM'
     initial_descriptors is)
Hproper': proper_equivocator_descriptors IM'
  final_descriptors'
  (finite_trace_last is tr')
Heq_final_descriptors': final_descriptors' =
equivocator_descriptors_update
  IM' final_descriptors
  (projT1
     (VLSM.l
        {|
          l := l;
          input := iom;
          destination := s;
          output := oom
        |}))
  (final_descriptors'
     (projT1
        (VLSM.l
           {|
             l := l;
             input := iom;
             destination := s;
             output := oom
           |})))
Hx: composite_valid IM' (VLSM.l item)
  (equivocators_state_project IM'
     final_descriptors'
     (finite_trace_last is tr'), 
   input item)
∧ composite_transition IM' 
    (VLSM.l item)
    (equivocators_state_project IM'
       final_descriptors'
       (finite_trace_last is tr'), 
     input item) =
  (destination item, output item)
Hl: ∃ Hex : existing_equivocator_label
          (IM' (projT1 l)) 
          (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM' (projT1 l)) 
       (projT2 l) Hex) = 
  VLSM.l item
Hinput: iom = input item
Houtput: oom = output item
Hdestination: equivocators_state_project IM'
  final_descriptors s =
destination item
Hprojectx: equivocators_transition_item_project IM'
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} =
Some (Some item, final_descriptors')
Htl: finite_valid_trace_from SeededXE s []
Hs: valid_state_prop SeededXE
  (finite_trace_last is tr')
Hiom: option_valid_message_prop SeededXE iom
Hc: no_equivocations_additional_constraint_with_preloaded
  (sub_IM (equivocator_IM IM) selection)
  (free_constraint
     (sub_IM (equivocator_IM IM) selection)) seed
  l (finite_trace_last is tr', iom)
Hlst_trX': valid_state_prop SeededX
  (finite_trace_last
     (equivocators_state_project IM'
        initial_descriptors is) trX')
finite_valid_trace_from SeededX
  (finite_trace_last
     (equivocators_state_project IM'
        initial_descriptors is) trX') [item]
    destruct Hx as [Hvx Htx].message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
is: composite_state
  (sub_IM (equivocator_IM IM) selection)
IM':= λ i : sub_index selection, IM (`i): sub_index selection → VLSM message
tr': list
  (composite_transition_item
     (sub_IM (equivocator_IM IM) selection))
s: state SeededXE
iom, oom: option message
l: label SeededXE
H: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}])
  → ∀ tr : list
             (composite_transition_item
                (sub_IM 
                   (equivocator_IM IM) selection)),
      finite_valid_trace SeededXE is tr
      → let final_state := 
          finite_trace_last is tr in
        y = length tr
        → ∀ final_descriptors : 
             equivocator_descriptors IM',
            proper_equivocator_descriptors IM'
              final_descriptors final_state
            → ∀ (initial_descriptors : 
                 equivocator_descriptors IM') 
                (trX : 
                 list
                   (composite_transition_item IM')),
                equivocators_trace_project IM'
                  final_descriptors tr =
                Some (trX, initial_descriptors)
                → finite_valid_trace SeededX
                    (equivocators_state_project
                      IM' initial_descriptors is)
                    trX
Htr: finite_valid_trace_from SeededXE is tr'
Hlst: finite_valid_trace_from SeededXE
  (finite_trace_last is tr')
  [{|
     l := l;
     input := iom;
     destination := s;
     output := oom
   |}]
Hinit: initial_state_prop is
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input := iom;
      destination := s;
      output := oom
    |}]): state
  (composite_type
     (sub_IM (equivocator_IM IM)
        selection))
final_descriptors: equivocator_descriptors IM'
Hproper: proper_equivocator_descriptors IM'
  final_descriptors
  (destination
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |})
initial_descriptors: equivocator_descriptors IM'
trX': list
  (composite_transition_item
     (λ i : sub_index selection, IM (`i)))
item: composite_transition_item IM'
final_descriptors': equivocator_descriptors IM'
Hpr_trX': equivocators_trace_project
  (λ i : sub_index selection, IM (`i))
  final_descriptors' tr' =
Some (trX', initial_descriptors)
HtrX': finite_valid_trace_from SeededX
  (equivocators_state_project IM'
     initial_descriptors is) trX'
HinitX: initial_state_prop
  (equivocators_state_project IM'
     initial_descriptors is)
Hproper': proper_equivocator_descriptors IM'
  final_descriptors'
  (finite_trace_last is tr')
Heq_final_descriptors': final_descriptors' =
equivocator_descriptors_update
  IM' final_descriptors
  (projT1
     (VLSM.l
        {|
          l := l;
          input := iom;
          destination := s;
          output := oom
        |}))
  (final_descriptors'
     (projT1
        (VLSM.l
           {|
             l := l;
             input := iom;
             destination := s;
             output := oom
           |})))
Hvx: composite_valid IM' (VLSM.l item)
  (equivocators_state_project IM'
     final_descriptors'
     (finite_trace_last is tr'), 
   input item)
Htx: composite_transition IM' 
  (VLSM.l item)
  (equivocators_state_project IM'
     final_descriptors'
     (finite_trace_last is tr'), 
   input item) = (destination item, output item)
Hl: ∃ Hex : existing_equivocator_label
          (IM' (projT1 l)) 
          (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM' (projT1 l)) 
       (projT2 l) Hex) = 
  VLSM.l item
Hinput: iom = input item
Houtput: oom = output item
Hdestination: equivocators_state_project IM'
  final_descriptors s =
destination item
Hprojectx: equivocators_transition_item_project IM'
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} =
Some (Some item, final_descriptors')
Htl: finite_valid_trace_from SeededXE s []
Hs: valid_state_prop SeededXE
  (finite_trace_last is tr')
Hiom: option_valid_message_prop SeededXE iom
Hc: no_equivocations_additional_constraint_with_preloaded
  (sub_IM (equivocator_IM IM) selection)
  (free_constraint
     (sub_IM (equivocator_IM IM) selection)) seed
  l (finite_trace_last is tr', iom)
Hlst_trX': valid_state_prop SeededX
  (finite_trace_last
     (equivocators_state_project IM'
        initial_descriptors is) trX')
finite_valid_trace_from SeededX
  (finite_trace_last
     (equivocators_state_project IM'
        initial_descriptors is) trX') [item]
    destruct item.message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
is: composite_state
  (sub_IM (equivocator_IM IM) selection)
IM':= λ i : sub_index selection, IM (`i): sub_index selection → VLSM message
tr': list
  (composite_transition_item
     (sub_IM (equivocator_IM IM) selection))
s: state SeededXE
iom, oom: option message
l: label SeededXE
H: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}])
  → ∀ tr : list
             (composite_transition_item
                (sub_IM 
                   (equivocator_IM IM) selection)),
      finite_valid_trace SeededXE is tr
      → let final_state := 
          finite_trace_last is tr in
        y = length tr
        → ∀ final_descriptors : 
             equivocator_descriptors IM',
            proper_equivocator_descriptors IM'
              final_descriptors final_state
            → ∀ (initial_descriptors : 
                 equivocator_descriptors IM') 
                (trX : 
                 list
                   (composite_transition_item IM')),
                equivocators_trace_project IM'
                  final_descriptors tr =
                Some (trX, initial_descriptors)
                → finite_valid_trace SeededX
                    (equivocators_state_project
                      IM' initial_descriptors is)
                    trX
Htr: finite_valid_trace_from SeededXE is tr'
Hlst: finite_valid_trace_from SeededXE
  (finite_trace_last is tr')
  [{|
     l := l;
     input := iom;
     destination := s;
     output := oom
   |}]
Hinit: initial_state_prop is
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input := iom;
      destination := s;
      output := oom
    |}]): state
  (composite_type
     (sub_IM (equivocator_IM IM)
        selection))
final_descriptors: equivocator_descriptors IM'
Hproper: proper_equivocator_descriptors IM'
  final_descriptors
  (VLSM.destination
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |})
initial_descriptors: equivocator_descriptors IM'
trX': list
  (composite_transition_item
     (λ i : sub_index selection, IM (`i)))
l0: label (composite_type IM')
input: option message
destination: state (composite_type IM')
output: option message
final_descriptors': equivocator_descriptors IM'
Hpr_trX': equivocators_trace_project
  (λ i : sub_index selection, IM (`i))
  final_descriptors' tr' =
Some (trX', initial_descriptors)
HtrX': finite_valid_trace_from SeededX
  (equivocators_state_project IM'
     initial_descriptors is) trX'
HinitX: initial_state_prop
  (equivocators_state_project IM'
     initial_descriptors is)
Hproper': proper_equivocator_descriptors IM'
  final_descriptors'
  (finite_trace_last is tr')
Heq_final_descriptors': final_descriptors' =
equivocator_descriptors_update
  IM' final_descriptors
  (projT1
     (VLSM.l
        {|
          l := l;
          input := iom;
          destination := s;
          output := oom
        |}))
  (final_descriptors'
     (projT1
        (VLSM.l
           {|
             l := l;
             input := iom;
             destination := s;
             output := oom
           |})))
Hvx: composite_valid IM'
  (VLSM.l
     {|
       l := l0;
       input := input;
       destination := destination;
       output := output
     |})
  (equivocators_state_project IM'
     final_descriptors'
     (finite_trace_last is tr'),
   VLSM.input
     {|
       l := l0;
       input := input;
       destination := destination;
       output := output
     |})
Htx: composite_transition IM'
  (VLSM.l
     {|
       l := l0;
       input := input;
       destination := destination;
       output := output
     |})
  (equivocators_state_project IM'
     final_descriptors'
     (finite_trace_last is tr'),
   VLSM.input
     {|
       l := l0;
       input := input;
       destination := destination;
       output := output
     |}) =
(VLSM.destination
   {|
     l := l0;
     input := input;
     destination := destination;
     output := output
   |},
 VLSM.output
   {|
     l := l0;
     input := input;
     destination := destination;
     output := output
   |})
Hl: ∃ Hex : existing_equivocator_label
          (IM' (projT1 l)) 
          (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM' (projT1 l)) 
       (projT2 l) Hex) =
  VLSM.l
    {|
      l := l0;
      input := input;
      destination := destination;
      output := output
    |}
Hinput: iom =
VLSM.input
  {|
    l := l0;
    input := input;
    destination := destination;
    output := output
  |}
Houtput: oom =
VLSM.output
  {|
    l := l0;
    input := input;
    destination := destination;
    output := output
  |}
Hdestination: equivocators_state_project IM'
  final_descriptors s =
VLSM.destination
  {|
    l := l0;
    input := input;
    destination := destination;
    output := output
  |}
Hprojectx: equivocators_transition_item_project IM'
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} =
Some
  (Some
     {|
       l := l0;
       input := input;
       destination := destination;
       output := output
     |}, final_descriptors')
Htl: finite_valid_trace_from SeededXE s []
Hs: valid_state_prop SeededXE
  (finite_trace_last is tr')
Hiom: option_valid_message_prop SeededXE iom
Hc: no_equivocations_additional_constraint_with_preloaded
  (sub_IM (equivocator_IM IM) selection)
  (free_constraint
     (sub_IM (equivocator_IM IM) selection)) seed
  l (finite_trace_last is tr', iom)
Hlst_trX': valid_state_prop SeededX
  (finite_trace_last
     (equivocators_state_project IM'
        initial_descriptors is) trX')
finite_valid_trace_from SeededX
  (finite_trace_last
     (equivocators_state_project IM'
        initial_descriptors is) trX')
  [{|
     l := l0;
     input := input;
     destination := destination;
     output := output
   |}] simpl in *.message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
is: composite_state
  (sub_IM (equivocator_IM IM) selection)
IM':= λ i : sub_index selection, IM (`i): sub_index selection → VLSM message
tr': list
  (composite_transition_item
     (sub_IM (equivocator_IM IM) selection))
s: composite_state
  (sub_IM (equivocator_IM IM) selection)
iom, oom: option message
l: composite_label
  (sub_IM (equivocator_IM IM) selection)
H: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}])
  → ∀ tr : list
             (composite_transition_item
                (sub_IM 
                   (equivocator_IM IM) selection)),
      finite_valid_trace SeededXE is tr
      → y = length tr
        → ∀ final_descriptors : 
             equivocator_descriptors IM',
            proper_equivocator_descriptors IM'
              final_descriptors
              (finite_trace_last is tr)
            → ∀ (initial_descriptors : 
                 equivocator_descriptors IM') 
                (trX : 
                 list
                   (composite_transition_item IM')),
                equivocators_trace_project IM'
                  final_descriptors tr =
                Some (trX, initial_descriptors)
                → finite_valid_trace SeededX
                    (equivocators_state_project
                      IM' initial_descriptors is)
                    trX
Htr: finite_valid_trace_from SeededXE is tr'
Hlst: finite_valid_trace_from SeededXE
  (finite_trace_last is tr')
  [{|
     l := l;
     input := iom;
     destination := s;
     output := oom
   |}]
Hinit: composite_initial_state_prop
  (sub_IM (equivocator_IM IM) selection) is
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input := iom;
      destination := s;
      output := oom
    |}]): composite_state
  (sub_IM (equivocator_IM IM) selection)
final_descriptors: equivocator_descriptors IM'
Hproper: proper_equivocator_descriptors IM'
  final_descriptors s
initial_descriptors: equivocator_descriptors IM'
trX': list
  (composite_transition_item
     (λ i : sub_index selection, IM (`i)))
l0: composite_label IM'
input: option message
destination: composite_state IM'
output: option message
final_descriptors': equivocator_descriptors IM'
Hpr_trX': equivocators_trace_project
  (λ i : sub_index selection, IM (`i))
  final_descriptors' tr' =
Some (trX', initial_descriptors)
HtrX': finite_valid_trace_from SeededX
  (equivocators_state_project IM'
     initial_descriptors is) trX'
HinitX: composite_initial_state_prop
  (sub_IM IM selection)
  (equivocators_state_project IM'
     initial_descriptors is)
Hproper': proper_equivocator_descriptors IM'
  final_descriptors'
  (finite_trace_last is tr')
Heq_final_descriptors': final_descriptors' =
equivocator_descriptors_update
  IM' final_descriptors
  (projT1 l)
  (final_descriptors'
     (projT1 l))
Hvx: let (i, li) := l0 in
valid li
  (equivocators_state_project IM'
     final_descriptors'
     (finite_trace_last is tr') i, input)
Htx: (let (i, li) := l0 in
 let (si', om') :=
   transition li
     (equivocators_state_project IM'
        final_descriptors'
        (finite_trace_last is tr') i, input) in
 (state_update IM'
    (equivocators_state_project IM'
       final_descriptors'
       (finite_trace_last is tr')) i si', om')) =
(destination, output)
Hl: ∃ Hex : existing_equivocator_label
          (IM' (projT1 l)) 
          (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM' (projT1 l)) 
       (projT2 l) Hex) = l0
Hinput: iom = input
Houtput: oom = output
Hdestination: equivocators_state_project IM'
  final_descriptors s = destination
Hprojectx: equivocators_transition_item_project IM'
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} =
Some
  (Some
     {|
       l := l0;
       input := input;
       destination := destination;
       output := output
     |}, final_descriptors')
Htl: finite_valid_trace_from SeededXE s []
Hs: valid_state_prop SeededXE
  (finite_trace_last is tr')
Hiom: option_valid_message_prop SeededXE iom
Hc: no_equivocations_additional_constraint_with_preloaded
  (sub_IM (equivocator_IM IM) selection)
  (free_constraint
     (sub_IM (equivocator_IM IM) selection)) seed
  l (finite_trace_last is tr', iom)
Hlst_trX': valid_state_prop SeededX
  (finite_trace_last
     (equivocators_state_project IM'
        initial_descriptors is) trX')
finite_valid_trace_from SeededX
  (finite_trace_last
     (equivocators_state_project IM'
        initial_descriptors is) trX')
  [{|
     l := l0;
     input := input;
     destination := destination;
     output := output
   |}] subst.message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
is: composite_state
  (sub_IM (equivocator_IM IM) selection)
IM':= λ i : sub_index selection, IM (`i): sub_index selection → VLSM message
tr': list
  (composite_transition_item
     (sub_IM (equivocator_IM IM) selection))
s: composite_state
  (sub_IM (equivocator_IM IM) selection)
l: composite_label
  (sub_IM (equivocator_IM IM) selection)
input, output: option message
H: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := input;
        destination := s;
        output := output
      |}])
  → ∀ tr : list
             (composite_transition_item
                (sub_IM 
                   (equivocator_IM IM) selection)),
      finite_valid_trace SeededXE is tr
      → y = length tr
        → ∀ final_descriptors : 
             equivocator_descriptors IM',
            proper_equivocator_descriptors IM'
              final_descriptors
              (finite_trace_last is tr)
            → ∀ (initial_descriptors : 
                 equivocator_descriptors IM') 
                (trX : 
                 list
                   (composite_transition_item IM')),
                equivocators_trace_project IM'
                  final_descriptors tr =
                Some (trX, initial_descriptors)
                → finite_valid_trace SeededX
                    (equivocators_state_project
                      IM' initial_descriptors is)
                    trX
Htr: finite_valid_trace_from SeededXE is tr'
Hlst: finite_valid_trace_from SeededXE
  (finite_trace_last is tr')
  [{|
     l := l;
     input := input;
     destination := s;
     output := output
   |}]
Hinit: composite_initial_state_prop
  (sub_IM (equivocator_IM IM) selection) is
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input := input;
      destination := s;
      output := output
    |}]): composite_state
  (sub_IM (equivocator_IM IM) selection)
final_descriptors: equivocator_descriptors IM'
Hproper: proper_equivocator_descriptors IM'
  final_descriptors s
initial_descriptors: equivocator_descriptors IM'
trX': list
  (composite_transition_item
     (λ i : sub_index selection, IM (`i)))
l0: composite_label IM'
final_descriptors': equivocator_descriptors IM'
Hpr_trX': equivocators_trace_project
  (λ i : sub_index selection, IM (`i))
  final_descriptors' tr' =
Some (trX', initial_descriptors)
HtrX': finite_valid_trace_from SeededX
  (equivocators_state_project IM'
     initial_descriptors is) trX'
HinitX: composite_initial_state_prop
  (sub_IM IM selection)
  (equivocators_state_project IM'
     initial_descriptors is)
Hproper': proper_equivocator_descriptors IM'
  final_descriptors'
  (finite_trace_last is tr')
Heq_final_descriptors': final_descriptors' =
equivocator_descriptors_update
  IM' final_descriptors
  (projT1 l)
  (final_descriptors'
     (projT1 l))
Hvx: let (i, li) := l0 in
valid li
  (equivocators_state_project IM'
     final_descriptors'
     (finite_trace_last is tr') i, input)
Htx: (let (i, li) := l0 in
 let (si', om') :=
   transition li
     (equivocators_state_project IM'
        final_descriptors'
        (finite_trace_last is tr') i, input) in
 (state_update IM'
    (equivocators_state_project IM'
       final_descriptors'
       (finite_trace_last is tr')) i si', om')) =
(equivocators_state_project IM'
   final_descriptors s, output)
Hl: ∃ Hex : existing_equivocator_label
          (IM' (projT1 l)) 
          (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM' (projT1 l)) 
       (projT2 l) Hex) = l0
Hprojectx: equivocators_transition_item_project IM'
  final_descriptors
  {|
    l := l;
    input := input;
    destination := s;
    output := output
  |} =
Some
  (Some
     {|
       l := l0;
       input := input;
       destination :=
         equivocators_state_project IM'
           final_descriptors s;
       output := output
     |}, final_descriptors')
Htl: finite_valid_trace_from SeededXE s []
Hs: valid_state_prop SeededXE
  (finite_trace_last is tr')
Hc: no_equivocations_additional_constraint_with_preloaded
  (sub_IM (equivocator_IM IM) selection)
  (free_constraint
     (sub_IM (equivocator_IM IM) selection)) seed
  l (finite_trace_last is tr', input)
Hiom: option_valid_message_prop SeededXE input
Hlst_trX': valid_state_prop SeededX
  (finite_trace_last
     (equivocators_state_project IM'
        initial_descriptors is) trX')
finite_valid_trace_from SeededX
  (finite_trace_last
     (equivocators_state_project IM'
        initial_descriptors is) trX')
  [{|
     l := l0;
     input := input;
     destination :=
       equivocators_state_project IM'
         final_descriptors s;
     output := output
   |}]
    apply finite_valid_trace_singleton.message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
is: composite_state
  (sub_IM (equivocator_IM IM) selection)
IM':= λ i : sub_index selection, IM (`i): sub_index selection → VLSM message
tr': list
  (composite_transition_item
     (sub_IM (equivocator_IM IM) selection))
s: composite_state
  (sub_IM (equivocator_IM IM) selection)
l: composite_label
  (sub_IM (equivocator_IM IM) selection)
input, output: option message
H: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := input;
        destination := s;
        output := output
      |}])
  → ∀ tr : list
             (composite_transition_item
                (sub_IM 
                   (equivocator_IM IM) selection)),
      finite_valid_trace SeededXE is tr
      → y = length tr
        → ∀ final_descriptors : 
             equivocator_descriptors IM',
            proper_equivocator_descriptors IM'
              final_descriptors
              (finite_trace_last is tr)
            → ∀ (initial_descriptors : 
                 equivocator_descriptors IM') 
                (trX : 
                 list
                   (composite_transition_item IM')),
                equivocators_trace_project IM'
                  final_descriptors tr =
                Some (trX, initial_descriptors)
                → finite_valid_trace SeededX
                    (equivocators_state_project
                      IM' initial_descriptors is)
                    trX
Htr: finite_valid_trace_from SeededXE is tr'
Hlst: finite_valid_trace_from SeededXE
  (finite_trace_last is tr')
  [{|
     l := l;
     input := input;
     destination := s;
     output := output
   |}]
Hinit: composite_initial_state_prop
  (sub_IM (equivocator_IM IM) selection) is
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input := input;
      destination := s;
      output := output
    |}]): composite_state
  (sub_IM (equivocator_IM IM) selection)
final_descriptors: equivocator_descriptors IM'
Hproper: proper_equivocator_descriptors IM'
  final_descriptors s
initial_descriptors: equivocator_descriptors IM'
trX': list
  (composite_transition_item
     (λ i : sub_index selection, IM (`i)))
l0: composite_label IM'
final_descriptors': equivocator_descriptors IM'
Hpr_trX': equivocators_trace_project
  (λ i : sub_index selection, IM (`i))
  final_descriptors' tr' =
Some (trX', initial_descriptors)
HtrX': finite_valid_trace_from SeededX
  (equivocators_state_project IM'
     initial_descriptors is) trX'
HinitX: composite_initial_state_prop
  (sub_IM IM selection)
  (equivocators_state_project IM'
     initial_descriptors is)
Hproper': proper_equivocator_descriptors IM'
  final_descriptors'
  (finite_trace_last is tr')
Heq_final_descriptors': final_descriptors' =
equivocator_descriptors_update
  IM' final_descriptors
  (projT1 l)
  (final_descriptors'
     (projT1 l))
Hvx: let (i, li) := l0 in
valid li
  (equivocators_state_project IM'
     final_descriptors'
     (finite_trace_last is tr') i, input)
Htx: (let (i, li) := l0 in
 let (si', om') :=
   transition li
     (equivocators_state_project IM'
        final_descriptors'
        (finite_trace_last is tr') i, input) in
 (state_update IM'
    (equivocators_state_project IM'
       final_descriptors'
       (finite_trace_last is tr')) i si', om')) =
(equivocators_state_project IM'
   final_descriptors s, output)
Hl: ∃ Hex : existing_equivocator_label
          (IM' (projT1 l)) 
          (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM' (projT1 l)) 
       (projT2 l) Hex) = l0
Hprojectx: equivocators_transition_item_project IM'
  final_descriptors
  {|
    l := l;
    input := input;
    destination := s;
    output := output
  |} =
Some
  (Some
     {|
       l := l0;
       input := input;
       destination :=
         equivocators_state_project IM'
           final_descriptors s;
       output := output
     |}, final_descriptors')
Htl: finite_valid_trace_from SeededXE s []
Hs: valid_state_prop SeededXE
  (finite_trace_last is tr')
Hc: no_equivocations_additional_constraint_with_preloaded
  (sub_IM (equivocator_IM IM) selection)
  (free_constraint
     (sub_IM (equivocator_IM IM) selection)) seed
  l (finite_trace_last is tr', input)
Hiom: option_valid_message_prop SeededXE input
Hlst_trX': valid_state_prop SeededX
  (finite_trace_last
     (equivocators_state_project IM'
        initial_descriptors is) trX')
input_valid_transition SeededX l0
  (finite_trace_last
     (equivocators_state_project IM'
        initial_descriptors is) trX', input)
  (equivocators_state_project IM' final_descriptors s,
   output)
    assert (Htr_to : finite_valid_trace_init_to SeededXE is (finite_trace_last is tr') tr')
      by (split; [apply finite_valid_trace_from_add_last |]; done).message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
is: composite_state
  (sub_IM (equivocator_IM IM) selection)
IM':= λ i : sub_index selection, IM (`i): sub_index selection → VLSM message
tr': list
  (composite_transition_item
     (sub_IM (equivocator_IM IM) selection))
s: composite_state
  (sub_IM (equivocator_IM IM) selection)
l: composite_label
  (sub_IM (equivocator_IM IM) selection)
input, output: option message
H: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := input;
        destination := s;
        output := output
      |}])
  → ∀ tr : list
             (composite_transition_item
                (sub_IM 
                   (equivocator_IM IM) selection)),
      finite_valid_trace SeededXE is tr
      → y = length tr
        → ∀ final_descriptors : 
             equivocator_descriptors IM',
            proper_equivocator_descriptors IM'
              final_descriptors
              (finite_trace_last is tr)
            → ∀ (initial_descriptors : 
                 equivocator_descriptors IM') 
                (trX : 
                 list
                   (composite_transition_item IM')),
                equivocators_trace_project IM'
                  final_descriptors tr =
                Some (trX, initial_descriptors)
                → finite_valid_trace SeededX
                    (equivocators_state_project
                      IM' initial_descriptors is)
                    trX
Htr: finite_valid_trace_from SeededXE is tr'
Hlst: finite_valid_trace_from SeededXE
  (finite_trace_last is tr')
  [{|
     l := l;
     input := input;
     destination := s;
     output := output
   |}]
Hinit: composite_initial_state_prop
  (sub_IM (equivocator_IM IM) selection) is
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input := input;
      destination := s;
      output := output
    |}]): composite_state
  (sub_IM (equivocator_IM IM) selection)
final_descriptors: equivocator_descriptors IM'
Hproper: proper_equivocator_descriptors IM'
  final_descriptors s
initial_descriptors: equivocator_descriptors IM'
trX': list
  (composite_transition_item
     (λ i : sub_index selection, IM (`i)))
l0: composite_label IM'
final_descriptors': equivocator_descriptors IM'
Hpr_trX': equivocators_trace_project
  (λ i : sub_index selection, IM (`i))
  final_descriptors' tr' =
Some (trX', initial_descriptors)
HtrX': finite_valid_trace_from SeededX
  (equivocators_state_project IM'
     initial_descriptors is) trX'
HinitX: composite_initial_state_prop
  (sub_IM IM selection)
  (equivocators_state_project IM'
     initial_descriptors is)
Hproper': proper_equivocator_descriptors IM'
  final_descriptors'
  (finite_trace_last is tr')
Heq_final_descriptors': final_descriptors' =
equivocator_descriptors_update
  IM' final_descriptors
  (projT1 l)
  (final_descriptors'
     (projT1 l))
Hvx: let (i, li) := l0 in
valid li
  (equivocators_state_project IM'
     final_descriptors'
     (finite_trace_last is tr') i, input)
Htx: (let (i, li) := l0 in
 let (si', om') :=
   transition li
     (equivocators_state_project IM'
        final_descriptors'
        (finite_trace_last is tr') i, input) in
 (state_update IM'
    (equivocators_state_project IM'
       final_descriptors'
       (finite_trace_last is tr')) i si', om')) =
(equivocators_state_project IM'
   final_descriptors s, output)
Hl: ∃ Hex : existing_equivocator_label
          (IM' (projT1 l)) 
          (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM' (projT1 l)) 
       (projT2 l) Hex) = l0
Hprojectx: equivocators_transition_item_project IM'
  final_descriptors
  {|
    l := l;
    input := input;
    destination := s;
    output := output
  |} =
Some
  (Some
     {|
       l := l0;
       input := input;
       destination :=
         equivocators_state_project IM'
           final_descriptors s;
       output := output
     |}, final_descriptors')
Htl: finite_valid_trace_from SeededXE s []
Hs: valid_state_prop SeededXE
  (finite_trace_last is tr')
Hc: no_equivocations_additional_constraint_with_preloaded
  (sub_IM (equivocator_IM IM) selection)
  (free_constraint
     (sub_IM (equivocator_IM IM) selection)) seed
  l (finite_trace_last is tr', input)
Hiom: option_valid_message_prop SeededXE input
Hlst_trX': valid_state_prop SeededX
  (finite_trace_last
     (equivocators_state_project IM'
        initial_descriptors is) trX')
Htr_to: finite_valid_trace_init_to SeededXE is
  (finite_trace_last is tr') tr'
input_valid_transition SeededX l0
  (finite_trace_last
     (equivocators_state_project IM'
        initial_descriptors is) trX', input)
  (equivocators_state_project IM' final_descriptors s,
   output)
    assert (Hpre_tr_to : finite_constrained_trace_init_to SubFreeE is (finite_trace_last is tr') tr').message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
is: composite_state
  (sub_IM (equivocator_IM IM) selection)
IM':= λ i : sub_index selection, IM (`i): sub_index selection → VLSM message
tr': list
  (composite_transition_item
     (sub_IM (equivocator_IM IM) selection))
s: composite_state
  (sub_IM (equivocator_IM IM) selection)
l: composite_label
  (sub_IM (equivocator_IM IM) selection)
input, output: option message
H: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := input;
        destination := s;
        output := output
      |}])
  → ∀ tr : list
             (composite_transition_item
                (sub_IM 
                   (equivocator_IM IM) selection)),
      finite_valid_trace SeededXE is tr
      → y = length tr
        → ∀ final_descriptors : 
             equivocator_descriptors IM',
            proper_equivocator_descriptors IM'
              final_descriptors
              (finite_trace_last is tr)
            → ∀ (initial_descriptors : 
                 equivocator_descriptors IM') 
                (trX : 
                 list
                   (composite_transition_item IM')),
                equivocators_trace_project IM'
                  final_descriptors tr =
                Some (trX, initial_descriptors)
                → finite_valid_trace SeededX
                    (equivocators_state_project
                      IM' initial_descriptors is)
                    trX
Htr: finite_valid_trace_from SeededXE is tr'
Hlst: finite_valid_trace_from SeededXE
  (finite_trace_last is tr')
  [{|
     l := l;
     input := input;
     destination := s;
     output := output
   |}]
Hinit: composite_initial_state_prop
  (sub_IM (equivocator_IM IM) selection) is
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input := input;
      destination := s;
      output := output
    |}]): composite_state
  (sub_IM (equivocator_IM IM) selection)
final_descriptors: equivocator_descriptors IM'
Hproper: proper_equivocator_descriptors IM'
  final_descriptors s
initial_descriptors: equivocator_descriptors IM'
trX': list
  (composite_transition_item
     (λ i : sub_index selection, IM (`i)))
l0: composite_label IM'
final_descriptors': equivocator_descriptors IM'
Hpr_trX': equivocators_trace_project
  (λ i : sub_index selection, IM (`i))
  final_descriptors' tr' =
Some (trX', initial_descriptors)
HtrX': finite_valid_trace_from SeededX
  (equivocators_state_project IM'
     initial_descriptors is) trX'
HinitX: composite_initial_state_prop
  (sub_IM IM selection)
  (equivocators_state_project IM'
     initial_descriptors is)
Hproper': proper_equivocator_descriptors IM'
  final_descriptors'
  (finite_trace_last is tr')
Heq_final_descriptors': final_descriptors' =
equivocator_descriptors_update
  IM' final_descriptors
  (projT1 l)
  (final_descriptors'
     (projT1 l))
Hvx: let (i, li) := l0 in
valid li
  (equivocators_state_project IM'
     final_descriptors'
     (finite_trace_last is tr') i, input)
Htx: (let (i, li) := l0 in
 let (si', om') :=
   transition li
     (equivocators_state_project IM'
        final_descriptors'
        (finite_trace_last is tr') i, input) in
 (state_update IM'
    (equivocators_state_project IM'
       final_descriptors'
       (finite_trace_last is tr')) i si', om')) =
(equivocators_state_project IM'
   final_descriptors s, output)
Hl: ∃ Hex : existing_equivocator_label
          (IM' (projT1 l)) 
          (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM' (projT1 l)) 
       (projT2 l) Hex) = l0
Hprojectx: equivocators_transition_item_project IM'
  final_descriptors
  {|
    l := l;
    input := input;
    destination := s;
    output := output
  |} =
Some
  (Some
     {|
       l := l0;
       input := input;
       destination :=
         equivocators_state_project IM'
           final_descriptors s;
       output := output
     |}, final_descriptors')
Htl: finite_valid_trace_from SeededXE s []
Hs: valid_state_prop SeededXE
  (finite_trace_last is tr')
Hc: no_equivocations_additional_constraint_with_preloaded
  (sub_IM (equivocator_IM IM) selection)
  (free_constraint
     (sub_IM (equivocator_IM IM) selection)) seed
  l (finite_trace_last is tr', input)
Hiom: option_valid_message_prop SeededXE input
Hlst_trX': valid_state_prop SeededX
  (finite_trace_last
     (equivocators_state_project IM'
        initial_descriptors is) trX')
Htr_to: finite_valid_trace_init_to SeededXE is
  (finite_trace_last is tr') tr'
finite_constrained_trace_init_to SubFreeE is
  (finite_trace_last is tr') tr'
message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
is: composite_state
  (sub_IM (equivocator_IM IM) selection)
IM':= λ i : sub_index selection, IM (`i): sub_index selection → VLSM message
tr': list
  (composite_transition_item
     (sub_IM (equivocator_IM IM) selection))
s: composite_state
  (sub_IM (equivocator_IM IM) selection)
l: composite_label
  (sub_IM (equivocator_IM IM) selection)
input, output: option message
H: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := input;
        destination := s;
        output := output
      |}])
  → ∀ tr : list
             (composite_transition_item
                (sub_IM 
                   (equivocator_IM IM) selection)),
      finite_valid_trace SeededXE is tr
      → y = length tr
        → ∀ final_descriptors : 
             equivocator_descriptors IM',
            proper_equivocator_descriptors IM'
              final_descriptors
              (finite_trace_last is tr)
            → ∀ (initial_descriptors : 
                 equivocator_descriptors IM') 
                (trX : 
                 list
                   (composite_transition_item IM')),
                equivocators_trace_project IM'
                  final_descriptors tr =
                Some (trX, initial_descriptors)
                → finite_valid_trace SeededX
                    (equivocators_state_project
                      IM' initial_descriptors is)
                    trX
Htr: finite_valid_trace_from SeededXE is tr'
Hlst: finite_valid_trace_from SeededXE
  (finite_trace_last is tr')
  [{|
     l := l;
     input := input;
     destination := s;
     output := output
   |}]
Hinit: composite_initial_state_prop
  (sub_IM (equivocator_IM IM) selection) is
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input := input;
      destination := s;
      output := output
    |}]): composite_state
  (sub_IM (equivocator_IM IM) selection)
final_descriptors: equivocator_descriptors IM'
Hproper: proper_equivocator_descriptors IM'
  final_descriptors s
initial_descriptors: equivocator_descriptors IM'
trX': list
  (composite_transition_item
     (λ i : sub_index selection, IM (`i)))
l0: composite_label IM'
final_descriptors': equivocator_descriptors IM'
Hpr_trX': equivocators_trace_project
  (λ i : sub_index selection, IM (`i))
  final_descriptors' tr' =
Some (trX', initial_descriptors)
HtrX': finite_valid_trace_from SeededX
  (equivocators_state_project IM'
     initial_descriptors is) trX'
HinitX: composite_initial_state_prop
  (sub_IM IM selection)
  (equivocators_state_project IM'
     initial_descriptors is)
Hproper': proper_equivocator_descriptors IM'
  final_descriptors'
  (finite_trace_last is tr')
Heq_final_descriptors': final_descriptors' =
equivocator_descriptors_update
  IM' final_descriptors
  (projT1 l)
  (final_descriptors'
     (projT1 l))
Hvx: let (i, li) := l0 in
valid li
  (equivocators_state_project IM'
     final_descriptors'
     (finite_trace_last is tr') i, input)
Htx: (let (i, li) := l0 in
 let (si', om') :=
   transition li
     (equivocators_state_project IM'
        final_descriptors'
        (finite_trace_last is tr') i, input) in
 (state_update IM'
    (equivocators_state_project IM'
       final_descriptors'
       (finite_trace_last is tr')) i si', om')) =
(equivocators_state_project IM'
   final_descriptors s, output)
Hl: ∃ Hex : existing_equivocator_label
          (IM' (projT1 l)) 
          (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM' (projT1 l)) 
       (projT2 l) Hex) = l0
Hprojectx: equivocators_transition_item_project IM'
  final_descriptors
  {|
    l := l;
    input := input;
    destination := s;
    output := output
  |} =
Some
  (Some
     {|
       l := l0;
       input := input;
       destination :=
         equivocators_state_project IM'
           final_descriptors s;
       output := output
     |}, final_descriptors')
Htl: finite_valid_trace_from SeededXE s []
Hs: valid_state_prop SeededXE
  (finite_trace_last is tr')
Hc: no_equivocations_additional_constraint_with_preloaded
  (sub_IM (equivocator_IM IM) selection)
  (free_constraint
     (sub_IM (equivocator_IM IM) selection)) seed
  l (finite_trace_last is tr', input)
Hiom: option_valid_message_prop SeededXE input
Hlst_trX': valid_state_prop SeededX
  (finite_trace_last
     (equivocators_state_project IM'
        initial_descriptors is) trX')
Htr_to: finite_valid_trace_init_to SeededXE is
  (finite_trace_last is tr') tr'
Hpre_tr_to: finite_constrained_trace_init_to SubFreeE
  is (finite_trace_last is tr') tr'
input_valid_transition SeededX l0
  (finite_trace_last
     (equivocators_state_project IM'
        initial_descriptors is) trX', input)
  (equivocators_state_project IM' final_descriptors s,
   output)
    {message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
is: composite_state
  (sub_IM (equivocator_IM IM) selection)
IM':= λ i : sub_index selection, IM (`i): sub_index selection → VLSM message
tr': list
  (composite_transition_item
     (sub_IM (equivocator_IM IM) selection))
s: composite_state
  (sub_IM (equivocator_IM IM) selection)
l: composite_label
  (sub_IM (equivocator_IM IM) selection)
input, output: option message
H: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := input;
        destination := s;
        output := output
      |}])
  → ∀ tr : list
             (composite_transition_item
                (sub_IM 
                   (equivocator_IM IM) selection)),
      finite_valid_trace SeededXE is tr
      → y = length tr
        → ∀ final_descriptors : 
             equivocator_descriptors IM',
            proper_equivocator_descriptors IM'
              final_descriptors
              (finite_trace_last is tr)
            → ∀ (initial_descriptors : 
                 equivocator_descriptors IM') 
                (trX : 
                 list
                   (composite_transition_item IM')),
                equivocators_trace_project IM'
                  final_descriptors tr =
                Some (trX, initial_descriptors)
                → finite_valid_trace SeededX
                    (equivocators_state_project
                      IM' initial_descriptors is)
                    trX
Htr: finite_valid_trace_from SeededXE is tr'
Hlst: finite_valid_trace_from SeededXE
  (finite_trace_last is tr')
  [{|
     l := l;
     input := input;
     destination := s;
     output := output
   |}]
Hinit: composite_initial_state_prop
  (sub_IM (equivocator_IM IM) selection) is
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input := input;
      destination := s;
      output := output
    |}]): composite_state
  (sub_IM (equivocator_IM IM) selection)
final_descriptors: equivocator_descriptors IM'
Hproper: proper_equivocator_descriptors IM'
  final_descriptors s
initial_descriptors: equivocator_descriptors IM'
trX': list
  (composite_transition_item
     (λ i : sub_index selection, IM (`i)))
l0: composite_label IM'
final_descriptors': equivocator_descriptors IM'
Hpr_trX': equivocators_trace_project
  (λ i : sub_index selection, IM (`i))
  final_descriptors' tr' =
Some (trX', initial_descriptors)
HtrX': finite_valid_trace_from SeededX
  (equivocators_state_project IM'
     initial_descriptors is) trX'
HinitX: composite_initial_state_prop
  (sub_IM IM selection)
  (equivocators_state_project IM'
     initial_descriptors is)
Hproper': proper_equivocator_descriptors IM'
  final_descriptors'
  (finite_trace_last is tr')
Heq_final_descriptors': final_descriptors' =
equivocator_descriptors_update
  IM' final_descriptors
  (projT1 l)
  (final_descriptors'
     (projT1 l))
Hvx: let (i, li) := l0 in
valid li
  (equivocators_state_project IM'
     final_descriptors'
     (finite_trace_last is tr') i, input)
Htx: (let (i, li) := l0 in
 let (si', om') :=
   transition li
     (equivocators_state_project IM'
        final_descriptors'
        (finite_trace_last is tr') i, input) in
 (state_update IM'
    (equivocators_state_project IM'
       final_descriptors'
       (finite_trace_last is tr')) i si', om')) =
(equivocators_state_project IM'
   final_descriptors s, output)
Hl: ∃ Hex : existing_equivocator_label
          (IM' (projT1 l)) 
          (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM' (projT1 l)) 
       (projT2 l) Hex) = l0
Hprojectx: equivocators_transition_item_project IM'
  final_descriptors
  {|
    l := l;
    input := input;
    destination := s;
    output := output
  |} =
Some
  (Some
     {|
       l := l0;
       input := input;
       destination :=
         equivocators_state_project IM'
           final_descriptors s;
       output := output
     |}, final_descriptors')
Htl: finite_valid_trace_from SeededXE s []
Hs: valid_state_prop SeededXE
  (finite_trace_last is tr')
Hc: no_equivocations_additional_constraint_with_preloaded
  (sub_IM (equivocator_IM IM) selection)
  (free_constraint
     (sub_IM (equivocator_IM IM) selection)) seed
  l (finite_trace_last is tr', input)
Hiom: option_valid_message_prop SeededXE input
Hlst_trX': valid_state_prop SeededX
  (finite_trace_last
     (equivocators_state_project IM'
        initial_descriptors is) trX')
Htr_to: finite_valid_trace_init_to SeededXE is
  (finite_trace_last is tr') tr'
finite_constrained_trace_init_to SubFreeE is
  (finite_trace_last is tr') tr'
      revert Htr_to; apply VLSM_incl_finite_valid_trace_init_to.message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
is: composite_state
  (sub_IM (equivocator_IM IM) selection)
IM':= λ i : sub_index selection, IM (`i): sub_index selection → VLSM message
tr': list
  (composite_transition_item
     (sub_IM (equivocator_IM IM) selection))
s: composite_state
  (sub_IM (equivocator_IM IM) selection)
l: composite_label
  (sub_IM (equivocator_IM IM) selection)
input, output: option message
H: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := input;
        destination := s;
        output := output
      |}])
  → ∀ tr : list
             (composite_transition_item
                (sub_IM 
                   (equivocator_IM IM) selection)),
      finite_valid_trace SeededXE is tr
      → y = length tr
        → ∀ final_descriptors : 
             equivocator_descriptors IM',
            proper_equivocator_descriptors IM'
              final_descriptors
              (finite_trace_last is tr)
            → ∀ (initial_descriptors : 
                 equivocator_descriptors IM') 
                (trX : 
                 list
                   (composite_transition_item IM')),
                equivocators_trace_project IM'
                  final_descriptors tr =
                Some (trX, initial_descriptors)
                → finite_valid_trace SeededX
                    (equivocators_state_project
                      IM' initial_descriptors is)
                    trX
Htr: finite_valid_trace_from SeededXE is tr'
Hlst: finite_valid_trace_from SeededXE
  (finite_trace_last is tr')
  [{|
     l := l;
     input := input;
     destination := s;
     output := output
   |}]
Hinit: composite_initial_state_prop
  (sub_IM (equivocator_IM IM) selection) is
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input := input;
      destination := s;
      output := output
    |}]): composite_state
  (sub_IM (equivocator_IM IM) selection)
final_descriptors: equivocator_descriptors IM'
Hproper: proper_equivocator_descriptors IM'
  final_descriptors s
initial_descriptors: equivocator_descriptors IM'
trX': list
  (composite_transition_item
     (λ i : sub_index selection, IM (`i)))
l0: composite_label IM'
final_descriptors': equivocator_descriptors IM'
Hpr_trX': equivocators_trace_project
  (λ i : sub_index selection, IM (`i))
  final_descriptors' tr' =
Some (trX', initial_descriptors)
HtrX': finite_valid_trace_from SeededX
  (equivocators_state_project IM'
     initial_descriptors is) trX'
HinitX: composite_initial_state_prop
  (sub_IM IM selection)
  (equivocators_state_project IM'
     initial_descriptors is)
Hproper': proper_equivocator_descriptors IM'
  final_descriptors'
  (finite_trace_last is tr')
Heq_final_descriptors': final_descriptors' =
equivocator_descriptors_update
  IM' final_descriptors
  (projT1 l)
  (final_descriptors'
     (projT1 l))
Hvx: let (i, li) := l0 in
valid li
  (equivocators_state_project IM'
     final_descriptors'
     (finite_trace_last is tr') i, input)
Htx: (let (i, li) := l0 in
 let (si', om') :=
   transition li
     (equivocators_state_project IM'
        final_descriptors'
        (finite_trace_last is tr') i, input) in
 (state_update IM'
    (equivocators_state_project IM'
       final_descriptors'
       (finite_trace_last is tr')) i si', om')) =
(equivocators_state_project IM'
   final_descriptors s, output)
Hl: ∃ Hex : existing_equivocator_label
          (IM' (projT1 l)) 
          (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM' (projT1 l)) 
       (projT2 l) Hex) = l0
Hprojectx: equivocators_transition_item_project IM'
  final_descriptors
  {|
    l := l;
    input := input;
    destination := s;
    output := output
  |} =
Some
  (Some
     {|
       l := l0;
       input := input;
       destination :=
         equivocators_state_project IM'
           final_descriptors s;
       output := output
     |}, final_descriptors')
Htl: finite_valid_trace_from SeededXE s []
Hs: valid_state_prop SeededXE
  (finite_trace_last is tr')
Hc: no_equivocations_additional_constraint_with_preloaded
  (sub_IM (equivocator_IM IM) selection)
  (free_constraint
     (sub_IM (equivocator_IM IM) selection)) seed
  l (finite_trace_last is tr', input)
Hiom: option_valid_message_prop SeededXE input
Hlst_trX': valid_state_prop SeededX
  (finite_trace_last
     (equivocators_state_project IM'
        initial_descriptors is) trX')
VLSM_incl_part
  (preloaded_vlsm_machine
     (composite_vlsm
        (sub_IM (equivocator_IM IM) selection)
        (no_equivocations_additional_constraint_with_preloaded
           (sub_IM (equivocator_IM IM) selection)
           (free_constraint
              (sub_IM (equivocator_IM IM) selection))
           seed)) seed)
  (preloaded_vlsm_machine SubFreeE
     (λ _ : message, True))
      by apply seeded_no_equivocation_incl_preloaded.
    }message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
is: composite_state
  (sub_IM (equivocator_IM IM) selection)
IM':= λ i : sub_index selection, IM (`i): sub_index selection → VLSM message
tr': list
  (composite_transition_item
     (sub_IM (equivocator_IM IM) selection))
s: composite_state
  (sub_IM (equivocator_IM IM) selection)
l: composite_label
  (sub_IM (equivocator_IM IM) selection)
input, output: option message
H: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := input;
        destination := s;
        output := output
      |}])
  → ∀ tr : list
             (composite_transition_item
                (sub_IM 
                   (equivocator_IM IM) selection)),
      finite_valid_trace SeededXE is tr
      → y = length tr
        → ∀ final_descriptors : 
             equivocator_descriptors IM',
            proper_equivocator_descriptors IM'
              final_descriptors
              (finite_trace_last is tr)
            → ∀ (initial_descriptors : 
                 equivocator_descriptors IM') 
                (trX : 
                 list
                   (composite_transition_item IM')),
                equivocators_trace_project IM'
                  final_descriptors tr =
                Some (trX, initial_descriptors)
                → finite_valid_trace SeededX
                    (equivocators_state_project
                      IM' initial_descriptors is)
                    trX
Htr: finite_valid_trace_from SeededXE is tr'
Hlst: finite_valid_trace_from SeededXE
  (finite_trace_last is tr')
  [{|
     l := l;
     input := input;
     destination := s;
     output := output
   |}]
Hinit: composite_initial_state_prop
  (sub_IM (equivocator_IM IM) selection) is
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input := input;
      destination := s;
      output := output
    |}]): composite_state
  (sub_IM (equivocator_IM IM) selection)
final_descriptors: equivocator_descriptors IM'
Hproper: proper_equivocator_descriptors IM'
  final_descriptors s
initial_descriptors: equivocator_descriptors IM'
trX': list
  (composite_transition_item
     (λ i : sub_index selection, IM (`i)))
l0: composite_label IM'
final_descriptors': equivocator_descriptors IM'
Hpr_trX': equivocators_trace_project
  (λ i : sub_index selection, IM (`i))
  final_descriptors' tr' =
Some (trX', initial_descriptors)
HtrX': finite_valid_trace_from SeededX
  (equivocators_state_project IM'
     initial_descriptors is) trX'
HinitX: composite_initial_state_prop
  (sub_IM IM selection)
  (equivocators_state_project IM'
     initial_descriptors is)
Hproper': proper_equivocator_descriptors IM'
  final_descriptors'
  (finite_trace_last is tr')
Heq_final_descriptors': final_descriptors' =
equivocator_descriptors_update
  IM' final_descriptors
  (projT1 l)
  (final_descriptors'
     (projT1 l))
Hvx: let (i, li) := l0 in
valid li
  (equivocators_state_project IM'
     final_descriptors'
     (finite_trace_last is tr') i, input)
Htx: (let (i, li) := l0 in
 let (si', om') :=
   transition li
     (equivocators_state_project IM'
        final_descriptors'
        (finite_trace_last is tr') i, input) in
 (state_update IM'
    (equivocators_state_project IM'
       final_descriptors'
       (finite_trace_last is tr')) i si', om')) =
(equivocators_state_project IM'
   final_descriptors s, output)
Hl: ∃ Hex : existing_equivocator_label
          (IM' (projT1 l)) 
          (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM' (projT1 l)) 
       (projT2 l) Hex) = l0
Hprojectx: equivocators_transition_item_project IM'
  final_descriptors
  {|
    l := l;
    input := input;
    destination := s;
    output := output
  |} =
Some
  (Some
     {|
       l := l0;
       input := input;
       destination :=
         equivocators_state_project IM'
           final_descriptors s;
       output := output
     |}, final_descriptors')
Htl: finite_valid_trace_from SeededXE s []
Hs: valid_state_prop SeededXE
  (finite_trace_last is tr')
Hc: no_equivocations_additional_constraint_with_preloaded
  (sub_IM (equivocator_IM IM) selection)
  (free_constraint
     (sub_IM (equivocator_IM IM) selection)) seed
  l (finite_trace_last is tr', input)
Hiom: option_valid_message_prop SeededXE input
Hlst_trX': valid_state_prop SeededX
  (finite_trace_last
     (equivocators_state_project IM'
        initial_descriptors is) trX')
Htr_to: finite_valid_trace_init_to SeededXE is
  (finite_trace_last is tr') tr'
Hpre_tr_to: finite_constrained_trace_init_to SubFreeE
  is (finite_trace_last is tr') tr'
input_valid_transition SeededX l0
  (finite_trace_last
     (equivocators_state_project IM'
        initial_descriptors is) trX', input)
  (equivocators_state_project IM' final_descriptors s,
   output)
    pose proof (pre_equivocators_valid_trace_project IM' _ _ _
     Hpre_tr_to final_descriptors' Hproper') as Hpr_tr'.message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
is: composite_state
  (sub_IM (equivocator_IM IM) selection)
IM':= λ i : sub_index selection, IM (`i): sub_index selection → VLSM message
tr': list
  (composite_transition_item
     (sub_IM (equivocator_IM IM) selection))
s: composite_state
  (sub_IM (equivocator_IM IM) selection)
l: composite_label
  (sub_IM (equivocator_IM IM) selection)
input, output: option message
H: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := input;
        destination := s;
        output := output
      |}])
  → ∀ tr : list
             (composite_transition_item
                (sub_IM 
                   (equivocator_IM IM) selection)),
      finite_valid_trace SeededXE is tr
      → y = length tr
        → ∀ final_descriptors : 
             equivocator_descriptors IM',
            proper_equivocator_descriptors IM'
              final_descriptors
              (finite_trace_last is tr)
            → ∀ (initial_descriptors : 
                 equivocator_descriptors IM') 
                (trX : 
                 list
                   (composite_transition_item IM')),
                equivocators_trace_project IM'
                  final_descriptors tr =
                Some (trX, initial_descriptors)
                → finite_valid_trace SeededX
                    (equivocators_state_project
                      IM' initial_descriptors is)
                    trX
Htr: finite_valid_trace_from SeededXE is tr'
Hlst: finite_valid_trace_from SeededXE
  (finite_trace_last is tr')
  [{|
     l := l;
     input := input;
     destination := s;
     output := output
   |}]
Hinit: composite_initial_state_prop
  (sub_IM (equivocator_IM IM) selection) is
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input := input;
      destination := s;
      output := output
    |}]): composite_state
  (sub_IM (equivocator_IM IM) selection)
final_descriptors: equivocator_descriptors IM'
Hproper: proper_equivocator_descriptors IM'
  final_descriptors s
initial_descriptors: equivocator_descriptors IM'
trX': list
  (composite_transition_item
     (λ i : sub_index selection, IM (`i)))
l0: composite_label IM'
final_descriptors': equivocator_descriptors IM'
Hpr_trX': equivocators_trace_project
  (λ i : sub_index selection, IM (`i))
  final_descriptors' tr' =
Some (trX', initial_descriptors)
HtrX': finite_valid_trace_from SeededX
  (equivocators_state_project IM'
     initial_descriptors is) trX'
HinitX: composite_initial_state_prop
  (sub_IM IM selection)
  (equivocators_state_project IM'
     initial_descriptors is)
Hproper': proper_equivocator_descriptors IM'
  final_descriptors'
  (finite_trace_last is tr')
Heq_final_descriptors': final_descriptors' =
equivocator_descriptors_update
  IM' final_descriptors
  (projT1 l)
  (final_descriptors'
     (projT1 l))
Hvx: let (i, li) := l0 in
valid li
  (equivocators_state_project IM'
     final_descriptors'
     (finite_trace_last is tr') i, input)
Htx: (let (i, li) := l0 in
 let (si', om') :=
   transition li
     (equivocators_state_project IM'
        final_descriptors'
        (finite_trace_last is tr') i, input) in
 (state_update IM'
    (equivocators_state_project IM'
       final_descriptors'
       (finite_trace_last is tr')) i si', om')) =
(equivocators_state_project IM'
   final_descriptors s, output)
Hl: ∃ Hex : existing_equivocator_label
          (IM' (projT1 l)) 
          (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM' (projT1 l)) 
       (projT2 l) Hex) = l0
Hprojectx: equivocators_transition_item_project IM'
  final_descriptors
  {|
    l := l;
    input := input;
    destination := s;
    output := output
  |} =
Some
  (Some
     {|
       l := l0;
       input := input;
       destination :=
         equivocators_state_project IM'
           final_descriptors s;
       output := output
     |}, final_descriptors')
Htl: finite_valid_trace_from SeededXE s []
Hs: valid_state_prop SeededXE
  (finite_trace_last is tr')
Hc: no_equivocations_additional_constraint_with_preloaded
  (sub_IM (equivocator_IM IM) selection)
  (free_constraint
     (sub_IM (equivocator_IM IM) selection)) seed
  l (finite_trace_last is tr', input)
Hiom: option_valid_message_prop SeededXE input
Hlst_trX': valid_state_prop SeededX
  (finite_trace_last
     (equivocators_state_project IM'
        initial_descriptors is) trX')
Htr_to: finite_valid_trace_init_to SeededXE is
  (finite_trace_last is tr') tr'
Hpre_tr_to: finite_constrained_trace_init_to SubFreeE
  is (finite_trace_last is tr') tr'
Hpr_tr': ∃ initial_descriptors : 
   equivocator_descriptors IM',
  proper_equivocator_descriptors IM'
    initial_descriptors is
  ∧ (let isX :=
       equivocators_state_project IM'
         initial_descriptors is in
     let final_stateX :=
       equivocators_state_project IM'
         final_descriptors'
         (finite_trace_last is tr') in
     ∃ trX : list
               (composite_transition_item
                  IM'),
       equivocators_trace_project IM'
         final_descriptors' tr' =
       Some (trX, initial_descriptors)
       ∧ finite_constrained_trace_init_to
           (free_composite_vlsm IM') isX
           final_stateX trX)
input_valid_transition SeededX l0
  (finite_trace_last
     (equivocators_state_project IM'
        initial_descriptors is) trX', input)
  (equivocators_state_project IM' final_descriptors s,
   output)
    destruct Hpr_tr' as [_initial_descriptors [_ [_trX' [_Hpr_trX' Heq_final_stateX']]]].message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
is: composite_state
  (sub_IM (equivocator_IM IM) selection)
IM':= λ i : sub_index selection, IM (`i): sub_index selection → VLSM message
tr': list
  (composite_transition_item
     (sub_IM (equivocator_IM IM) selection))
s: composite_state
  (sub_IM (equivocator_IM IM) selection)
l: composite_label
  (sub_IM (equivocator_IM IM) selection)
input, output: option message
H: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := input;
        destination := s;
        output := output
      |}])
  → ∀ tr : list
             (composite_transition_item
                (sub_IM 
                   (equivocator_IM IM) selection)),
      finite_valid_trace SeededXE is tr
      → y = length tr
        → ∀ final_descriptors : 
             equivocator_descriptors IM',
            proper_equivocator_descriptors IM'
              final_descriptors
              (finite_trace_last is tr)
            → ∀ (initial_descriptors : 
                 equivocator_descriptors IM') 
                (trX : 
                 list
                   (composite_transition_item IM')),
                equivocators_trace_project IM'
                  final_descriptors tr =
                Some (trX, initial_descriptors)
                → finite_valid_trace SeededX
                    (equivocators_state_project
                      IM' initial_descriptors is)
                    trX
Htr: finite_valid_trace_from SeededXE is tr'
Hlst: finite_valid_trace_from SeededXE
  (finite_trace_last is tr')
  [{|
     l := l;
     input := input;
     destination := s;
     output := output
   |}]
Hinit: composite_initial_state_prop
  (sub_IM (equivocator_IM IM) selection) is
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input := input;
      destination := s;
      output := output
    |}]): composite_state
  (sub_IM (equivocator_IM IM) selection)
final_descriptors: equivocator_descriptors IM'
Hproper: proper_equivocator_descriptors IM'
  final_descriptors s
initial_descriptors: equivocator_descriptors IM'
trX': list
  (composite_transition_item
     (λ i : sub_index selection, IM (`i)))
l0: composite_label IM'
final_descriptors': equivocator_descriptors IM'
Hpr_trX': equivocators_trace_project
  (λ i : sub_index selection, IM (`i))
  final_descriptors' tr' =
Some (trX', initial_descriptors)
HtrX': finite_valid_trace_from SeededX
  (equivocators_state_project IM'
     initial_descriptors is) trX'
HinitX: composite_initial_state_prop
  (sub_IM IM selection)
  (equivocators_state_project IM'
     initial_descriptors is)
Hproper': proper_equivocator_descriptors IM'
  final_descriptors'
  (finite_trace_last is tr')
Heq_final_descriptors': final_descriptors' =
equivocator_descriptors_update
  IM' final_descriptors
  (projT1 l)
  (final_descriptors'
     (projT1 l))
Hvx: let (i, li) := l0 in
valid li
  (equivocators_state_project IM'
     final_descriptors'
     (finite_trace_last is tr') i, input)
Htx: (let (i, li) := l0 in
 let (si', om') :=
   transition li
     (equivocators_state_project IM'
        final_descriptors'
        (finite_trace_last is tr') i, input) in
 (state_update IM'
    (equivocators_state_project IM'
       final_descriptors'
       (finite_trace_last is tr')) i si', om')) =
(equivocators_state_project IM'
   final_descriptors s, output)
Hl: ∃ Hex : existing_equivocator_label
          (IM' (projT1 l)) 
          (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM' (projT1 l)) 
       (projT2 l) Hex) = l0
Hprojectx: equivocators_transition_item_project IM'
  final_descriptors
  {|
    l := l;
    input := input;
    destination := s;
    output := output
  |} =
Some
  (Some
     {|
       l := l0;
       input := input;
       destination :=
         equivocators_state_project IM'
           final_descriptors s;
       output := output
     |}, final_descriptors')
Htl: finite_valid_trace_from SeededXE s []
Hs: valid_state_prop SeededXE
  (finite_trace_last is tr')
Hc: no_equivocations_additional_constraint_with_preloaded
  (sub_IM (equivocator_IM IM) selection)
  (free_constraint
     (sub_IM (equivocator_IM IM) selection)) seed
  l (finite_trace_last is tr', input)
Hiom: option_valid_message_prop SeededXE input
Hlst_trX': valid_state_prop SeededX
  (finite_trace_last
     (equivocators_state_project IM'
        initial_descriptors is) trX')
Htr_to: finite_valid_trace_init_to SeededXE is
  (finite_trace_last is tr') tr'
Hpre_tr_to: finite_constrained_trace_init_to SubFreeE
  is (finite_trace_last is tr') tr'
_initial_descriptors: equivocator_descriptors IM'
_trX': list (composite_transition_item IM')
_Hpr_trX': equivocators_trace_project IM'
  final_descriptors' tr' =
Some (_trX', _initial_descriptors)
Heq_final_stateX': finite_constrained_trace_init_to
  (free_composite_vlsm IM')
  (equivocators_state_project IM'
     _initial_descriptors is)
  (equivocators_state_project IM'
     final_descriptors'
     (finite_trace_last is tr'))
  _trX'
input_valid_transition SeededX l0
  (finite_trace_last
     (equivocators_state_project IM'
        initial_descriptors is) trX', input)
  (equivocators_state_project IM' final_descriptors s,
   output)
    replace (equivocators_trace_project _ _ _) with (Some (trX', initial_descriptors))
      in _Hpr_trX'.message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
is: composite_state
  (sub_IM (equivocator_IM IM) selection)
IM':= λ i : sub_index selection, IM (`i): sub_index selection → VLSM message
tr': list
  (composite_transition_item
     (sub_IM (equivocator_IM IM) selection))
s: composite_state
  (sub_IM (equivocator_IM IM) selection)
l: composite_label
  (sub_IM (equivocator_IM IM) selection)
input, output: option message
H: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := input;
        destination := s;
        output := output
      |}])
  → ∀ tr : list
             (composite_transition_item
                (sub_IM 
                   (equivocator_IM IM) selection)),
      finite_valid_trace SeededXE is tr
      → y = length tr
        → ∀ final_descriptors : 
             equivocator_descriptors IM',
            proper_equivocator_descriptors IM'
              final_descriptors
              (finite_trace_last is tr)
            → ∀ (initial_descriptors : 
                 equivocator_descriptors IM') 
                (trX : 
                 list
                   (composite_transition_item IM')),
                equivocators_trace_project IM'
                  final_descriptors tr =
                Some (trX, initial_descriptors)
                → finite_valid_trace SeededX
                    (equivocators_state_project
                      IM' initial_descriptors is)
                    trX
Htr: finite_valid_trace_from SeededXE is tr'
Hlst: finite_valid_trace_from SeededXE
  (finite_trace_last is tr')
  [{|
     l := l;
     input := input;
     destination := s;
     output := output
   |}]
Hinit: composite_initial_state_prop
  (sub_IM (equivocator_IM IM) selection) is
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input := input;
      destination := s;
      output := output
    |}]): composite_state
  (sub_IM (equivocator_IM IM) selection)
final_descriptors: equivocator_descriptors IM'
Hproper: proper_equivocator_descriptors IM'
  final_descriptors s
initial_descriptors: equivocator_descriptors IM'
trX': list
  (composite_transition_item
     (λ i : sub_index selection, IM (`i)))
l0: composite_label IM'
final_descriptors': equivocator_descriptors IM'
Hpr_trX': equivocators_trace_project
  (λ i : sub_index selection, IM (`i))
  final_descriptors' tr' =
Some (trX', initial_descriptors)
HtrX': finite_valid_trace_from SeededX
  (equivocators_state_project IM'
     initial_descriptors is) trX'
HinitX: composite_initial_state_prop
  (sub_IM IM selection)
  (equivocators_state_project IM'
     initial_descriptors is)
Hproper': proper_equivocator_descriptors IM'
  final_descriptors'
  (finite_trace_last is tr')
Heq_final_descriptors': final_descriptors' =
equivocator_descriptors_update
  IM' final_descriptors
  (projT1 l)
  (final_descriptors'
     (projT1 l))
Hvx: let (i, li) := l0 in
valid li
  (equivocators_state_project IM'
     final_descriptors'
     (finite_trace_last is tr') i, input)
Htx: (let (i, li) := l0 in
 let (si', om') :=
   transition li
     (equivocators_state_project IM'
        final_descriptors'
        (finite_trace_last is tr') i, input) in
 (state_update IM'
    (equivocators_state_project IM'
       final_descriptors'
       (finite_trace_last is tr')) i si', om')) =
(equivocators_state_project IM'
   final_descriptors s, output)
Hl: ∃ Hex : existing_equivocator_label
          (IM' (projT1 l)) 
          (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM' (projT1 l)) 
       (projT2 l) Hex) = l0
Hprojectx: equivocators_transition_item_project IM'
  final_descriptors
  {|
    l := l;
    input := input;
    destination := s;
    output := output
  |} =
Some
  (Some
     {|
       l := l0;
       input := input;
       destination :=
         equivocators_state_project IM'
           final_descriptors s;
       output := output
     |}, final_descriptors')
Htl: finite_valid_trace_from SeededXE s []
Hs: valid_state_prop SeededXE
  (finite_trace_last is tr')
Hc: no_equivocations_additional_constraint_with_preloaded
  (sub_IM (equivocator_IM IM) selection)
  (free_constraint
     (sub_IM (equivocator_IM IM) selection)) seed
  l (finite_trace_last is tr', input)
Hiom: option_valid_message_prop SeededXE input
Hlst_trX': valid_state_prop SeededX
  (finite_trace_last
     (equivocators_state_project IM'
        initial_descriptors is) trX')
Htr_to: finite_valid_trace_init_to SeededXE is
  (finite_trace_last is tr') tr'
Hpre_tr_to: finite_constrained_trace_init_to SubFreeE
  is (finite_trace_last is tr') tr'
_initial_descriptors: equivocator_descriptors IM'
_trX': list (composite_transition_item IM')
_Hpr_trX': Some (trX', initial_descriptors) =
Some (_trX', _initial_descriptors)
Heq_final_stateX': finite_constrained_trace_init_to
  (free_composite_vlsm IM')
  (equivocators_state_project IM'
     _initial_descriptors is)
  (equivocators_state_project IM'
     final_descriptors'
     (finite_trace_last is tr'))
  _trX'
input_valid_transition SeededX l0
  (finite_trace_last
     (equivocators_state_project IM'
        initial_descriptors is) trX', input)
  (equivocators_state_project IM' final_descriptors s,
   output)
    inversion _Hpr_trX'.message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
is: composite_state
  (sub_IM (equivocator_IM IM) selection)
IM':= λ i : sub_index selection, IM (`i): sub_index selection → VLSM message
tr': list
  (composite_transition_item
     (sub_IM (equivocator_IM IM) selection))
s: composite_state
  (sub_IM (equivocator_IM IM) selection)
l: composite_label
  (sub_IM (equivocator_IM IM) selection)
input, output: option message
H: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := input;
        destination := s;
        output := output
      |}])
  → ∀ tr : list
             (composite_transition_item
                (sub_IM 
                   (equivocator_IM IM) selection)),
      finite_valid_trace SeededXE is tr
      → y = length tr
        → ∀ final_descriptors : 
             equivocator_descriptors IM',
            proper_equivocator_descriptors IM'
              final_descriptors
              (finite_trace_last is tr)
            → ∀ (initial_descriptors : 
                 equivocator_descriptors IM') 
                (trX : 
                 list
                   (composite_transition_item IM')),
                equivocators_trace_project IM'
                  final_descriptors tr =
                Some (trX, initial_descriptors)
                → finite_valid_trace SeededX
                    (equivocators_state_project
                      IM' initial_descriptors is)
                    trX
Htr: finite_valid_trace_from SeededXE is tr'
Hlst: finite_valid_trace_from SeededXE
  (finite_trace_last is tr')
  [{|
     l := l;
     input := input;
     destination := s;
     output := output
   |}]
Hinit: composite_initial_state_prop
  (sub_IM (equivocator_IM IM) selection) is
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input := input;
      destination := s;
      output := output
    |}]): composite_state
  (sub_IM (equivocator_IM IM) selection)
final_descriptors: equivocator_descriptors IM'
Hproper: proper_equivocator_descriptors IM'
  final_descriptors s
initial_descriptors: equivocator_descriptors IM'
trX': list
  (composite_transition_item
     (λ i : sub_index selection, IM (`i)))
l0: composite_label IM'
final_descriptors': equivocator_descriptors IM'
Hpr_trX': equivocators_trace_project
  (λ i : sub_index selection, IM (`i))
  final_descriptors' tr' =
Some (trX', initial_descriptors)
HtrX': finite_valid_trace_from SeededX
  (equivocators_state_project IM'
     initial_descriptors is) trX'
HinitX: composite_initial_state_prop
  (sub_IM IM selection)
  (equivocators_state_project IM'
     initial_descriptors is)
Hproper': proper_equivocator_descriptors IM'
  final_descriptors'
  (finite_trace_last is tr')
Heq_final_descriptors': final_descriptors' =
equivocator_descriptors_update
  IM' final_descriptors
  (projT1 l)
  (final_descriptors'
     (projT1 l))
Hvx: let (i, li) := l0 in
valid li
  (equivocators_state_project IM'
     final_descriptors'
     (finite_trace_last is tr') i, input)
Htx: (let (i, li) := l0 in
 let (si', om') :=
   transition li
     (equivocators_state_project IM'
        final_descriptors'
        (finite_trace_last is tr') i, input) in
 (state_update IM'
    (equivocators_state_project IM'
       final_descriptors'
       (finite_trace_last is tr')) i si', om')) =
(equivocators_state_project IM'
   final_descriptors s, output)
Hl: ∃ Hex : existing_equivocator_label
          (IM' (projT1 l)) 
          (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM' (projT1 l)) 
       (projT2 l) Hex) = l0
Hprojectx: equivocators_transition_item_project IM'
  final_descriptors
  {|
    l := l;
    input := input;
    destination := s;
    output := output
  |} =
Some
  (Some
     {|
       l := l0;
       input := input;
       destination :=
         equivocators_state_project IM'
           final_descriptors s;
       output := output
     |}, final_descriptors')
Htl: finite_valid_trace_from SeededXE s []
Hs: valid_state_prop SeededXE
  (finite_trace_last is tr')
Hc: no_equivocations_additional_constraint_with_preloaded
  (sub_IM (equivocator_IM IM) selection)
  (free_constraint
     (sub_IM (equivocator_IM IM) selection)) seed
  l (finite_trace_last is tr', input)
Hiom: option_valid_message_prop SeededXE input
Hlst_trX': valid_state_prop SeededX
  (finite_trace_last
     (equivocators_state_project IM'
        initial_descriptors is) trX')
Htr_to: finite_valid_trace_init_to SeededXE is
  (finite_trace_last is tr') tr'
Hpre_tr_to: finite_constrained_trace_init_to SubFreeE
  is (finite_trace_last is tr') tr'
_initial_descriptors: equivocator_descriptors IM'
_trX': list (composite_transition_item IM')
_Hpr_trX': Some (trX', initial_descriptors) =
Some (_trX', _initial_descriptors)
Heq_final_stateX': finite_constrained_trace_init_to
  (free_composite_vlsm IM')
  (equivocators_state_project IM'
     _initial_descriptors is)
  (equivocators_state_project IM'
     final_descriptors'
     (finite_trace_last is tr'))
  _trX'
H2: trX' = _trX'
H3: initial_descriptors = _initial_descriptors
input_valid_transition SeededX l0
  (finite_trace_last
     (equivocators_state_project IM'
        _initial_descriptors is) _trX', input)
  (equivocators_state_project IM' final_descriptors s,
   output) subst _initial_descriptors _trX'.message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
is: composite_state
  (sub_IM (equivocator_IM IM) selection)
IM':= λ i : sub_index selection, IM (`i): sub_index selection → VLSM message
tr': list
  (composite_transition_item
     (sub_IM (equivocator_IM IM) selection))
s: composite_state
  (sub_IM (equivocator_IM IM) selection)
l: composite_label
  (sub_IM (equivocator_IM IM) selection)
input, output: option message
H: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := input;
        destination := s;
        output := output
      |}])
  → ∀ tr : list
             (composite_transition_item
                (sub_IM 
                   (equivocator_IM IM) selection)),
      finite_valid_trace SeededXE is tr
      → y = length tr
        → ∀ final_descriptors : 
             equivocator_descriptors IM',
            proper_equivocator_descriptors IM'
              final_descriptors
              (finite_trace_last is tr)
            → ∀ (initial_descriptors : 
                 equivocator_descriptors IM') 
                (trX : 
                 list
                   (composite_transition_item IM')),
                equivocators_trace_project IM'
                  final_descriptors tr =
                Some (trX, initial_descriptors)
                → finite_valid_trace SeededX
                    (equivocators_state_project
                      IM' initial_descriptors is)
                    trX
Htr: finite_valid_trace_from SeededXE is tr'
Hlst: finite_valid_trace_from SeededXE
  (finite_trace_last is tr')
  [{|
     l := l;
     input := input;
     destination := s;
     output := output
   |}]
Hinit: composite_initial_state_prop
  (sub_IM (equivocator_IM IM) selection) is
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input := input;
      destination := s;
      output := output
    |}]): composite_state
  (sub_IM (equivocator_IM IM) selection)
final_descriptors: equivocator_descriptors IM'
Hproper: proper_equivocator_descriptors IM'
  final_descriptors s
initial_descriptors: equivocator_descriptors IM'
trX': list
  (composite_transition_item
     (λ i : sub_index selection, IM (`i)))
l0: composite_label IM'
final_descriptors': equivocator_descriptors IM'
Hpr_trX': equivocators_trace_project
  (λ i : sub_index selection, IM (`i))
  final_descriptors' tr' =
Some (trX', initial_descriptors)
HtrX': finite_valid_trace_from SeededX
  (equivocators_state_project IM'
     initial_descriptors is) trX'
HinitX: composite_initial_state_prop
  (sub_IM IM selection)
  (equivocators_state_project IM'
     initial_descriptors is)
Hproper': proper_equivocator_descriptors IM'
  final_descriptors'
  (finite_trace_last is tr')
Heq_final_descriptors': final_descriptors' =
equivocator_descriptors_update
  IM' final_descriptors
  (projT1 l)
  (final_descriptors'
     (projT1 l))
Hvx: let (i, li) := l0 in
valid li
  (equivocators_state_project IM'
     final_descriptors'
     (finite_trace_last is tr') i, input)
Htx: (let (i, li) := l0 in
 let (si', om') :=
   transition li
     (equivocators_state_project IM'
        final_descriptors'
        (finite_trace_last is tr') i, input) in
 (state_update IM'
    (equivocators_state_project IM'
       final_descriptors'
       (finite_trace_last is tr')) i si', om')) =
(equivocators_state_project IM'
   final_descriptors s, output)
Hl: ∃ Hex : existing_equivocator_label
          (IM' (projT1 l)) 
          (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM' (projT1 l)) 
       (projT2 l) Hex) = l0
Hprojectx: equivocators_transition_item_project IM'
  final_descriptors
  {|
    l := l;
    input := input;
    destination := s;
    output := output
  |} =
Some
  (Some
     {|
       l := l0;
       input := input;
       destination :=
         equivocators_state_project IM'
           final_descriptors s;
       output := output
     |}, final_descriptors')
Htl: finite_valid_trace_from SeededXE s []
Hs: valid_state_prop SeededXE
  (finite_trace_last is tr')
Hc: no_equivocations_additional_constraint_with_preloaded
  (sub_IM (equivocator_IM IM) selection)
  (free_constraint
     (sub_IM (equivocator_IM IM) selection)) seed
  l (finite_trace_last is tr', input)
Hiom: option_valid_message_prop SeededXE input
Hlst_trX': valid_state_prop SeededX
  (finite_trace_last
     (equivocators_state_project IM'
        initial_descriptors is) trX')
Htr_to: finite_valid_trace_init_to SeededXE is
  (finite_trace_last is tr') tr'
Hpre_tr_to: finite_constrained_trace_init_to SubFreeE
  is (finite_trace_last is tr') tr'
_Hpr_trX': Some (trX', initial_descriptors) =
Some (trX', initial_descriptors)
Heq_final_stateX': finite_constrained_trace_init_to
  (free_composite_vlsm IM')
  (equivocators_state_project IM'
     initial_descriptors is)
  (equivocators_state_project IM'
     final_descriptors'
     (finite_trace_last is tr'))
  trX'
input_valid_transition SeededX l0
  (finite_trace_last
     (equivocators_state_project IM'
        initial_descriptors is) trX', input)
  (equivocators_state_project IM' final_descriptors s,
   output)
    apply finite_valid_trace_init_to_last in Heq_final_stateX'.message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
is: composite_state
  (sub_IM (equivocator_IM IM) selection)
IM':= λ i : sub_index selection, IM (`i): sub_index selection → VLSM message
tr': list
  (composite_transition_item
     (sub_IM (equivocator_IM IM) selection))
s: composite_state
  (sub_IM (equivocator_IM IM) selection)
l: composite_label
  (sub_IM (equivocator_IM IM) selection)
input, output: option message
H: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := input;
        destination := s;
        output := output
      |}])
  → ∀ tr : list
             (composite_transition_item
                (sub_IM 
                   (equivocator_IM IM) selection)),
      finite_valid_trace SeededXE is tr
      → y = length tr
        → ∀ final_descriptors : 
             equivocator_descriptors IM',
            proper_equivocator_descriptors IM'
              final_descriptors
              (finite_trace_last is tr)
            → ∀ (initial_descriptors : 
                 equivocator_descriptors IM') 
                (trX : 
                 list
                   (composite_transition_item IM')),
                equivocators_trace_project IM'
                  final_descriptors tr =
                Some (trX, initial_descriptors)
                → finite_valid_trace SeededX
                    (equivocators_state_project
                      IM' initial_descriptors is)
                    trX
Htr: finite_valid_trace_from SeededXE is tr'
Hlst: finite_valid_trace_from SeededXE
  (finite_trace_last is tr')
  [{|
     l := l;
     input := input;
     destination := s;
     output := output
   |}]
Hinit: composite_initial_state_prop
  (sub_IM (equivocator_IM IM) selection) is
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input := input;
      destination := s;
      output := output
    |}]): composite_state
  (sub_IM (equivocator_IM IM) selection)
final_descriptors: equivocator_descriptors IM'
Hproper: proper_equivocator_descriptors IM'
  final_descriptors s
initial_descriptors: equivocator_descriptors IM'
trX': list
  (composite_transition_item
     (λ i : sub_index selection, IM (`i)))
l0: composite_label IM'
final_descriptors': equivocator_descriptors IM'
Hpr_trX': equivocators_trace_project
  (λ i : sub_index selection, IM (`i))
  final_descriptors' tr' =
Some (trX', initial_descriptors)
HtrX': finite_valid_trace_from SeededX
  (equivocators_state_project IM'
     initial_descriptors is) trX'
HinitX: composite_initial_state_prop
  (sub_IM IM selection)
  (equivocators_state_project IM'
     initial_descriptors is)
Hproper': proper_equivocator_descriptors IM'
  final_descriptors'
  (finite_trace_last is tr')
Heq_final_descriptors': final_descriptors' =
equivocator_descriptors_update
  IM' final_descriptors
  (projT1 l)
  (final_descriptors'
     (projT1 l))
Hvx: let (i, li) := l0 in
valid li
  (equivocators_state_project IM'
     final_descriptors'
     (finite_trace_last is tr') i, input)
Htx: (let (i, li) := l0 in
 let (si', om') :=
   transition li
     (equivocators_state_project IM'
        final_descriptors'
        (finite_trace_last is tr') i, input) in
 (state_update IM'
    (equivocators_state_project IM'
       final_descriptors'
       (finite_trace_last is tr')) i si', om')) =
(equivocators_state_project IM'
   final_descriptors s, output)
Hl: ∃ Hex : existing_equivocator_label
          (IM' (projT1 l)) 
          (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM' (projT1 l)) 
       (projT2 l) Hex) = l0
Hprojectx: equivocators_transition_item_project IM'
  final_descriptors
  {|
    l := l;
    input := input;
    destination := s;
    output := output
  |} =
Some
  (Some
     {|
       l := l0;
       input := input;
       destination :=
         equivocators_state_project IM'
           final_descriptors s;
       output := output
     |}, final_descriptors')
Htl: finite_valid_trace_from SeededXE s []
Hs: valid_state_prop SeededXE
  (finite_trace_last is tr')
Hc: no_equivocations_additional_constraint_with_preloaded
  (sub_IM (equivocator_IM IM) selection)
  (free_constraint
     (sub_IM (equivocator_IM IM) selection)) seed
  l (finite_trace_last is tr', input)
Hiom: option_valid_message_prop SeededXE input
Hlst_trX': valid_state_prop SeededX
  (finite_trace_last
     (equivocators_state_project IM'
        initial_descriptors is) trX')
Htr_to: finite_valid_trace_init_to SeededXE is
  (finite_trace_last is tr') tr'
Hpre_tr_to: finite_constrained_trace_init_to SubFreeE
  is (finite_trace_last is tr') tr'
_Hpr_trX': Some (trX', initial_descriptors) =
Some (trX', initial_descriptors)
Heq_final_stateX': finite_trace_last
  (equivocators_state_project IM'
     initial_descriptors is) trX' =
equivocators_state_project IM'
  final_descriptors'
  (finite_trace_last is tr')
input_valid_transition SeededX l0
  (finite_trace_last
     (equivocators_state_project IM'
        initial_descriptors is) trX', input)
  (equivocators_state_project IM' final_descriptors s,
   output)
    simpl in *.message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
is: composite_state
  (sub_IM (equivocator_IM IM) selection)
IM':= λ i : sub_index selection, IM (`i): sub_index selection → VLSM message
tr': list
  (composite_transition_item
     (sub_IM (equivocator_IM IM) selection))
s: composite_state
  (sub_IM (equivocator_IM IM) selection)
l: composite_label
  (sub_IM (equivocator_IM IM) selection)
input, output: option message
H: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := input;
        destination := s;
        output := output
      |}])
  → ∀ tr : list
             (composite_transition_item
                (sub_IM 
                   (equivocator_IM IM) selection)),
      finite_valid_trace SeededXE is tr
      → y = length tr
        → ∀ final_descriptors : 
             equivocator_descriptors IM',
            proper_equivocator_descriptors IM'
              final_descriptors
              (finite_trace_last is tr)
            → ∀ (initial_descriptors : 
                 equivocator_descriptors IM') 
                (trX : 
                 list
                   (composite_transition_item IM')),
                equivocators_trace_project IM'
                  final_descriptors tr =
                Some (trX, initial_descriptors)
                → finite_valid_trace SeededX
                    (equivocators_state_project
                      IM' initial_descriptors is)
                    trX
Htr: finite_valid_trace_from SeededXE is tr'
Hlst: finite_valid_trace_from SeededXE
  (finite_trace_last is tr')
  [{|
     l := l;
     input := input;
     destination := s;
     output := output
   |}]
Hinit: composite_initial_state_prop
  (sub_IM (equivocator_IM IM) selection) is
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input := input;
      destination := s;
      output := output
    |}]): composite_state
  (sub_IM (equivocator_IM IM) selection)
final_descriptors: equivocator_descriptors IM'
Hproper: proper_equivocator_descriptors IM'
  final_descriptors s
initial_descriptors: equivocator_descriptors IM'
trX': list
  (composite_transition_item
     (λ i : sub_index selection, IM (`i)))
l0: composite_label IM'
final_descriptors': equivocator_descriptors IM'
Hpr_trX': equivocators_trace_project
  (λ i : sub_index selection, IM (`i))
  final_descriptors' tr' =
Some (trX', initial_descriptors)
HtrX': finite_valid_trace_from SeededX
  (equivocators_state_project IM'
     initial_descriptors is) trX'
HinitX: composite_initial_state_prop
  (sub_IM IM selection)
  (equivocators_state_project IM'
     initial_descriptors is)
Hproper': proper_equivocator_descriptors IM'
  final_descriptors'
  (finite_trace_last is tr')
Heq_final_descriptors': final_descriptors' =
equivocator_descriptors_update
  IM' final_descriptors
  (projT1 l)
  (final_descriptors'
     (projT1 l))
Hvx: let (i, li) := l0 in
valid li
  (equivocators_state_project IM'
     final_descriptors'
     (finite_trace_last is tr') i, input)
Htx: (let (i, li) := l0 in
 let (si', om') :=
   transition li
     (equivocators_state_project IM'
        final_descriptors'
        (finite_trace_last is tr') i, input) in
 (state_update IM'
    (equivocators_state_project IM'
       final_descriptors'
       (finite_trace_last is tr')) i si', om')) =
(equivocators_state_project IM'
   final_descriptors s, output)
Hl: ∃ Hex : existing_equivocator_label
          (IM' (projT1 l)) 
          (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM' (projT1 l)) 
       (projT2 l) Hex) = l0
Hprojectx: equivocators_transition_item_project IM'
  final_descriptors
  {|
    l := l;
    input := input;
    destination := s;
    output := output
  |} =
Some
  (Some
     {|
       l := l0;
       input := input;
       destination :=
         equivocators_state_project IM'
           final_descriptors s;
       output := output
     |}, final_descriptors')
Htl: finite_valid_trace_from SeededXE s []
Hs: valid_state_prop SeededXE
  (finite_trace_last is tr')
Hc: no_equivocations_additional_constraint_with_preloaded
  (sub_IM (equivocator_IM IM) selection)
  (free_constraint
     (sub_IM (equivocator_IM IM) selection)) seed
  l (finite_trace_last is tr', input)
Hiom: option_valid_message_prop SeededXE input
Hlst_trX': valid_state_prop SeededX
  (finite_trace_last
     (equivocators_state_project IM'
        initial_descriptors is) trX')
Htr_to: finite_valid_trace_init_to SeededXE is
  (finite_trace_last is tr') tr'
Hpre_tr_to: finite_constrained_trace_init_to SubFreeE
  is (finite_trace_last is tr') tr'
_Hpr_trX': Some (trX', initial_descriptors) =
Some (trX', initial_descriptors)
Heq_final_stateX': finite_trace_last
  (equivocators_state_project IM'
     initial_descriptors is) trX' =
equivocators_state_project IM'
  final_descriptors'
  (finite_trace_last is tr')
input_valid_transition SeededX l0
  (finite_trace_last
     (equivocators_state_project IM'
        initial_descriptors is) trX', input)
  (equivocators_state_project IM' final_descriptors s,
   output)
    rewrite <- Heq_final_stateX' in Htx, Hvx.message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
is: composite_state
  (sub_IM (equivocator_IM IM) selection)
IM':= λ i : sub_index selection, IM (`i): sub_index selection → VLSM message
tr': list
  (composite_transition_item
     (sub_IM (equivocator_IM IM) selection))
s: composite_state
  (sub_IM (equivocator_IM IM) selection)
l: composite_label
  (sub_IM (equivocator_IM IM) selection)
input, output: option message
H: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := input;
        destination := s;
        output := output
      |}])
  → ∀ tr : list
             (composite_transition_item
                (sub_IM 
                   (equivocator_IM IM) selection)),
      finite_valid_trace SeededXE is tr
      → y = length tr
        → ∀ final_descriptors : 
             equivocator_descriptors IM',
            proper_equivocator_descriptors IM'
              final_descriptors
              (finite_trace_last is tr)
            → ∀ (initial_descriptors : 
                 equivocator_descriptors IM') 
                (trX : 
                 list
                   (composite_transition_item IM')),
                equivocators_trace_project IM'
                  final_descriptors tr =
                Some (trX, initial_descriptors)
                → finite_valid_trace SeededX
                    (equivocators_state_project
                      IM' initial_descriptors is)
                    trX
Htr: finite_valid_trace_from SeededXE is tr'
Hlst: finite_valid_trace_from SeededXE
  (finite_trace_last is tr')
  [{|
     l := l;
     input := input;
     destination := s;
     output := output
   |}]
Hinit: composite_initial_state_prop
  (sub_IM (equivocator_IM IM) selection) is
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input := input;
      destination := s;
      output := output
    |}]): composite_state
  (sub_IM (equivocator_IM IM) selection)
final_descriptors: equivocator_descriptors IM'
Hproper: proper_equivocator_descriptors IM'
  final_descriptors s
initial_descriptors: equivocator_descriptors IM'
trX': list
  (composite_transition_item
     (λ i : sub_index selection, IM (`i)))
l0: composite_label IM'
final_descriptors': equivocator_descriptors IM'
Hpr_trX': equivocators_trace_project
  (λ i : sub_index selection, IM (`i))
  final_descriptors' tr' =
Some (trX', initial_descriptors)
HtrX': finite_valid_trace_from SeededX
  (equivocators_state_project IM'
     initial_descriptors is) trX'
HinitX: composite_initial_state_prop
  (sub_IM IM selection)
  (equivocators_state_project IM'
     initial_descriptors is)
Hproper': proper_equivocator_descriptors IM'
  final_descriptors'
  (finite_trace_last is tr')
Heq_final_descriptors': final_descriptors' =
equivocator_descriptors_update
  IM' final_descriptors
  (projT1 l)
  (final_descriptors'
     (projT1 l))
Hvx: let (i, li) := l0 in
valid li
  (finite_trace_last
     (equivocators_state_project IM'
        initial_descriptors is) trX' i, input)
Htx: (let (i, li) := l0 in
 let (si', om') :=
   transition li
     (finite_trace_last
        (equivocators_state_project IM'
           initial_descriptors is) trX' i, input) in
 (state_update IM'
    (finite_trace_last
       (equivocators_state_project IM'
          initial_descriptors is) trX') i si',
  om')) =
(equivocators_state_project IM'
   final_descriptors s, output)
Hl: ∃ Hex : existing_equivocator_label
          (IM' (projT1 l)) 
          (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM' (projT1 l)) 
       (projT2 l) Hex) = l0
Hprojectx: equivocators_transition_item_project IM'
  final_descriptors
  {|
    l := l;
    input := input;
    destination := s;
    output := output
  |} =
Some
  (Some
     {|
       l := l0;
       input := input;
       destination :=
         equivocators_state_project IM'
           final_descriptors s;
       output := output
     |}, final_descriptors')
Htl: finite_valid_trace_from SeededXE s []
Hs: valid_state_prop SeededXE
  (finite_trace_last is tr')
Hc: no_equivocations_additional_constraint_with_preloaded
  (sub_IM (equivocator_IM IM) selection)
  (free_constraint
     (sub_IM (equivocator_IM IM) selection)) seed
  l (finite_trace_last is tr', input)
Hiom: option_valid_message_prop SeededXE input
Hlst_trX': valid_state_prop SeededX
  (finite_trace_last
     (equivocators_state_project IM'
        initial_descriptors is) trX')
Htr_to: finite_valid_trace_init_to SeededXE is
  (finite_trace_last is tr') tr'
Hpre_tr_to: finite_constrained_trace_init_to SubFreeE
  is (finite_trace_last is tr') tr'
_Hpr_trX': Some (trX', initial_descriptors) =
Some (trX', initial_descriptors)
Heq_final_stateX': finite_trace_last
  (equivocators_state_project IM'
     initial_descriptors is) trX' =
equivocators_state_project IM'
  final_descriptors'
  (finite_trace_last is tr')
input_valid_transition SeededX l0
  (finite_trace_last
     (equivocators_state_project IM'
        initial_descriptors is) trX', input)
  (equivocators_state_project IM' final_descriptors s,
   output)
    repeat split; [done | | done | done].message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
is: composite_state
  (sub_IM (equivocator_IM IM) selection)
IM':= λ i : sub_index selection, IM (`i): sub_index selection → VLSM message
tr': list
  (composite_transition_item
     (sub_IM (equivocator_IM IM) selection))
s: composite_state
  (sub_IM (equivocator_IM IM) selection)
l: composite_label
  (sub_IM (equivocator_IM IM) selection)
input, output: option message
H: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := input;
        destination := s;
        output := output
      |}])
  → ∀ tr : list
             (composite_transition_item
                (sub_IM 
                   (equivocator_IM IM) selection)),
      finite_valid_trace SeededXE is tr
      → y = length tr
        → ∀ final_descriptors : 
             equivocator_descriptors IM',
            proper_equivocator_descriptors IM'
              final_descriptors
              (finite_trace_last is tr)
            → ∀ (initial_descriptors : 
                 equivocator_descriptors IM') 
                (trX : 
                 list
                   (composite_transition_item IM')),
                equivocators_trace_project IM'
                  final_descriptors tr =
                Some (trX, initial_descriptors)
                → finite_valid_trace SeededX
                    (equivocators_state_project
                      IM' initial_descriptors is)
                    trX
Htr: finite_valid_trace_from SeededXE is tr'
Hlst: finite_valid_trace_from SeededXE
  (finite_trace_last is tr')
  [{|
     l := l;
     input := input;
     destination := s;
     output := output
   |}]
Hinit: composite_initial_state_prop
  (sub_IM (equivocator_IM IM) selection) is
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input := input;
      destination := s;
      output := output
    |}]): composite_state
  (sub_IM (equivocator_IM IM) selection)
final_descriptors: equivocator_descriptors IM'
Hproper: proper_equivocator_descriptors IM'
  final_descriptors s
initial_descriptors: equivocator_descriptors IM'
trX': list
  (composite_transition_item
     (λ i : sub_index selection, IM (`i)))
l0: composite_label IM'
final_descriptors': equivocator_descriptors IM'
Hpr_trX': equivocators_trace_project
  (λ i : sub_index selection, IM (`i))
  final_descriptors' tr' =
Some (trX', initial_descriptors)
HtrX': finite_valid_trace_from SeededX
  (equivocators_state_project IM'
     initial_descriptors is) trX'
HinitX: composite_initial_state_prop
  (sub_IM IM selection)
  (equivocators_state_project IM'
     initial_descriptors is)
Hproper': proper_equivocator_descriptors IM'
  final_descriptors'
  (finite_trace_last is tr')
Heq_final_descriptors': final_descriptors' =
equivocator_descriptors_update
  IM' final_descriptors
  (projT1 l)
  (final_descriptors'
     (projT1 l))
Hvx: let (i, li) := l0 in
valid li
  (finite_trace_last
     (equivocators_state_project IM'
        initial_descriptors is) trX' i, input)
Htx: (let (i, li) := l0 in
 let (si', om') :=
   transition li
     (finite_trace_last
        (equivocators_state_project IM'
           initial_descriptors is) trX' i, input) in
 (state_update IM'
    (finite_trace_last
       (equivocators_state_project IM'
          initial_descriptors is) trX') i si',
  om')) =
(equivocators_state_project IM'
   final_descriptors s, output)
Hl: ∃ Hex : existing_equivocator_label
          (IM' (projT1 l)) 
          (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM' (projT1 l)) 
       (projT2 l) Hex) = l0
Hprojectx: equivocators_transition_item_project IM'
  final_descriptors
  {|
    l := l;
    input := input;
    destination := s;
    output := output
  |} =
Some
  (Some
     {|
       l := l0;
       input := input;
       destination :=
         equivocators_state_project IM'
           final_descriptors s;
       output := output
     |}, final_descriptors')
Htl: finite_valid_trace_from SeededXE s []
Hs: valid_state_prop SeededXE
  (finite_trace_last is tr')
Hc: no_equivocations_additional_constraint_with_preloaded
  (sub_IM (equivocator_IM IM) selection)
  (free_constraint
     (sub_IM (equivocator_IM IM) selection)) seed
  l (finite_trace_last is tr', input)
Hiom: option_valid_message_prop SeededXE input
Hlst_trX': valid_state_prop SeededX
  (finite_trace_last
     (equivocators_state_project IM'
        initial_descriptors is) trX')
Htr_to: finite_valid_trace_init_to SeededXE is
  (finite_trace_last is tr') tr'
Hpre_tr_to: finite_constrained_trace_init_to SubFreeE
  is (finite_trace_last is tr') tr'
_Hpr_trX': Some (trX', initial_descriptors) =
Some (trX', initial_descriptors)
Heq_final_stateX': finite_trace_last
  (equivocators_state_project IM'
     initial_descriptors is) trX' =
equivocators_state_project IM'
  final_descriptors'
  (finite_trace_last is tr')
option_valid_message_prop SeededX input

    destruct input as [input |]; [| by apply option_valid_message_None].message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
is: composite_state
  (sub_IM (equivocator_IM IM) selection)
IM':= λ i : sub_index selection, IM (`i): sub_index selection → VLSM message
tr': list
  (composite_transition_item
     (sub_IM (equivocator_IM IM) selection))
s: composite_state
  (sub_IM (equivocator_IM IM) selection)
l: composite_label
  (sub_IM (equivocator_IM IM) selection)
input: message
output: option message
H: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := Some input;
        destination := s;
        output := output
      |}])
  → ∀ tr : list
             (composite_transition_item
                (sub_IM 
                   (equivocator_IM IM) selection)),
      finite_valid_trace SeededXE is tr
      → y = length tr
        → ∀ final_descriptors : 
             equivocator_descriptors IM',
            proper_equivocator_descriptors IM'
              final_descriptors
              (finite_trace_last is tr)
            → ∀ (initial_descriptors : 
                 equivocator_descriptors IM') 
                (trX : 
                 list
                   (composite_transition_item IM')),
                equivocators_trace_project IM'
                  final_descriptors tr =
                Some (trX, initial_descriptors)
                → finite_valid_trace SeededX
                    (equivocators_state_project
                      IM' initial_descriptors is)
                    trX
Htr: finite_valid_trace_from SeededXE is tr'
Hlst: finite_valid_trace_from SeededXE
  (finite_trace_last is tr')
  [{|
     l := l;
     input := Some input;
     destination := s;
     output := output
   |}]
Hinit: composite_initial_state_prop
  (sub_IM (equivocator_IM IM) selection) is
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input := Some input;
      destination := s;
      output := output
    |}]): composite_state
  (sub_IM (equivocator_IM IM) selection)
final_descriptors: equivocator_descriptors IM'
Hproper: proper_equivocator_descriptors IM'
  final_descriptors s
initial_descriptors: equivocator_descriptors IM'
trX': list
  (composite_transition_item
     (λ i : sub_index selection, IM (`i)))
l0: composite_label IM'
final_descriptors': equivocator_descriptors IM'
Hpr_trX': equivocators_trace_project
  (λ i : sub_index selection, IM (`i))
  final_descriptors' tr' =
Some (trX', initial_descriptors)
HtrX': finite_valid_trace_from SeededX
  (equivocators_state_project IM'
     initial_descriptors is) trX'
HinitX: composite_initial_state_prop
  (sub_IM IM selection)
  (equivocators_state_project IM'
     initial_descriptors is)
Hproper': proper_equivocator_descriptors IM'
  final_descriptors'
  (finite_trace_last is tr')
Heq_final_descriptors': final_descriptors' =
equivocator_descriptors_update
  IM' final_descriptors
  (projT1 l)
  (final_descriptors'
     (projT1 l))
Hvx: let (i, li) := l0 in
valid li
  (finite_trace_last
     (equivocators_state_project IM'
        initial_descriptors is) trX' i,
   Some input)
Htx: (let (i, li) := l0 in
 let (si', om') :=
   transition li
     (finite_trace_last
        (equivocators_state_project IM'
           initial_descriptors is) trX' i,
      Some input) in
 (state_update IM'
    (finite_trace_last
       (equivocators_state_project IM'
          initial_descriptors is) trX') i si',
  om')) =
(equivocators_state_project IM'
   final_descriptors s, output)
Hl: ∃ Hex : existing_equivocator_label
          (IM' (projT1 l)) 
          (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM' (projT1 l)) 
       (projT2 l) Hex) = l0
Hprojectx: equivocators_transition_item_project IM'
  final_descriptors
  {|
    l := l;
    input := Some input;
    destination := s;
    output := output
  |} =
Some
  (Some
     {|
       l := l0;
       input := Some input;
       destination :=
         equivocators_state_project IM'
           final_descriptors s;
       output := output
     |}, final_descriptors')
Htl: finite_valid_trace_from SeededXE s []
Hs: valid_state_prop SeededXE
  (finite_trace_last is tr')
Hc: no_equivocations_additional_constraint_with_preloaded
  (sub_IM (equivocator_IM IM) selection)
  (free_constraint
     (sub_IM (equivocator_IM IM) selection)) seed
  l (finite_trace_last is tr', Some input)
Hiom: option_valid_message_prop SeededXE (Some input)
Hlst_trX': valid_state_prop SeededX
  (finite_trace_last
     (equivocators_state_project IM'
        initial_descriptors is) trX')
Htr_to: finite_valid_trace_init_to SeededXE is
  (finite_trace_last is tr') tr'
Hpre_tr_to: finite_constrained_trace_init_to SubFreeE
  is (finite_trace_last is tr') tr'
_Hpr_trX': Some (trX', initial_descriptors) =
Some (trX', initial_descriptors)
Heq_final_stateX': finite_trace_last
  (equivocators_state_project IM'
     initial_descriptors is) trX' =
equivocators_state_project IM'
  final_descriptors'
  (finite_trace_last is tr')
option_valid_message_prop SeededX (Some input)
    apply proj1 in Hc.message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
is: composite_state
  (sub_IM (equivocator_IM IM) selection)
IM':= λ i : sub_index selection, IM (`i): sub_index selection → VLSM message
tr': list
  (composite_transition_item
     (sub_IM (equivocator_IM IM) selection))
s: composite_state
  (sub_IM (equivocator_IM IM) selection)
l: composite_label
  (sub_IM (equivocator_IM IM) selection)
input: message
output: option message
H: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := Some input;
        destination := s;
        output := output
      |}])
  → ∀ tr : list
             (composite_transition_item
                (sub_IM 
                   (equivocator_IM IM) selection)),
      finite_valid_trace SeededXE is tr
      → y = length tr
        → ∀ final_descriptors : 
             equivocator_descriptors IM',
            proper_equivocator_descriptors IM'
              final_descriptors
              (finite_trace_last is tr)
            → ∀ (initial_descriptors : 
                 equivocator_descriptors IM') 
                (trX : 
                 list
                   (composite_transition_item IM')),
                equivocators_trace_project IM'
                  final_descriptors tr =
                Some (trX, initial_descriptors)
                → finite_valid_trace SeededX
                    (equivocators_state_project
                      IM' initial_descriptors is)
                    trX
Htr: finite_valid_trace_from SeededXE is tr'
Hlst: finite_valid_trace_from SeededXE
  (finite_trace_last is tr')
  [{|
     l := l;
     input := Some input;
     destination := s;
     output := output
   |}]
Hinit: composite_initial_state_prop
  (sub_IM (equivocator_IM IM) selection) is
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input := Some input;
      destination := s;
      output := output
    |}]): composite_state
  (sub_IM (equivocator_IM IM) selection)
final_descriptors: equivocator_descriptors IM'
Hproper: proper_equivocator_descriptors IM'
  final_descriptors s
initial_descriptors: equivocator_descriptors IM'
trX': list
  (composite_transition_item
     (λ i : sub_index selection, IM (`i)))
l0: composite_label IM'
final_descriptors': equivocator_descriptors IM'
Hpr_trX': equivocators_trace_project
  (λ i : sub_index selection, IM (`i))
  final_descriptors' tr' =
Some (trX', initial_descriptors)
HtrX': finite_valid_trace_from SeededX
  (equivocators_state_project IM'
     initial_descriptors is) trX'
HinitX: composite_initial_state_prop
  (sub_IM IM selection)
  (equivocators_state_project IM'
     initial_descriptors is)
Hproper': proper_equivocator_descriptors IM'
  final_descriptors'
  (finite_trace_last is tr')
Heq_final_descriptors': final_descriptors' =
equivocator_descriptors_update
  IM' final_descriptors
  (projT1 l)
  (final_descriptors'
     (projT1 l))
Hvx: let (i, li) := l0 in
valid li
  (finite_trace_last
     (equivocators_state_project IM'
        initial_descriptors is) trX' i,
   Some input)
Htx: (let (i, li) := l0 in
 let (si', om') :=
   transition li
     (finite_trace_last
        (equivocators_state_project IM'
           initial_descriptors is) trX' i,
      Some input) in
 (state_update IM'
    (finite_trace_last
       (equivocators_state_project IM'
          initial_descriptors is) trX') i si',
  om')) =
(equivocators_state_project IM'
   final_descriptors s, output)
Hl: ∃ Hex : existing_equivocator_label
          (IM' (projT1 l)) 
          (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM' (projT1 l)) 
       (projT2 l) Hex) = l0
Hprojectx: equivocators_transition_item_project IM'
  final_descriptors
  {|
    l := l;
    input := Some input;
    destination := s;
    output := output
  |} =
Some
  (Some
     {|
       l := l0;
       input := Some input;
       destination :=
         equivocators_state_project IM'
           final_descriptors s;
       output := output
     |}, final_descriptors')
Htl: finite_valid_trace_from SeededXE s []
Hs: valid_state_prop SeededXE
  (finite_trace_last is tr')
Hc: composite_no_equivocations_except_from
  (sub_IM (equivocator_IM IM) selection) seed l
  (finite_trace_last is tr', Some input)
Hiom: option_valid_message_prop SeededXE (Some input)
Hlst_trX': valid_state_prop SeededX
  (finite_trace_last
     (equivocators_state_project IM'
        initial_descriptors is) trX')
Htr_to: finite_valid_trace_init_to SeededXE is
  (finite_trace_last is tr') tr'
Hpre_tr_to: finite_constrained_trace_init_to SubFreeE
  is (finite_trace_last is tr') tr'
_Hpr_trX': Some (trX', initial_descriptors) =
Some (trX', initial_descriptors)
Heq_final_stateX': finite_trace_last
  (equivocators_state_project IM'
     initial_descriptors is) trX' =
equivocators_state_project IM'
  final_descriptors'
  (finite_trace_last is tr')
option_valid_message_prop SeededX (Some input) simpl in Hc.message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
is: composite_state
  (sub_IM (equivocator_IM IM) selection)
IM':= λ i : sub_index selection, IM (`i): sub_index selection → VLSM message
tr': list
  (composite_transition_item
     (sub_IM (equivocator_IM IM) selection))
s: composite_state
  (sub_IM (equivocator_IM IM) selection)
l: composite_label
  (sub_IM (equivocator_IM IM) selection)
input: message
output: option message
H: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := Some input;
        destination := s;
        output := output
      |}])
  → ∀ tr : list
             (composite_transition_item
                (sub_IM 
                   (equivocator_IM IM) selection)),
      finite_valid_trace SeededXE is tr
      → y = length tr
        → ∀ final_descriptors : 
             equivocator_descriptors IM',
            proper_equivocator_descriptors IM'
              final_descriptors
              (finite_trace_last is tr)
            → ∀ (initial_descriptors : 
                 equivocator_descriptors IM') 
                (trX : 
                 list
                   (composite_transition_item IM')),
                equivocators_trace_project IM'
                  final_descriptors tr =
                Some (trX, initial_descriptors)
                → finite_valid_trace SeededX
                    (equivocators_state_project
                      IM' initial_descriptors is)
                    trX
Htr: finite_valid_trace_from SeededXE is tr'
Hlst: finite_valid_trace_from SeededXE
  (finite_trace_last is tr')
  [{|
     l := l;
     input := Some input;
     destination := s;
     output := output
   |}]
Hinit: composite_initial_state_prop
  (sub_IM (equivocator_IM IM) selection) is
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input := Some input;
      destination := s;
      output := output
    |}]): composite_state
  (sub_IM (equivocator_IM IM) selection)
final_descriptors: equivocator_descriptors IM'
Hproper: proper_equivocator_descriptors IM'
  final_descriptors s
initial_descriptors: equivocator_descriptors IM'
trX': list
  (composite_transition_item
     (λ i : sub_index selection, IM (`i)))
l0: composite_label IM'
final_descriptors': equivocator_descriptors IM'
Hpr_trX': equivocators_trace_project
  (λ i : sub_index selection, IM (`i))
  final_descriptors' tr' =
Some (trX', initial_descriptors)
HtrX': finite_valid_trace_from SeededX
  (equivocators_state_project IM'
     initial_descriptors is) trX'
HinitX: composite_initial_state_prop
  (sub_IM IM selection)
  (equivocators_state_project IM'
     initial_descriptors is)
Hproper': proper_equivocator_descriptors IM'
  final_descriptors'
  (finite_trace_last is tr')
Heq_final_descriptors': final_descriptors' =
equivocator_descriptors_update
  IM' final_descriptors
  (projT1 l)
  (final_descriptors'
     (projT1 l))
Hvx: let (i, li) := l0 in
valid li
  (finite_trace_last
     (equivocators_state_project IM'
        initial_descriptors is) trX' i,
   Some input)
Htx: (let (i, li) := l0 in
 let (si', om') :=
   transition li
     (finite_trace_last
        (equivocators_state_project IM'
           initial_descriptors is) trX' i,
      Some input) in
 (state_update IM'
    (finite_trace_last
       (equivocators_state_project IM'
          initial_descriptors is) trX') i si',
  om')) =
(equivocators_state_project IM'
   final_descriptors s, output)
Hl: ∃ Hex : existing_equivocator_label
          (IM' (projT1 l)) 
          (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM' (projT1 l)) 
       (projT2 l) Hex) = l0
Hprojectx: equivocators_transition_item_project IM'
  final_descriptors
  {|
    l := l;
    input := Some input;
    destination := s;
    output := output
  |} =
Some
  (Some
     {|
       l := l0;
       input := Some input;
       destination :=
         equivocators_state_project IM'
           final_descriptors s;
       output := output
     |}, final_descriptors')
Htl: finite_valid_trace_from SeededXE s []
Hs: valid_state_prop SeededXE
  (finite_trace_last is tr')
Hc: composite_no_equivocations_except_from
  (sub_IM (equivocator_IM IM) selection) seed l
  (finite_trace_last is tr', Some input)
Hiom: option_valid_message_prop SeededXE (Some input)
Hlst_trX': valid_state_prop SeededX
  (finite_trace_last
     (equivocators_state_project IM'
        initial_descriptors is) trX')
Htr_to: finite_valid_trace_init_to SeededXE is
  (finite_trace_last is tr') tr'
Hpre_tr_to: finite_constrained_trace_init_to SubFreeE
  is (finite_trace_last is tr') tr'
_Hpr_trX': Some (trX', initial_descriptors) =
Some (trX', initial_descriptors)
Heq_final_stateX': finite_trace_last
  (equivocators_state_project IM'
     initial_descriptors is) trX' =
equivocators_state_project IM'
  final_descriptors'
  (finite_trace_last is tr')
option_valid_message_prop SeededX (Some input)
    apply or_comm in Hc.message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
is: composite_state
  (sub_IM (equivocator_IM IM) selection)
IM':= λ i : sub_index selection, IM (`i): sub_index selection → VLSM message
tr': list
  (composite_transition_item
     (sub_IM (equivocator_IM IM) selection))
s: composite_state
  (sub_IM (equivocator_IM IM) selection)
l: composite_label
  (sub_IM (equivocator_IM IM) selection)
input: message
output: option message
H: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := Some input;
        destination := s;
        output := output
      |}])
  → ∀ tr : list
             (composite_transition_item
                (sub_IM 
                   (equivocator_IM IM) selection)),
      finite_valid_trace SeededXE is tr
      → y = length tr
        → ∀ final_descriptors : 
             equivocator_descriptors IM',
            proper_equivocator_descriptors IM'
              final_descriptors
              (finite_trace_last is tr)
            → ∀ (initial_descriptors : 
                 equivocator_descriptors IM') 
                (trX : 
                 list
                   (composite_transition_item IM')),
                equivocators_trace_project IM'
                  final_descriptors tr =
                Some (trX, initial_descriptors)
                → finite_valid_trace SeededX
                    (equivocators_state_project
                      IM' initial_descriptors is)
                    trX
Htr: finite_valid_trace_from SeededXE is tr'
Hlst: finite_valid_trace_from SeededXE
  (finite_trace_last is tr')
  [{|
     l := l;
     input := Some input;
     destination := s;
     output := output
   |}]
Hinit: composite_initial_state_prop
  (sub_IM (equivocator_IM IM) selection) is
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input := Some input;
      destination := s;
      output := output
    |}]): composite_state
  (sub_IM (equivocator_IM IM) selection)
final_descriptors: equivocator_descriptors IM'
Hproper: proper_equivocator_descriptors IM'
  final_descriptors s
initial_descriptors: equivocator_descriptors IM'
trX': list
  (composite_transition_item
     (λ i : sub_index selection, IM (`i)))
l0: composite_label IM'
final_descriptors': equivocator_descriptors IM'
Hpr_trX': equivocators_trace_project
  (λ i : sub_index selection, IM (`i))
  final_descriptors' tr' =
Some (trX', initial_descriptors)
HtrX': finite_valid_trace_from SeededX
  (equivocators_state_project IM'
     initial_descriptors is) trX'
HinitX: composite_initial_state_prop
  (sub_IM IM selection)
  (equivocators_state_project IM'
     initial_descriptors is)
Hproper': proper_equivocator_descriptors IM'
  final_descriptors'
  (finite_trace_last is tr')
Heq_final_descriptors': final_descriptors' =
equivocator_descriptors_update
  IM' final_descriptors
  (projT1 l)
  (final_descriptors'
     (projT1 l))
Hvx: let (i, li) := l0 in
valid li
  (finite_trace_last
     (equivocators_state_project IM'
        initial_descriptors is) trX' i,
   Some input)
Htx: (let (i, li) := l0 in
 let (si', om') :=
   transition li
     (finite_trace_last
        (equivocators_state_project IM'
           initial_descriptors is) trX' i,
      Some input) in
 (state_update IM'
    (finite_trace_last
       (equivocators_state_project IM'
          initial_descriptors is) trX') i si',
  om')) =
(equivocators_state_project IM'
   final_descriptors s, output)
Hl: ∃ Hex : existing_equivocator_label
          (IM' (projT1 l)) 
          (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM' (projT1 l)) 
       (projT2 l) Hex) = l0
Hprojectx: equivocators_transition_item_project IM'
  final_descriptors
  {|
    l := l;
    input := Some input;
    destination := s;
    output := output
  |} =
Some
  (Some
     {|
       l := l0;
       input := Some input;
       destination :=
         equivocators_state_project IM'
           final_descriptors s;
       output := output
     |}, final_descriptors')
Htl: finite_valid_trace_from SeededXE s []
Hs: valid_state_prop SeededXE
  (finite_trace_last is tr')
Hc: seed input
∨ composite_has_been_sent
    (sub_IM (equivocator_IM IM) selection)
    (finite_trace_last is tr', Some input).1
    input
Hiom: option_valid_message_prop SeededXE (Some input)
Hlst_trX': valid_state_prop SeededX
  (finite_trace_last
     (equivocators_state_project IM'
        initial_descriptors is) trX')
Htr_to: finite_valid_trace_init_to SeededXE is
  (finite_trace_last is tr') tr'
Hpre_tr_to: finite_constrained_trace_init_to SubFreeE
  is (finite_trace_last is tr') tr'
_Hpr_trX': Some (trX', initial_descriptors) =
Some (trX', initial_descriptors)
Heq_final_stateX': finite_trace_last
  (equivocators_state_project IM'
     initial_descriptors is) trX' =
equivocators_state_project IM'
  final_descriptors'
  (finite_trace_last is tr')
option_valid_message_prop SeededX (Some input)
    destruct Hc as [Hinit_input | Hno_equiv]
    ; [by apply initial_message_is_valid, seeded_equivocators_initial_message; right |].message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
is: composite_state
  (sub_IM (equivocator_IM IM) selection)
IM':= λ i : sub_index selection, IM (`i): sub_index selection → VLSM message
tr': list
  (composite_transition_item
     (sub_IM (equivocator_IM IM) selection))
s: composite_state
  (sub_IM (equivocator_IM IM) selection)
l: composite_label
  (sub_IM (equivocator_IM IM) selection)
input: message
output: option message
H: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := Some input;
        destination := s;
        output := output
      |}])
  → ∀ tr : list
             (composite_transition_item
                (sub_IM 
                   (equivocator_IM IM) selection)),
      finite_valid_trace SeededXE is tr
      → y = length tr
        → ∀ final_descriptors : 
             equivocator_descriptors IM',
            proper_equivocator_descriptors IM'
              final_descriptors
              (finite_trace_last is tr)
            → ∀ (initial_descriptors : 
                 equivocator_descriptors IM') 
                (trX : 
                 list
                   (composite_transition_item IM')),
                equivocators_trace_project IM'
                  final_descriptors tr =
                Some (trX, initial_descriptors)
                → finite_valid_trace SeededX
                    (equivocators_state_project
                      IM' initial_descriptors is)
                    trX
Htr: finite_valid_trace_from SeededXE is tr'
Hlst: finite_valid_trace_from SeededXE
  (finite_trace_last is tr')
  [{|
     l := l;
     input := Some input;
     destination := s;
     output := output
   |}]
Hinit: composite_initial_state_prop
  (sub_IM (equivocator_IM IM) selection) is
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input := Some input;
      destination := s;
      output := output
    |}]): composite_state
  (sub_IM (equivocator_IM IM) selection)
final_descriptors: equivocator_descriptors IM'
Hproper: proper_equivocator_descriptors IM'
  final_descriptors s
initial_descriptors: equivocator_descriptors IM'
trX': list
  (composite_transition_item
     (λ i : sub_index selection, IM (`i)))
l0: composite_label IM'
final_descriptors': equivocator_descriptors IM'
Hpr_trX': equivocators_trace_project
  (λ i : sub_index selection, IM (`i))
  final_descriptors' tr' =
Some (trX', initial_descriptors)
HtrX': finite_valid_trace_from SeededX
  (equivocators_state_project IM'
     initial_descriptors is) trX'
HinitX: composite_initial_state_prop
  (sub_IM IM selection)
  (equivocators_state_project IM'
     initial_descriptors is)
Hproper': proper_equivocator_descriptors IM'
  final_descriptors'
  (finite_trace_last is tr')
Heq_final_descriptors': final_descriptors' =
equivocator_descriptors_update
  IM' final_descriptors
  (projT1 l)
  (final_descriptors'
     (projT1 l))
Hvx: let (i, li) := l0 in
valid li
  (finite_trace_last
     (equivocators_state_project IM'
        initial_descriptors is) trX' i,
   Some input)
Htx: (let (i, li) := l0 in
 let (si', om') :=
   transition li
     (finite_trace_last
        (equivocators_state_project IM'
           initial_descriptors is) trX' i,
      Some input) in
 (state_update IM'
    (finite_trace_last
       (equivocators_state_project IM'
          initial_descriptors is) trX') i si',
  om')) =
(equivocators_state_project IM'
   final_descriptors s, output)
Hl: ∃ Hex : existing_equivocator_label
          (IM' (projT1 l)) 
          (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM' (projT1 l)) 
       (projT2 l) Hex) = l0
Hprojectx: equivocators_transition_item_project IM'
  final_descriptors
  {|
    l := l;
    input := Some input;
    destination := s;
    output := output
  |} =
Some
  (Some
     {|
       l := l0;
       input := Some input;
       destination :=
         equivocators_state_project IM'
           final_descriptors s;
       output := output
     |}, final_descriptors')
Htl: finite_valid_trace_from SeededXE s []
Hs: valid_state_prop SeededXE
  (finite_trace_last is tr')
Hno_equiv: composite_has_been_sent
  (sub_IM (equivocator_IM IM) selection)
  (finite_trace_last is tr', Some input).1
  input
Hiom: option_valid_message_prop SeededXE (Some input)
Hlst_trX': valid_state_prop SeededX
  (finite_trace_last
     (equivocators_state_project IM'
        initial_descriptors is) trX')
Htr_to: finite_valid_trace_init_to SeededXE is
  (finite_trace_last is tr') tr'
Hpre_tr_to: finite_constrained_trace_init_to SubFreeE
  is (finite_trace_last is tr') tr'
_Hpr_trX': Some (trX', initial_descriptors) =
Some (trX', initial_descriptors)
Heq_final_stateX': finite_trace_last
  (equivocators_state_project IM'
     initial_descriptors is) trX' =
equivocators_state_project IM'
  final_descriptors'
  (finite_trace_last is tr')
option_valid_message_prop SeededX (Some input)
    assert (Hs_free : constrained_state_prop SubFreeE (finite_trace_last is tr'))
      by (apply proj1, finite_valid_trace_from_to_last_pstate in Hpre_tr_to; done).message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
is: composite_state
  (sub_IM (equivocator_IM IM) selection)
IM':= λ i : sub_index selection, IM (`i): sub_index selection → VLSM message
tr': list
  (composite_transition_item
     (sub_IM (equivocator_IM IM) selection))
s: composite_state
  (sub_IM (equivocator_IM IM) selection)
l: composite_label
  (sub_IM (equivocator_IM IM) selection)
input: message
output: option message
H: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := Some input;
        destination := s;
        output := output
      |}])
  → ∀ tr : list
             (composite_transition_item
                (sub_IM 
                   (equivocator_IM IM) selection)),
      finite_valid_trace SeededXE is tr
      → y = length tr
        → ∀ final_descriptors : 
             equivocator_descriptors IM',
            proper_equivocator_descriptors IM'
              final_descriptors
              (finite_trace_last is tr)
            → ∀ (initial_descriptors : 
                 equivocator_descriptors IM') 
                (trX : 
                 list
                   (composite_transition_item IM')),
                equivocators_trace_project IM'
                  final_descriptors tr =
                Some (trX, initial_descriptors)
                → finite_valid_trace SeededX
                    (equivocators_state_project
                      IM' initial_descriptors is)
                    trX
Htr: finite_valid_trace_from SeededXE is tr'
Hlst: finite_valid_trace_from SeededXE
  (finite_trace_last is tr')
  [{|
     l := l;
     input := Some input;
     destination := s;
     output := output
   |}]
Hinit: composite_initial_state_prop
  (sub_IM (equivocator_IM IM) selection) is
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input := Some input;
      destination := s;
      output := output
    |}]): composite_state
  (sub_IM (equivocator_IM IM) selection)
final_descriptors: equivocator_descriptors IM'
Hproper: proper_equivocator_descriptors IM'
  final_descriptors s
initial_descriptors: equivocator_descriptors IM'
trX': list
  (composite_transition_item
     (λ i : sub_index selection, IM (`i)))
l0: composite_label IM'
final_descriptors': equivocator_descriptors IM'
Hpr_trX': equivocators_trace_project
  (λ i : sub_index selection, IM (`i))
  final_descriptors' tr' =
Some (trX', initial_descriptors)
HtrX': finite_valid_trace_from SeededX
  (equivocators_state_project IM'
     initial_descriptors is) trX'
HinitX: composite_initial_state_prop
  (sub_IM IM selection)
  (equivocators_state_project IM'
     initial_descriptors is)
Hproper': proper_equivocator_descriptors IM'
  final_descriptors'
  (finite_trace_last is tr')
Heq_final_descriptors': final_descriptors' =
equivocator_descriptors_update
  IM' final_descriptors
  (projT1 l)
  (final_descriptors'
     (projT1 l))
Hvx: let (i, li) := l0 in
valid li
  (finite_trace_last
     (equivocators_state_project IM'
        initial_descriptors is) trX' i,
   Some input)
Htx: (let (i, li) := l0 in
 let (si', om') :=
   transition li
     (finite_trace_last
        (equivocators_state_project IM'
           initial_descriptors is) trX' i,
      Some input) in
 (state_update IM'
    (finite_trace_last
       (equivocators_state_project IM'
          initial_descriptors is) trX') i si',
  om')) =
(equivocators_state_project IM'
   final_descriptors s, output)
Hl: ∃ Hex : existing_equivocator_label
          (IM' (projT1 l)) 
          (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM' (projT1 l)) 
       (projT2 l) Hex) = l0
Hprojectx: equivocators_transition_item_project IM'
  final_descriptors
  {|
    l := l;
    input := Some input;
    destination := s;
    output := output
  |} =
Some
  (Some
     {|
       l := l0;
       input := Some input;
       destination :=
         equivocators_state_project IM'
           final_descriptors s;
       output := output
     |}, final_descriptors')
Htl: finite_valid_trace_from SeededXE s []
Hs: valid_state_prop SeededXE
  (finite_trace_last is tr')
Hno_equiv: composite_has_been_sent
  (sub_IM (equivocator_IM IM) selection)
  (finite_trace_last is tr', Some input).1
  input
Hiom: option_valid_message_prop SeededXE (Some input)
Hlst_trX': valid_state_prop SeededX
  (finite_trace_last
     (equivocators_state_project IM'
        initial_descriptors is) trX')
Htr_to: finite_valid_trace_init_to SeededXE is
  (finite_trace_last is tr') tr'
Hpre_tr_to: finite_constrained_trace_init_to SubFreeE
  is (finite_trace_last is tr') tr'
_Hpr_trX': Some (trX', initial_descriptors) =
Some (trX', initial_descriptors)
Heq_final_stateX': finite_trace_last
  (equivocators_state_project IM'
     initial_descriptors is) trX' =
equivocators_state_project IM'
  final_descriptors'
  (finite_trace_last is tr')
Hs_free: constrained_state_prop SubFreeE
  (finite_trace_last is tr')
option_valid_message_prop SeededX (Some input)
    apply (composite_proper_sent _ _ Hs_free) in Hno_equiv.message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
is: composite_state
  (sub_IM (equivocator_IM IM) selection)
IM':= λ i : sub_index selection, IM (`i): sub_index selection → VLSM message
tr': list
  (composite_transition_item
     (sub_IM (equivocator_IM IM) selection))
s: composite_state
  (sub_IM (equivocator_IM IM) selection)
l: composite_label
  (sub_IM (equivocator_IM IM) selection)
input: message
output: option message
H: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := Some input;
        destination := s;
        output := output
      |}])
  → ∀ tr : list
             (composite_transition_item
                (sub_IM 
                   (equivocator_IM IM) selection)),
      finite_valid_trace SeededXE is tr
      → y = length tr
        → ∀ final_descriptors : 
             equivocator_descriptors IM',
            proper_equivocator_descriptors IM'
              final_descriptors
              (finite_trace_last is tr)
            → ∀ (initial_descriptors : 
                 equivocator_descriptors IM') 
                (trX : 
                 list
                   (composite_transition_item IM')),
                equivocators_trace_project IM'
                  final_descriptors tr =
                Some (trX, initial_descriptors)
                → finite_valid_trace SeededX
                    (equivocators_state_project
                      IM' initial_descriptors is)
                    trX
Htr: finite_valid_trace_from SeededXE is tr'
Hlst: finite_valid_trace_from SeededXE
  (finite_trace_last is tr')
  [{|
     l := l;
     input := Some input;
     destination := s;
     output := output
   |}]
Hinit: composite_initial_state_prop
  (sub_IM (equivocator_IM IM) selection) is
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input := Some input;
      destination := s;
      output := output
    |}]): composite_state
  (sub_IM (equivocator_IM IM) selection)
final_descriptors: equivocator_descriptors IM'
Hproper: proper_equivocator_descriptors IM'
  final_descriptors s
initial_descriptors: equivocator_descriptors IM'
trX': list
  (composite_transition_item
     (λ i : sub_index selection, IM (`i)))
l0: composite_label IM'
final_descriptors': equivocator_descriptors IM'
Hpr_trX': equivocators_trace_project
  (λ i : sub_index selection, IM (`i))
  final_descriptors' tr' =
Some (trX', initial_descriptors)
HtrX': finite_valid_trace_from SeededX
  (equivocators_state_project IM'
     initial_descriptors is) trX'
HinitX: composite_initial_state_prop
  (sub_IM IM selection)
  (equivocators_state_project IM'
     initial_descriptors is)
Hproper': proper_equivocator_descriptors IM'
  final_descriptors'
  (finite_trace_last is tr')
Heq_final_descriptors': final_descriptors' =
equivocator_descriptors_update
  IM' final_descriptors
  (projT1 l)
  (final_descriptors'
     (projT1 l))
Hvx: let (i, li) := l0 in
valid li
  (finite_trace_last
     (equivocators_state_project IM'
        initial_descriptors is) trX' i,
   Some input)
Htx: (let (i, li) := l0 in
 let (si', om') :=
   transition li
     (finite_trace_last
        (equivocators_state_project IM'
           initial_descriptors is) trX' i,
      Some input) in
 (state_update IM'
    (finite_trace_last
       (equivocators_state_project IM'
          initial_descriptors is) trX') i si',
  om')) =
(equivocators_state_project IM'
   final_descriptors s, output)
Hl: ∃ Hex : existing_equivocator_label
          (IM' (projT1 l)) 
          (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM' (projT1 l)) 
       (projT2 l) Hex) = l0
Hprojectx: equivocators_transition_item_project IM'
  final_descriptors
  {|
    l := l;
    input := Some input;
    destination := s;
    output := output
  |} =
Some
  (Some
     {|
       l := l0;
       input := Some input;
       destination :=
         equivocators_state_project IM'
           final_descriptors s;
       output := output
     |}, final_descriptors')
Htl: finite_valid_trace_from SeededXE s []
Hs: valid_state_prop SeededXE
  (finite_trace_last is tr')
Hno_equiv: selected_message_exists_in_all_preloaded_traces
  (free_composite_vlsm
     (sub_IM (equivocator_IM IM) selection))
  (field_selector VLSM.output)
  (finite_trace_last is tr') input
Hiom: option_valid_message_prop SeededXE (Some input)
Hlst_trX': valid_state_prop SeededX
  (finite_trace_last
     (equivocators_state_project IM'
        initial_descriptors is) trX')
Htr_to: finite_valid_trace_init_to SeededXE is
  (finite_trace_last is tr') tr'
Hpre_tr_to: finite_constrained_trace_init_to SubFreeE
  is (finite_trace_last is tr') tr'
_Hpr_trX': Some (trX', initial_descriptors) =
Some (trX', initial_descriptors)
Heq_final_stateX': finite_trace_last
  (equivocators_state_project IM'
     initial_descriptors is) trX' =
equivocators_state_project IM'
  final_descriptors'
  (finite_trace_last is tr')
Hs_free: constrained_state_prop SubFreeE
  (finite_trace_last is tr')
option_valid_message_prop SeededX (Some input)
    specialize (Hno_equiv is tr' Hpre_tr_to).message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
is: composite_state
  (sub_IM (equivocator_IM IM) selection)
IM':= λ i : sub_index selection, IM (`i): sub_index selection → VLSM message
tr': list
  (composite_transition_item
     (sub_IM (equivocator_IM IM) selection))
s: composite_state
  (sub_IM (equivocator_IM IM) selection)
l: composite_label
  (sub_IM (equivocator_IM IM) selection)
input: message
output: option message
H: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := Some input;
        destination := s;
        output := output
      |}])
  → ∀ tr : list
             (composite_transition_item
                (sub_IM 
                   (equivocator_IM IM) selection)),
      finite_valid_trace SeededXE is tr
      → y = length tr
        → ∀ final_descriptors : 
             equivocator_descriptors IM',
            proper_equivocator_descriptors IM'
              final_descriptors
              (finite_trace_last is tr)
            → ∀ (initial_descriptors : 
                 equivocator_descriptors IM') 
                (trX : 
                 list
                   (composite_transition_item IM')),
                equivocators_trace_project IM'
                  final_descriptors tr =
                Some (trX, initial_descriptors)
                → finite_valid_trace SeededX
                    (equivocators_state_project
                      IM' initial_descriptors is)
                    trX
Htr: finite_valid_trace_from SeededXE is tr'
Hlst: finite_valid_trace_from SeededXE
  (finite_trace_last is tr')
  [{|
     l := l;
     input := Some input;
     destination := s;
     output := output
   |}]
Hinit: composite_initial_state_prop
  (sub_IM (equivocator_IM IM) selection) is
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input := Some input;
      destination := s;
      output := output
    |}]): composite_state
  (sub_IM (equivocator_IM IM) selection)
final_descriptors: equivocator_descriptors IM'
Hproper: proper_equivocator_descriptors IM'
  final_descriptors s
initial_descriptors: equivocator_descriptors IM'
trX': list
  (composite_transition_item
     (λ i : sub_index selection, IM (`i)))
l0: composite_label IM'
final_descriptors': equivocator_descriptors IM'
Hpr_trX': equivocators_trace_project
  (λ i : sub_index selection, IM (`i))
  final_descriptors' tr' =
Some (trX', initial_descriptors)
HtrX': finite_valid_trace_from SeededX
  (equivocators_state_project IM'
     initial_descriptors is) trX'
HinitX: composite_initial_state_prop
  (sub_IM IM selection)
  (equivocators_state_project IM'
     initial_descriptors is)
Hproper': proper_equivocator_descriptors IM'
  final_descriptors'
  (finite_trace_last is tr')
Heq_final_descriptors': final_descriptors' =
equivocator_descriptors_update
  IM' final_descriptors
  (projT1 l)
  (final_descriptors'
     (projT1 l))
Hvx: let (i, li) := l0 in
valid li
  (finite_trace_last
     (equivocators_state_project IM'
        initial_descriptors is) trX' i,
   Some input)
Htx: (let (i, li) := l0 in
 let (si', om') :=
   transition li
     (finite_trace_last
        (equivocators_state_project IM'
           initial_descriptors is) trX' i,
      Some input) in
 (state_update IM'
    (finite_trace_last
       (equivocators_state_project IM'
          initial_descriptors is) trX') i si',
  om')) =
(equivocators_state_project IM'
   final_descriptors s, output)
Hl: ∃ Hex : existing_equivocator_label
          (IM' (projT1 l)) 
          (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM' (projT1 l)) 
       (projT2 l) Hex) = l0
Hprojectx: equivocators_transition_item_project IM'
  final_descriptors
  {|
    l := l;
    input := Some input;
    destination := s;
    output := output
  |} =
Some
  (Some
     {|
       l := l0;
       input := Some input;
       destination :=
         equivocators_state_project IM'
           final_descriptors s;
       output := output
     |}, final_descriptors')
Htl: finite_valid_trace_from SeededXE s []
Hs: valid_state_prop SeededXE
  (finite_trace_last is tr')
Hno_equiv: trace_has_message
  (field_selector VLSM.output) input tr'
Hiom: option_valid_message_prop SeededXE (Some input)
Hlst_trX': valid_state_prop SeededX
  (finite_trace_last
     (equivocators_state_project IM'
        initial_descriptors is) trX')
Htr_to: finite_valid_trace_init_to SeededXE is
  (finite_trace_last is tr') tr'
Hpre_tr_to: finite_constrained_trace_init_to SubFreeE
  is (finite_trace_last is tr') tr'
_Hpr_trX': Some (trX', initial_descriptors) =
Some (trX', initial_descriptors)
Heq_final_stateX': finite_trace_last
  (equivocators_state_project IM'
     initial_descriptors is) trX' =
equivocators_state_project IM'
  final_descriptors'
  (finite_trace_last is tr')
Hs_free: constrained_state_prop SubFreeE
  (finite_trace_last is tr')
option_valid_message_prop SeededX (Some input)
    apply finite_valid_trace_init_to_forget_last in Hpre_tr_to as Hpre_tr.message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
is: composite_state
  (sub_IM (equivocator_IM IM) selection)
IM':= λ i : sub_index selection, IM (`i): sub_index selection → VLSM message
tr': list
  (composite_transition_item
     (sub_IM (equivocator_IM IM) selection))
s: composite_state
  (sub_IM (equivocator_IM IM) selection)
l: composite_label
  (sub_IM (equivocator_IM IM) selection)
input: message
output: option message
H: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := Some input;
        destination := s;
        output := output
      |}])
  → ∀ tr : list
             (composite_transition_item
                (sub_IM 
                   (equivocator_IM IM) selection)),
      finite_valid_trace SeededXE is tr
      → y = length tr
        → ∀ final_descriptors : 
             equivocator_descriptors IM',
            proper_equivocator_descriptors IM'
              final_descriptors
              (finite_trace_last is tr)
            → ∀ (initial_descriptors : 
                 equivocator_descriptors IM') 
                (trX : 
                 list
                   (composite_transition_item IM')),
                equivocators_trace_project IM'
                  final_descriptors tr =
                Some (trX, initial_descriptors)
                → finite_valid_trace SeededX
                    (equivocators_state_project
                      IM' initial_descriptors is)
                    trX
Htr: finite_valid_trace_from SeededXE is tr'
Hlst: finite_valid_trace_from SeededXE
  (finite_trace_last is tr')
  [{|
     l := l;
     input := Some input;
     destination := s;
     output := output
   |}]
Hinit: composite_initial_state_prop
  (sub_IM (equivocator_IM IM) selection) is
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input := Some input;
      destination := s;
      output := output
    |}]): composite_state
  (sub_IM (equivocator_IM IM) selection)
final_descriptors: equivocator_descriptors IM'
Hproper: proper_equivocator_descriptors IM'
  final_descriptors s
initial_descriptors: equivocator_descriptors IM'
trX': list
  (composite_transition_item
     (λ i : sub_index selection, IM (`i)))
l0: composite_label IM'
final_descriptors': equivocator_descriptors IM'
Hpr_trX': equivocators_trace_project
  (λ i : sub_index selection, IM (`i))
  final_descriptors' tr' =
Some (trX', initial_descriptors)
HtrX': finite_valid_trace_from SeededX
  (equivocators_state_project IM'
     initial_descriptors is) trX'
HinitX: composite_initial_state_prop
  (sub_IM IM selection)
  (equivocators_state_project IM'
     initial_descriptors is)
Hproper': proper_equivocator_descriptors IM'
  final_descriptors'
  (finite_trace_last is tr')
Heq_final_descriptors': final_descriptors' =
equivocator_descriptors_update
  IM' final_descriptors
  (projT1 l)
  (final_descriptors'
     (projT1 l))
Hvx: let (i, li) := l0 in
valid li
  (finite_trace_last
     (equivocators_state_project IM'
        initial_descriptors is) trX' i,
   Some input)
Htx: (let (i, li) := l0 in
 let (si', om') :=
   transition li
     (finite_trace_last
        (equivocators_state_project IM'
           initial_descriptors is) trX' i,
      Some input) in
 (state_update IM'
    (finite_trace_last
       (equivocators_state_project IM'
          initial_descriptors is) trX') i si',
  om')) =
(equivocators_state_project IM'
   final_descriptors s, output)
Hl: ∃ Hex : existing_equivocator_label
          (IM' (projT1 l)) 
          (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM' (projT1 l)) 
       (projT2 l) Hex) = l0
Hprojectx: equivocators_transition_item_project IM'
  final_descriptors
  {|
    l := l;
    input := Some input;
    destination := s;
    output := output
  |} =
Some
  (Some
     {|
       l := l0;
       input := Some input;
       destination :=
         equivocators_state_project IM'
           final_descriptors s;
       output := output
     |}, final_descriptors')
Htl: finite_valid_trace_from SeededXE s []
Hs: valid_state_prop SeededXE
  (finite_trace_last is tr')
Hno_equiv: trace_has_message
  (field_selector VLSM.output) input tr'
Hiom: option_valid_message_prop SeededXE (Some input)
Hlst_trX': valid_state_prop SeededX
  (finite_trace_last
     (equivocators_state_project IM'
        initial_descriptors is) trX')
Htr_to: finite_valid_trace_init_to SeededXE is
  (finite_trace_last is tr') tr'
Hpre_tr_to: finite_constrained_trace_init_to SubFreeE
  is (finite_trace_last is tr') tr'
_Hpr_trX': Some (trX', initial_descriptors) =
Some (trX', initial_descriptors)
Heq_final_stateX': finite_trace_last
  (equivocators_state_project IM'
     initial_descriptors is) trX' =
equivocators_state_project IM'
  final_descriptors'
  (finite_trace_last is tr')
Hs_free: constrained_state_prop SubFreeE
  (finite_trace_last is tr')
Hpre_tr: finite_valid_trace
  (preloaded_with_all_messages_vlsm SubFreeE)
  is tr'
option_valid_message_prop SeededX (Some input)
    destruct (equivocators_trace_project_output_reflecting_inv _ _ _ (proj1 Hpre_tr) _ Hno_equiv)
      as [final_descriptors_m [initial_descriptors_m [trXm [Hfinal_descriptors_m
          [Hproject_trXm Hex]]]]].message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
is: composite_state
  (sub_IM (equivocator_IM IM) selection)
IM':= λ i : sub_index selection, IM (`i): sub_index selection → VLSM message
tr': list
  (composite_transition_item
     (sub_IM (equivocator_IM IM) selection))
s: composite_state
  (sub_IM (equivocator_IM IM) selection)
l: composite_label
  (sub_IM (equivocator_IM IM) selection)
input: message
output: option message
H: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := Some input;
        destination := s;
        output := output
      |}])
  → ∀ tr : list
             (composite_transition_item
                (sub_IM 
                   (equivocator_IM IM) selection)),
      finite_valid_trace SeededXE is tr
      → y = length tr
        → ∀ final_descriptors : 
             equivocator_descriptors IM',
            proper_equivocator_descriptors IM'
              final_descriptors
              (finite_trace_last is tr)
            → ∀ (initial_descriptors : 
                 equivocator_descriptors IM') 
                (trX : 
                 list
                   (composite_transition_item IM')),
                equivocators_trace_project IM'
                  final_descriptors tr =
                Some (trX, initial_descriptors)
                → finite_valid_trace SeededX
                    (equivocators_state_project
                      IM' initial_descriptors is)
                    trX
Htr: finite_valid_trace_from SeededXE is tr'
Hlst: finite_valid_trace_from SeededXE
  (finite_trace_last is tr')
  [{|
     l := l;
     input := Some input;
     destination := s;
     output := output
   |}]
Hinit: composite_initial_state_prop
  (sub_IM (equivocator_IM IM) selection) is
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input := Some input;
      destination := s;
      output := output
    |}]): composite_state
  (sub_IM (equivocator_IM IM) selection)
final_descriptors: equivocator_descriptors IM'
Hproper: proper_equivocator_descriptors IM'
  final_descriptors s
initial_descriptors: equivocator_descriptors IM'
trX': list
  (composite_transition_item
     (λ i : sub_index selection, IM (`i)))
l0: composite_label IM'
final_descriptors': equivocator_descriptors IM'
Hpr_trX': equivocators_trace_project
  (λ i : sub_index selection, IM (`i))
  final_descriptors' tr' =
Some (trX', initial_descriptors)
HtrX': finite_valid_trace_from SeededX
  (equivocators_state_project IM'
     initial_descriptors is) trX'
HinitX: composite_initial_state_prop
  (sub_IM IM selection)
  (equivocators_state_project IM'
     initial_descriptors is)
Hproper': proper_equivocator_descriptors IM'
  final_descriptors'
  (finite_trace_last is tr')
Heq_final_descriptors': final_descriptors' =
equivocator_descriptors_update
  IM' final_descriptors
  (projT1 l)
  (final_descriptors'
     (projT1 l))
Hvx: let (i, li) := l0 in
valid li
  (finite_trace_last
     (equivocators_state_project IM'
        initial_descriptors is) trX' i,
   Some input)
Htx: (let (i, li) := l0 in
 let (si', om') :=
   transition li
     (finite_trace_last
        (equivocators_state_project IM'
           initial_descriptors is) trX' i,
      Some input) in
 (state_update IM'
    (finite_trace_last
       (equivocators_state_project IM'
          initial_descriptors is) trX') i si',
  om')) =
(equivocators_state_project IM'
   final_descriptors s, output)
Hl: ∃ Hex : existing_equivocator_label
          (IM' (projT1 l)) 
          (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM' (projT1 l)) 
       (projT2 l) Hex) = l0
Hprojectx: equivocators_transition_item_project IM'
  final_descriptors
  {|
    l := l;
    input := Some input;
    destination := s;
    output := output
  |} =
Some
  (Some
     {|
       l := l0;
       input := Some input;
       destination :=
         equivocators_state_project IM'
           final_descriptors s;
       output := output
     |}, final_descriptors')
Htl: finite_valid_trace_from SeededXE s []
Hs: valid_state_prop SeededXE
  (finite_trace_last is tr')
Hno_equiv: trace_has_message
  (field_selector VLSM.output) input tr'
Hiom: option_valid_message_prop SeededXE (Some input)
Hlst_trX': valid_state_prop SeededX
  (finite_trace_last
     (equivocators_state_project IM'
        initial_descriptors is) trX')
Htr_to: finite_valid_trace_init_to SeededXE is
  (finite_trace_last is tr') tr'
Hpre_tr_to: finite_constrained_trace_init_to SubFreeE
  is (finite_trace_last is tr') tr'
_Hpr_trX': Some (trX', initial_descriptors) =
Some (trX', initial_descriptors)
Heq_final_stateX': finite_trace_last
  (equivocators_state_project IM'
     initial_descriptors is) trX' =
equivocators_state_project IM'
  final_descriptors'
  (finite_trace_last is tr')
Hs_free: constrained_state_prop SubFreeE
  (finite_trace_last is tr')
Hpre_tr: finite_valid_trace
  (preloaded_with_all_messages_vlsm SubFreeE)
  is tr'
final_descriptors_m, initial_descriptors_m: equivocator_descriptors
  (λ ei : sub_index selection,
     IM (`ei))
trXm: list
  (composite_transition_item
     (λ ei : sub_index selection, IM (`ei)))
Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  (λ ei : sub_index selection,
     IM (`ei))
  final_descriptors_m
  (finite_trace_last is tr')
Hproject_trXm: equivocators_trace_project
  (λ ei : sub_index selection,
     IM (`ei)) final_descriptors_m tr' =
Some (trXm, initial_descriptors_m)
Hex: Exists (field_selector VLSM.output input) trXm
option_valid_message_prop SeededX (Some input)
    specialize (H (length tr')).message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
is: composite_state
  (sub_IM (equivocator_IM IM) selection)
IM':= λ i : sub_index selection, IM (`i): sub_index selection → VLSM message
tr': list
  (composite_transition_item
     (sub_IM (equivocator_IM IM) selection))
s: composite_state
  (sub_IM (equivocator_IM IM) selection)
l: composite_label
  (sub_IM (equivocator_IM IM) selection)
input: message
output: option message
H: length tr' <
length
  (tr' ++
   [{|
      l := l;
      input := Some input;
      destination := s;
      output := output
    |}])
→ ∀ tr : list
           (composite_transition_item
              (sub_IM (equivocator_IM IM)
                 selection)),
    finite_valid_trace SeededXE is tr
    → length tr' = length tr
      → ∀ final_descriptors : 
           equivocator_descriptors IM',
          proper_equivocator_descriptors IM'
            final_descriptors
            (finite_trace_last is tr)
          → ∀ (initial_descriptors : 
               equivocator_descriptors IM') 
              (trX : list
                      (composite_transition_item
                      IM')),
              equivocators_trace_project IM'
                final_descriptors tr =
              Some (trX, initial_descriptors)
              → finite_valid_trace SeededX
                  (equivocators_state_project IM'
                     initial_descriptors is) trX
Htr: finite_valid_trace_from SeededXE is tr'
Hlst: finite_valid_trace_from SeededXE
  (finite_trace_last is tr')
  [{|
     l := l;
     input := Some input;
     destination := s;
     output := output
   |}]
Hinit: composite_initial_state_prop
  (sub_IM (equivocator_IM IM) selection) is
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input := Some input;
      destination := s;
      output := output
    |}]): composite_state
  (sub_IM (equivocator_IM IM) selection)
final_descriptors: equivocator_descriptors IM'
Hproper: proper_equivocator_descriptors IM'
  final_descriptors s
initial_descriptors: equivocator_descriptors IM'
trX': list
  (composite_transition_item
     (λ i : sub_index selection, IM (`i)))
l0: composite_label IM'
final_descriptors': equivocator_descriptors IM'
Hpr_trX': equivocators_trace_project
  (λ i : sub_index selection, IM (`i))
  final_descriptors' tr' =
Some (trX', initial_descriptors)
HtrX': finite_valid_trace_from SeededX
  (equivocators_state_project IM'
     initial_descriptors is) trX'
HinitX: composite_initial_state_prop
  (sub_IM IM selection)
  (equivocators_state_project IM'
     initial_descriptors is)
Hproper': proper_equivocator_descriptors IM'
  final_descriptors'
  (finite_trace_last is tr')
Heq_final_descriptors': final_descriptors' =
equivocator_descriptors_update
  IM' final_descriptors
  (projT1 l)
  (final_descriptors'
     (projT1 l))
Hvx: let (i, li) := l0 in
valid li
  (finite_trace_last
     (equivocators_state_project IM'
        initial_descriptors is) trX' i,
   Some input)
Htx: (let (i, li) := l0 in
 let (si', om') :=
   transition li
     (finite_trace_last
        (equivocators_state_project IM'
           initial_descriptors is) trX' i,
      Some input) in
 (state_update IM'
    (finite_trace_last
       (equivocators_state_project IM'
          initial_descriptors is) trX') i si',
  om')) =
(equivocators_state_project IM'
   final_descriptors s, output)
Hl: ∃ Hex : existing_equivocator_label
          (IM' (projT1 l)) 
          (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM' (projT1 l)) 
       (projT2 l) Hex) = l0
Hprojectx: equivocators_transition_item_project IM'
  final_descriptors
  {|
    l := l;
    input := Some input;
    destination := s;
    output := output
  |} =
Some
  (Some
     {|
       l := l0;
       input := Some input;
       destination :=
         equivocators_state_project IM'
           final_descriptors s;
       output := output
     |}, final_descriptors')
Htl: finite_valid_trace_from SeededXE s []
Hs: valid_state_prop SeededXE
  (finite_trace_last is tr')
Hno_equiv: trace_has_message
  (field_selector VLSM.output) input tr'
Hiom: option_valid_message_prop SeededXE (Some input)
Hlst_trX': valid_state_prop SeededX
  (finite_trace_last
     (equivocators_state_project IM'
        initial_descriptors is) trX')
Htr_to: finite_valid_trace_init_to SeededXE is
  (finite_trace_last is tr') tr'
Hpre_tr_to: finite_constrained_trace_init_to SubFreeE
  is (finite_trace_last is tr') tr'
_Hpr_trX': Some (trX', initial_descriptors) =
Some (trX', initial_descriptors)
Heq_final_stateX': finite_trace_last
  (equivocators_state_project IM'
     initial_descriptors is) trX' =
equivocators_state_project IM'
  final_descriptors'
  (finite_trace_last is tr')
Hs_free: constrained_state_prop SubFreeE
  (finite_trace_last is tr')
Hpre_tr: finite_valid_trace
  (preloaded_with_all_messages_vlsm SubFreeE)
  is tr'
final_descriptors_m, initial_descriptors_m: equivocator_descriptors
  (λ ei : sub_index selection,
     IM (`ei))
trXm: list
  (composite_transition_item
     (λ ei : sub_index selection, IM (`ei)))
Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  (λ ei : sub_index selection,
     IM (`ei))
  final_descriptors_m
  (finite_trace_last is tr')
Hproject_trXm: equivocators_trace_project
  (λ ei : sub_index selection,
     IM (`ei)) final_descriptors_m tr' =
Some (trXm, initial_descriptors_m)
Hex: Exists (field_selector VLSM.output input) trXm
option_valid_message_prop SeededX (Some input)
    spec H; [by rewrite app_length; cbn; lia |].message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
is: composite_state
  (sub_IM (equivocator_IM IM) selection)
IM':= λ i : sub_index selection, IM (`i): sub_index selection → VLSM message
tr': list
  (composite_transition_item
     (sub_IM (equivocator_IM IM) selection))
s: composite_state
  (sub_IM (equivocator_IM IM) selection)
l: composite_label
  (sub_IM (equivocator_IM IM) selection)
input: message
output: option message
Htr: finite_valid_trace_from SeededXE is tr'
Hlst: finite_valid_trace_from SeededXE
  (finite_trace_last is tr')
  [{|
     l := l;
     input := Some input;
     destination := s;
     output := output
   |}]
Hinit: composite_initial_state_prop
  (sub_IM (equivocator_IM IM) selection) is
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input := Some input;
      destination := s;
      output := output
    |}]): composite_state
  (sub_IM (equivocator_IM IM) selection)
final_descriptors: equivocator_descriptors IM'
Hproper: proper_equivocator_descriptors IM'
  final_descriptors s
initial_descriptors: equivocator_descriptors IM'
trX': list
  (composite_transition_item
     (λ i : sub_index selection, IM (`i)))
l0: composite_label IM'
final_descriptors': equivocator_descriptors IM'
Hpr_trX': equivocators_trace_project
  (λ i : sub_index selection, IM (`i))
  final_descriptors' tr' =
Some (trX', initial_descriptors)
HtrX': finite_valid_trace_from SeededX
  (equivocators_state_project IM'
     initial_descriptors is) trX'
HinitX: composite_initial_state_prop
  (sub_IM IM selection)
  (equivocators_state_project IM'
     initial_descriptors is)
Hproper': proper_equivocator_descriptors IM'
  final_descriptors'
  (finite_trace_last is tr')
Heq_final_descriptors': final_descriptors' =
equivocator_descriptors_update
  IM' final_descriptors
  (projT1 l)
  (final_descriptors'
     (projT1 l))
Hvx: let (i, li) := l0 in
valid li
  (finite_trace_last
     (equivocators_state_project IM'
        initial_descriptors is) trX' i,
   Some input)
Htx: (let (i, li) := l0 in
 let (si', om') :=
   transition li
     (finite_trace_last
        (equivocators_state_project IM'
           initial_descriptors is) trX' i,
      Some input) in
 (state_update IM'
    (finite_trace_last
       (equivocators_state_project IM'
          initial_descriptors is) trX') i si',
  om')) =
(equivocators_state_project IM'
   final_descriptors s, output)
Hl: ∃ Hex : existing_equivocator_label
          (IM' (projT1 l)) 
          (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM' (projT1 l)) 
       (projT2 l) Hex) = l0
Hprojectx: equivocators_transition_item_project IM'
  final_descriptors
  {|
    l := l;
    input := Some input;
    destination := s;
    output := output
  |} =
Some
  (Some
     {|
       l := l0;
       input := Some input;
       destination :=
         equivocators_state_project IM'
           final_descriptors s;
       output := output
     |}, final_descriptors')
Htl: finite_valid_trace_from SeededXE s []
Hs: valid_state_prop SeededXE
  (finite_trace_last is tr')
Hno_equiv: trace_has_message
  (field_selector VLSM.output) input tr'
Hiom: option_valid_message_prop SeededXE (Some input)
Hlst_trX': valid_state_prop SeededX
  (finite_trace_last
     (equivocators_state_project IM'
        initial_descriptors is) trX')
Htr_to: finite_valid_trace_init_to SeededXE is
  (finite_trace_last is tr') tr'
Hpre_tr_to: finite_constrained_trace_init_to SubFreeE
  is (finite_trace_last is tr') tr'
_Hpr_trX': Some (trX', initial_descriptors) =
Some (trX', initial_descriptors)
Heq_final_stateX': finite_trace_last
  (equivocators_state_project IM'
     initial_descriptors is) trX' =
equivocators_state_project IM'
  final_descriptors'
  (finite_trace_last is tr')
Hs_free: constrained_state_prop SubFreeE
  (finite_trace_last is tr')
Hpre_tr: finite_valid_trace
  (preloaded_with_all_messages_vlsm SubFreeE)
  is tr'
final_descriptors_m, initial_descriptors_m: equivocator_descriptors
  (λ ei : sub_index selection,
     IM (`ei))
trXm: list
  (composite_transition_item
     (λ ei : sub_index selection, IM (`ei)))
Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  (λ ei : sub_index selection,
     IM (`ei))
  final_descriptors_m
  (finite_trace_last is tr')
Hproject_trXm: equivocators_trace_project
  (λ ei : sub_index selection,
     IM (`ei)) final_descriptors_m tr' =
Some (trXm, initial_descriptors_m)
Hex: Exists (field_selector VLSM.output input) trXm
H: ∀ tr : list
         (composite_transition_item
            (sub_IM (equivocator_IM IM) selection)),
  finite_valid_trace SeededXE is tr
  → length tr' = length tr
    → ∀ final_descriptors : 
         equivocator_descriptors IM',
        proper_equivocator_descriptors IM'
          final_descriptors
          (finite_trace_last is tr)
        → ∀ (initial_descriptors : 
             equivocator_descriptors IM') 
            (trX : list
                     (composite_transition_item
                      IM')),
            equivocators_trace_project IM'
              final_descriptors tr =
            Some (trX, initial_descriptors)
            → finite_valid_trace SeededX
                (equivocators_state_project IM'
                   initial_descriptors is) trX
option_valid_message_prop SeededX (Some input)
    specialize (H tr' (conj Htr Hinit) eq_refl).message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
is: composite_state
  (sub_IM (equivocator_IM IM) selection)
IM':= λ i : sub_index selection, IM (`i): sub_index selection → VLSM message
tr': list
  (composite_transition_item
     (sub_IM (equivocator_IM IM) selection))
s: composite_state
  (sub_IM (equivocator_IM IM) selection)
l: composite_label
  (sub_IM (equivocator_IM IM) selection)
input: message
output: option message
Htr: finite_valid_trace_from SeededXE is tr'
Hlst: finite_valid_trace_from SeededXE
  (finite_trace_last is tr')
  [{|
     l := l;
     input := Some input;
     destination := s;
     output := output
   |}]
Hinit: composite_initial_state_prop
  (sub_IM (equivocator_IM IM) selection) is
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input := Some input;
      destination := s;
      output := output
    |}]): composite_state
  (sub_IM (equivocator_IM IM) selection)
final_descriptors: equivocator_descriptors IM'
Hproper: proper_equivocator_descriptors IM'
  final_descriptors s
initial_descriptors: equivocator_descriptors IM'
trX': list
  (composite_transition_item
     (λ i : sub_index selection, IM (`i)))
l0: composite_label IM'
final_descriptors': equivocator_descriptors IM'
Hpr_trX': equivocators_trace_project
  (λ i : sub_index selection, IM (`i))
  final_descriptors' tr' =
Some (trX', initial_descriptors)
HtrX': finite_valid_trace_from SeededX
  (equivocators_state_project IM'
     initial_descriptors is) trX'
HinitX: composite_initial_state_prop
  (sub_IM IM selection)
  (equivocators_state_project IM'
     initial_descriptors is)
Hproper': proper_equivocator_descriptors IM'
  final_descriptors'
  (finite_trace_last is tr')
Heq_final_descriptors': final_descriptors' =
equivocator_descriptors_update
  IM' final_descriptors
  (projT1 l)
  (final_descriptors'
     (projT1 l))
Hvx: let (i, li) := l0 in
valid li
  (finite_trace_last
     (equivocators_state_project IM'
        initial_descriptors is) trX' i,
   Some input)
Htx: (let (i, li) := l0 in
 let (si', om') :=
   transition li
     (finite_trace_last
        (equivocators_state_project IM'
           initial_descriptors is) trX' i,
      Some input) in
 (state_update IM'
    (finite_trace_last
       (equivocators_state_project IM'
          initial_descriptors is) trX') i si',
  om')) =
(equivocators_state_project IM'
   final_descriptors s, output)
Hl: ∃ Hex : existing_equivocator_label
          (IM' (projT1 l)) 
          (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM' (projT1 l)) 
       (projT2 l) Hex) = l0
Hprojectx: equivocators_transition_item_project IM'
  final_descriptors
  {|
    l := l;
    input := Some input;
    destination := s;
    output := output
  |} =
Some
  (Some
     {|
       l := l0;
       input := Some input;
       destination :=
         equivocators_state_project IM'
           final_descriptors s;
       output := output
     |}, final_descriptors')
Htl: finite_valid_trace_from SeededXE s []
Hs: valid_state_prop SeededXE
  (finite_trace_last is tr')
Hno_equiv: trace_has_message
  (field_selector VLSM.output) input tr'
Hiom: option_valid_message_prop SeededXE (Some input)
Hlst_trX': valid_state_prop SeededX
  (finite_trace_last
     (equivocators_state_project IM'
        initial_descriptors is) trX')
Htr_to: finite_valid_trace_init_to SeededXE is
  (finite_trace_last is tr') tr'
Hpre_tr_to: finite_constrained_trace_init_to SubFreeE
  is (finite_trace_last is tr') tr'
_Hpr_trX': Some (trX', initial_descriptors) =
Some (trX', initial_descriptors)
Heq_final_stateX': finite_trace_last
  (equivocators_state_project IM'
     initial_descriptors is) trX' =
equivocators_state_project IM'
  final_descriptors'
  (finite_trace_last is tr')
Hs_free: constrained_state_prop SubFreeE
  (finite_trace_last is tr')
Hpre_tr: finite_valid_trace
  (preloaded_with_all_messages_vlsm SubFreeE)
  is tr'
final_descriptors_m, initial_descriptors_m: equivocator_descriptors
  (λ ei : sub_index selection,
     IM (`ei))
trXm: list
  (composite_transition_item
     (λ ei : sub_index selection, IM (`ei)))
Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  (λ ei : sub_index selection,
     IM (`ei))
  final_descriptors_m
  (finite_trace_last is tr')
Hproject_trXm: equivocators_trace_project
  (λ ei : sub_index selection,
     IM (`ei)) final_descriptors_m tr' =
Some (trXm, initial_descriptors_m)
Hex: Exists (field_selector VLSM.output input) trXm
H: ∀ final_descriptors : equivocator_descriptors IM',
  proper_equivocator_descriptors IM'
    final_descriptors (finite_trace_last is tr')
  → ∀ (initial_descriptors : 
       equivocator_descriptors IM') 
      (trX : list (composite_transition_item IM')),
      equivocators_trace_project IM'
        final_descriptors tr' =
      Some (trX, initial_descriptors)
      → finite_valid_trace SeededX
          (equivocators_state_project IM'
             initial_descriptors is) trX
option_valid_message_prop SeededX (Some input)
    assert (Hfinal_descriptors_m_proper :
      proper_equivocator_descriptors _ final_descriptors_m (finite_trace_last is tr'))
      by (apply not_equivocating_equivocator_descriptors_proper; done).message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
is: composite_state
  (sub_IM (equivocator_IM IM) selection)
IM':= λ i : sub_index selection, IM (`i): sub_index selection → VLSM message
tr': list
  (composite_transition_item
     (sub_IM (equivocator_IM IM) selection))
s: composite_state
  (sub_IM (equivocator_IM IM) selection)
l: composite_label
  (sub_IM (equivocator_IM IM) selection)
input: message
output: option message
Htr: finite_valid_trace_from SeededXE is tr'
Hlst: finite_valid_trace_from SeededXE
  (finite_trace_last is tr')
  [{|
     l := l;
     input := Some input;
     destination := s;
     output := output
   |}]
Hinit: composite_initial_state_prop
  (sub_IM (equivocator_IM IM) selection) is
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input := Some input;
      destination := s;
      output := output
    |}]): composite_state
  (sub_IM (equivocator_IM IM) selection)
final_descriptors: equivocator_descriptors IM'
Hproper: proper_equivocator_descriptors IM'
  final_descriptors s
initial_descriptors: equivocator_descriptors IM'
trX': list
  (composite_transition_item
     (λ i : sub_index selection, IM (`i)))
l0: composite_label IM'
final_descriptors': equivocator_descriptors IM'
Hpr_trX': equivocators_trace_project
  (λ i : sub_index selection, IM (`i))
  final_descriptors' tr' =
Some (trX', initial_descriptors)
HtrX': finite_valid_trace_from SeededX
  (equivocators_state_project IM'
     initial_descriptors is) trX'
HinitX: composite_initial_state_prop
  (sub_IM IM selection)
  (equivocators_state_project IM'
     initial_descriptors is)
Hproper': proper_equivocator_descriptors IM'
  final_descriptors'
  (finite_trace_last is tr')
Heq_final_descriptors': final_descriptors' =
equivocator_descriptors_update
  IM' final_descriptors
  (projT1 l)
  (final_descriptors'
     (projT1 l))
Hvx: let (i, li) := l0 in
valid li
  (finite_trace_last
     (equivocators_state_project IM'
        initial_descriptors is) trX' i,
   Some input)
Htx: (let (i, li) := l0 in
 let (si', om') :=
   transition li
     (finite_trace_last
        (equivocators_state_project IM'
           initial_descriptors is) trX' i,
      Some input) in
 (state_update IM'
    (finite_trace_last
       (equivocators_state_project IM'
          initial_descriptors is) trX') i si',
  om')) =
(equivocators_state_project IM'
   final_descriptors s, output)
Hl: ∃ Hex : existing_equivocator_label
          (IM' (projT1 l)) 
          (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM' (projT1 l)) 
       (projT2 l) Hex) = l0
Hprojectx: equivocators_transition_item_project IM'
  final_descriptors
  {|
    l := l;
    input := Some input;
    destination := s;
    output := output
  |} =
Some
  (Some
     {|
       l := l0;
       input := Some input;
       destination :=
         equivocators_state_project IM'
           final_descriptors s;
       output := output
     |}, final_descriptors')
Htl: finite_valid_trace_from SeededXE s []
Hs: valid_state_prop SeededXE
  (finite_trace_last is tr')
Hno_equiv: trace_has_message
  (field_selector VLSM.output) input tr'
Hiom: option_valid_message_prop SeededXE (Some input)
Hlst_trX': valid_state_prop SeededX
  (finite_trace_last
     (equivocators_state_project IM'
        initial_descriptors is) trX')
Htr_to: finite_valid_trace_init_to SeededXE is
  (finite_trace_last is tr') tr'
Hpre_tr_to: finite_constrained_trace_init_to SubFreeE
  is (finite_trace_last is tr') tr'
_Hpr_trX': Some (trX', initial_descriptors) =
Some (trX', initial_descriptors)
Heq_final_stateX': finite_trace_last
  (equivocators_state_project IM'
     initial_descriptors is) trX' =
equivocators_state_project IM'
  final_descriptors'
  (finite_trace_last is tr')
Hs_free: constrained_state_prop SubFreeE
  (finite_trace_last is tr')
Hpre_tr: finite_valid_trace
  (preloaded_with_all_messages_vlsm SubFreeE)
  is tr'
final_descriptors_m, initial_descriptors_m: equivocator_descriptors
  (λ ei : sub_index selection,
     IM (`ei))
trXm: list
  (composite_transition_item
     (λ ei : sub_index selection, IM (`ei)))
Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  (λ ei : sub_index selection,
     IM (`ei))
  final_descriptors_m
  (finite_trace_last is tr')
Hproject_trXm: equivocators_trace_project
  (λ ei : sub_index selection,
     IM (`ei)) final_descriptors_m tr' =
Some (trXm, initial_descriptors_m)
Hex: Exists (field_selector VLSM.output input) trXm
H: ∀ final_descriptors : equivocator_descriptors IM',
  proper_equivocator_descriptors IM'
    final_descriptors (finite_trace_last is tr')
  → ∀ (initial_descriptors : 
       equivocator_descriptors IM') 
      (trX : list (composite_transition_item IM')),
      equivocators_trace_project IM'
        final_descriptors tr' =
      Some (trX, initial_descriptors)
      → finite_valid_trace SeededX
          (equivocators_state_project IM'
             initial_descriptors is) trX
Hfinal_descriptors_m_proper: proper_equivocator_descriptors
  (λ ei : sub_index
            selection,
     IM (`ei))
  final_descriptors_m
  (finite_trace_last is
     tr')
option_valid_message_prop SeededX (Some input)
    specialize (H final_descriptors_m Hfinal_descriptors_m_proper).message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
is: composite_state
  (sub_IM (equivocator_IM IM) selection)
IM':= λ i : sub_index selection, IM (`i): sub_index selection → VLSM message
tr': list
  (composite_transition_item
     (sub_IM (equivocator_IM IM) selection))
s: composite_state
  (sub_IM (equivocator_IM IM) selection)
l: composite_label
  (sub_IM (equivocator_IM IM) selection)
input: message
output: option message
Htr: finite_valid_trace_from SeededXE is tr'
Hlst: finite_valid_trace_from SeededXE
  (finite_trace_last is tr')
  [{|
     l := l;
     input := Some input;
     destination := s;
     output := output
   |}]
Hinit: composite_initial_state_prop
  (sub_IM (equivocator_IM IM) selection) is
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input := Some input;
      destination := s;
      output := output
    |}]): composite_state
  (sub_IM (equivocator_IM IM) selection)
final_descriptors: equivocator_descriptors IM'
Hproper: proper_equivocator_descriptors IM'
  final_descriptors s
initial_descriptors: equivocator_descriptors IM'
trX': list
  (composite_transition_item
     (λ i : sub_index selection, IM (`i)))
l0: composite_label IM'
final_descriptors': equivocator_descriptors IM'
Hpr_trX': equivocators_trace_project
  (λ i : sub_index selection, IM (`i))
  final_descriptors' tr' =
Some (trX', initial_descriptors)
HtrX': finite_valid_trace_from SeededX
  (equivocators_state_project IM'
     initial_descriptors is) trX'
HinitX: composite_initial_state_prop
  (sub_IM IM selection)
  (equivocators_state_project IM'
     initial_descriptors is)
Hproper': proper_equivocator_descriptors IM'
  final_descriptors'
  (finite_trace_last is tr')
Heq_final_descriptors': final_descriptors' =
equivocator_descriptors_update
  IM' final_descriptors
  (projT1 l)
  (final_descriptors'
     (projT1 l))
Hvx: let (i, li) := l0 in
valid li
  (finite_trace_last
     (equivocators_state_project IM'
        initial_descriptors is) trX' i,
   Some input)
Htx: (let (i, li) := l0 in
 let (si', om') :=
   transition li
     (finite_trace_last
        (equivocators_state_project IM'
           initial_descriptors is) trX' i,
      Some input) in
 (state_update IM'
    (finite_trace_last
       (equivocators_state_project IM'
          initial_descriptors is) trX') i si',
  om')) =
(equivocators_state_project IM'
   final_descriptors s, output)
Hl: ∃ Hex : existing_equivocator_label
          (IM' (projT1 l)) 
          (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM' (projT1 l)) 
       (projT2 l) Hex) = l0
Hprojectx: equivocators_transition_item_project IM'
  final_descriptors
  {|
    l := l;
    input := Some input;
    destination := s;
    output := output
  |} =
Some
  (Some
     {|
       l := l0;
       input := Some input;
       destination :=
         equivocators_state_project IM'
           final_descriptors s;
       output := output
     |}, final_descriptors')
Htl: finite_valid_trace_from SeededXE s []
Hs: valid_state_prop SeededXE
  (finite_trace_last is tr')
Hno_equiv: trace_has_message
  (field_selector VLSM.output) input tr'
Hiom: option_valid_message_prop SeededXE (Some input)
Hlst_trX': valid_state_prop SeededX
  (finite_trace_last
     (equivocators_state_project IM'
        initial_descriptors is) trX')
Htr_to: finite_valid_trace_init_to SeededXE is
  (finite_trace_last is tr') tr'
Hpre_tr_to: finite_constrained_trace_init_to SubFreeE
  is (finite_trace_last is tr') tr'
_Hpr_trX': Some (trX', initial_descriptors) =
Some (trX', initial_descriptors)
Heq_final_stateX': finite_trace_last
  (equivocators_state_project IM'
     initial_descriptors is) trX' =
equivocators_state_project IM'
  final_descriptors'
  (finite_trace_last is tr')
Hs_free: constrained_state_prop SubFreeE
  (finite_trace_last is tr')
Hpre_tr: finite_valid_trace
  (preloaded_with_all_messages_vlsm SubFreeE)
  is tr'
final_descriptors_m, initial_descriptors_m: equivocator_descriptors
  (λ ei : sub_index selection,
     IM (`ei))
trXm: list
  (composite_transition_item
     (λ ei : sub_index selection, IM (`ei)))
Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  (λ ei : sub_index selection,
     IM (`ei))
  final_descriptors_m
  (finite_trace_last is tr')
Hproject_trXm: equivocators_trace_project
  (λ ei : sub_index selection,
     IM (`ei)) final_descriptors_m tr' =
Some (trXm, initial_descriptors_m)
Hex: Exists (field_selector VLSM.output input) trXm
H: ∀ (initial_descriptors : 
   equivocator_descriptors IM') 
  (trX : list (composite_transition_item IM')),
  equivocators_trace_project IM'
    final_descriptors_m tr' =
  Some (trX, initial_descriptors)
  → finite_valid_trace SeededX
      (equivocators_state_project IM'
         initial_descriptors is) trX
Hfinal_descriptors_m_proper: proper_equivocator_descriptors
  (λ ei : sub_index
            selection,
     IM (`ei))
  final_descriptors_m
  (finite_trace_last is
     tr')
option_valid_message_prop SeededX (Some input)
    pose proof (pre_equivocators_valid_trace_project _ _ _ _
     Hpre_tr_to final_descriptors_m Hfinal_descriptors_m_proper) as Hpr_tr'.message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
is: composite_state
  (sub_IM (equivocator_IM IM) selection)
IM':= λ i : sub_index selection, IM (`i): sub_index selection → VLSM message
tr': list
  (composite_transition_item
     (sub_IM (equivocator_IM IM) selection))
s: composite_state
  (sub_IM (equivocator_IM IM) selection)
l: composite_label
  (sub_IM (equivocator_IM IM) selection)
input: message
output: option message
Htr: finite_valid_trace_from SeededXE is tr'
Hlst: finite_valid_trace_from SeededXE
  (finite_trace_last is tr')
  [{|
     l := l;
     input := Some input;
     destination := s;
     output := output
   |}]
Hinit: composite_initial_state_prop
  (sub_IM (equivocator_IM IM) selection) is
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input := Some input;
      destination := s;
      output := output
    |}]): composite_state
  (sub_IM (equivocator_IM IM) selection)
final_descriptors: equivocator_descriptors IM'
Hproper: proper_equivocator_descriptors IM'
  final_descriptors s
initial_descriptors: equivocator_descriptors IM'
trX': list
  (composite_transition_item
     (λ i : sub_index selection, IM (`i)))
l0: composite_label IM'
final_descriptors': equivocator_descriptors IM'
Hpr_trX': equivocators_trace_project
  (λ i : sub_index selection, IM (`i))
  final_descriptors' tr' =
Some (trX', initial_descriptors)
HtrX': finite_valid_trace_from SeededX
  (equivocators_state_project IM'
     initial_descriptors is) trX'
HinitX: composite_initial_state_prop
  (sub_IM IM selection)
  (equivocators_state_project IM'
     initial_descriptors is)
Hproper': proper_equivocator_descriptors IM'
  final_descriptors'
  (finite_trace_last is tr')
Heq_final_descriptors': final_descriptors' =
equivocator_descriptors_update
  IM' final_descriptors
  (projT1 l)
  (final_descriptors'
     (projT1 l))
Hvx: let (i, li) := l0 in
valid li
  (finite_trace_last
     (equivocators_state_project IM'
        initial_descriptors is) trX' i,
   Some input)
Htx: (let (i, li) := l0 in
 let (si', om') :=
   transition li
     (finite_trace_last
        (equivocators_state_project IM'
           initial_descriptors is) trX' i,
      Some input) in
 (state_update IM'
    (finite_trace_last
       (equivocators_state_project IM'
          initial_descriptors is) trX') i si',
  om')) =
(equivocators_state_project IM'
   final_descriptors s, output)
Hl: ∃ Hex : existing_equivocator_label
          (IM' (projT1 l)) 
          (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM' (projT1 l)) 
       (projT2 l) Hex) = l0
Hprojectx: equivocators_transition_item_project IM'
  final_descriptors
  {|
    l := l;
    input := Some input;
    destination := s;
    output := output
  |} =
Some
  (Some
     {|
       l := l0;
       input := Some input;
       destination :=
         equivocators_state_project IM'
           final_descriptors s;
       output := output
     |}, final_descriptors')
Htl: finite_valid_trace_from SeededXE s []
Hs: valid_state_prop SeededXE
  (finite_trace_last is tr')
Hno_equiv: trace_has_message
  (field_selector VLSM.output) input tr'
Hiom: option_valid_message_prop SeededXE (Some input)
Hlst_trX': valid_state_prop SeededX
  (finite_trace_last
     (equivocators_state_project IM'
        initial_descriptors is) trX')
Htr_to: finite_valid_trace_init_to SeededXE is
  (finite_trace_last is tr') tr'
Hpre_tr_to: finite_constrained_trace_init_to SubFreeE
  is (finite_trace_last is tr') tr'
_Hpr_trX': Some (trX', initial_descriptors) =
Some (trX', initial_descriptors)
Heq_final_stateX': finite_trace_last
  (equivocators_state_project IM'
     initial_descriptors is) trX' =
equivocators_state_project IM'
  final_descriptors'
  (finite_trace_last is tr')
Hs_free: constrained_state_prop SubFreeE
  (finite_trace_last is tr')
Hpre_tr: finite_valid_trace
  (preloaded_with_all_messages_vlsm SubFreeE)
  is tr'
final_descriptors_m, initial_descriptors_m: equivocator_descriptors
  (λ ei : sub_index selection,
     IM (`ei))
trXm: list
  (composite_transition_item
     (λ ei : sub_index selection, IM (`ei)))
Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  (λ ei : sub_index selection,
     IM (`ei))
  final_descriptors_m
  (finite_trace_last is tr')
Hproject_trXm: equivocators_trace_project
  (λ ei : sub_index selection,
     IM (`ei)) final_descriptors_m tr' =
Some (trXm, initial_descriptors_m)
Hex: Exists (field_selector VLSM.output input) trXm
H: ∀ (initial_descriptors : 
   equivocator_descriptors IM') 
  (trX : list (composite_transition_item IM')),
  equivocators_trace_project IM'
    final_descriptors_m tr' =
  Some (trX, initial_descriptors)
  → finite_valid_trace SeededX
      (equivocators_state_project IM'
         initial_descriptors is) trX
Hfinal_descriptors_m_proper: proper_equivocator_descriptors
  (λ ei : sub_index
            selection,
     IM (`ei))
  final_descriptors_m
  (finite_trace_last is
     tr')
Hpr_tr': ∃ initial_descriptors : 
   equivocator_descriptors
     (λ ei : sub_index selection, IM (`ei)),
  proper_equivocator_descriptors
    (λ ei : sub_index selection, IM (`ei))
    initial_descriptors is
  ∧ (let isX :=
       equivocators_state_project
         (λ ei : sub_index selection,
            IM (`ei)) initial_descriptors is
       in
     let final_stateX :=
       equivocators_state_project
         (λ ei : sub_index selection,
            IM (`ei)) final_descriptors_m
         (finite_trace_last is tr') in
     ∃ trX : list
               (composite_transition_item
                  (λ ei : 
                      sub_index selection,
                     IM (`ei))),
       equivocators_trace_project
         (λ ei : sub_index selection,
            IM (`ei)) final_descriptors_m
         tr' =
       Some (trX, initial_descriptors)
       ∧ finite_constrained_trace_init_to
           (free_composite_vlsm
              (λ ei : sub_index selection,
                 IM (`ei))) isX final_stateX
           trX)
option_valid_message_prop SeededX (Some input)
    destruct Hpr_tr' as [initial_descriptors_m' [Hproper_initial_m [trXm' [Hproject_trXm' HtrXm]]]].message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
is: composite_state
  (sub_IM (equivocator_IM IM) selection)
IM':= λ i : sub_index selection, IM (`i): sub_index selection → VLSM message
tr': list
  (composite_transition_item
     (sub_IM (equivocator_IM IM) selection))
s: composite_state
  (sub_IM (equivocator_IM IM) selection)
l: composite_label
  (sub_IM (equivocator_IM IM) selection)
input: message
output: option message
Htr: finite_valid_trace_from SeededXE is tr'
Hlst: finite_valid_trace_from SeededXE
  (finite_trace_last is tr')
  [{|
     l := l;
     input := Some input;
     destination := s;
     output := output
   |}]
Hinit: composite_initial_state_prop
  (sub_IM (equivocator_IM IM) selection) is
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input := Some input;
      destination := s;
      output := output
    |}]): composite_state
  (sub_IM (equivocator_IM IM) selection)
final_descriptors: equivocator_descriptors IM'
Hproper: proper_equivocator_descriptors IM'
  final_descriptors s
initial_descriptors: equivocator_descriptors IM'
trX': list
  (composite_transition_item
     (λ i : sub_index selection, IM (`i)))
l0: composite_label IM'
final_descriptors': equivocator_descriptors IM'
Hpr_trX': equivocators_trace_project
  (λ i : sub_index selection, IM (`i))
  final_descriptors' tr' =
Some (trX', initial_descriptors)
HtrX': finite_valid_trace_from SeededX
  (equivocators_state_project IM'
     initial_descriptors is) trX'
HinitX: composite_initial_state_prop
  (sub_IM IM selection)
  (equivocators_state_project IM'
     initial_descriptors is)
Hproper': proper_equivocator_descriptors IM'
  final_descriptors'
  (finite_trace_last is tr')
Heq_final_descriptors': final_descriptors' =
equivocator_descriptors_update
  IM' final_descriptors
  (projT1 l)
  (final_descriptors'
     (projT1 l))
Hvx: let (i, li) := l0 in
valid li
  (finite_trace_last
     (equivocators_state_project IM'
        initial_descriptors is) trX' i,
   Some input)
Htx: (let (i, li) := l0 in
 let (si', om') :=
   transition li
     (finite_trace_last
        (equivocators_state_project IM'
           initial_descriptors is) trX' i,
      Some input) in
 (state_update IM'
    (finite_trace_last
       (equivocators_state_project IM'
          initial_descriptors is) trX') i si',
  om')) =
(equivocators_state_project IM'
   final_descriptors s, output)
Hl: ∃ Hex : existing_equivocator_label
          (IM' (projT1 l)) 
          (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM' (projT1 l)) 
       (projT2 l) Hex) = l0
Hprojectx: equivocators_transition_item_project IM'
  final_descriptors
  {|
    l := l;
    input := Some input;
    destination := s;
    output := output
  |} =
Some
  (Some
     {|
       l := l0;
       input := Some input;
       destination :=
         equivocators_state_project IM'
           final_descriptors s;
       output := output
     |}, final_descriptors')
Htl: finite_valid_trace_from SeededXE s []
Hs: valid_state_prop SeededXE
  (finite_trace_last is tr')
Hno_equiv: trace_has_message
  (field_selector VLSM.output) input tr'
Hiom: option_valid_message_prop SeededXE (Some input)
Hlst_trX': valid_state_prop SeededX
  (finite_trace_last
     (equivocators_state_project IM'
        initial_descriptors is) trX')
Htr_to: finite_valid_trace_init_to SeededXE is
  (finite_trace_last is tr') tr'
Hpre_tr_to: finite_constrained_trace_init_to SubFreeE
  is (finite_trace_last is tr') tr'
_Hpr_trX': Some (trX', initial_descriptors) =
Some (trX', initial_descriptors)
Heq_final_stateX': finite_trace_last
  (equivocators_state_project IM'
     initial_descriptors is) trX' =
equivocators_state_project IM'
  final_descriptors'
  (finite_trace_last is tr')
Hs_free: constrained_state_prop SubFreeE
  (finite_trace_last is tr')
Hpre_tr: finite_valid_trace
  (preloaded_with_all_messages_vlsm SubFreeE)
  is tr'
final_descriptors_m, initial_descriptors_m: equivocator_descriptors
  (λ ei : sub_index selection,
     IM (`ei))
trXm: list
  (composite_transition_item
     (λ ei : sub_index selection, IM (`ei)))
Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  (λ ei : sub_index selection,
     IM (`ei))
  final_descriptors_m
  (finite_trace_last is tr')
Hproject_trXm: equivocators_trace_project
  (λ ei : sub_index selection,
     IM (`ei)) final_descriptors_m tr' =
Some (trXm, initial_descriptors_m)
Hex: Exists (field_selector VLSM.output input) trXm
H: ∀ (initial_descriptors : 
   equivocator_descriptors IM') 
  (trX : list (composite_transition_item IM')),
  equivocators_trace_project IM'
    final_descriptors_m tr' =
  Some (trX, initial_descriptors)
  → finite_valid_trace SeededX
      (equivocators_state_project IM'
         initial_descriptors is) trX
Hfinal_descriptors_m_proper: proper_equivocator_descriptors
  (λ ei : sub_index
            selection,
     IM (`ei))
  final_descriptors_m
  (finite_trace_last is
     tr')
initial_descriptors_m': equivocator_descriptors
  (λ ei : sub_index selection,
     IM (`ei))
Hproper_initial_m: proper_equivocator_descriptors
  (λ ei : sub_index selection,
     IM (`ei))
  initial_descriptors_m' is
trXm': list
  (composite_transition_item
     (λ ei : sub_index selection, IM (`ei)))
Hproject_trXm': equivocators_trace_project
  (λ ei : sub_index selection,
     IM (`ei)) final_descriptors_m
  tr' =
Some (trXm', initial_descriptors_m')
HtrXm: finite_constrained_trace_init_to
  (free_composite_vlsm
     (λ ei : sub_index selection, IM (`ei)))
  (equivocators_state_project
     (λ ei : sub_index selection, IM (`ei))
     initial_descriptors_m' is)
  (equivocators_state_project
     (λ ei : sub_index selection, IM (`ei))
     final_descriptors_m
     (finite_trace_last is tr')) trXm'
option_valid_message_prop SeededX (Some input)
    specialize (H _ _ Hproject_trXm').message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
is: composite_state
  (sub_IM (equivocator_IM IM) selection)
IM':= λ i : sub_index selection, IM (`i): sub_index selection → VLSM message
tr': list
  (composite_transition_item
     (sub_IM (equivocator_IM IM) selection))
s: composite_state
  (sub_IM (equivocator_IM IM) selection)
l: composite_label
  (sub_IM (equivocator_IM IM) selection)
input: message
output: option message
Htr: finite_valid_trace_from SeededXE is tr'
Hlst: finite_valid_trace_from SeededXE
  (finite_trace_last is tr')
  [{|
     l := l;
     input := Some input;
     destination := s;
     output := output
   |}]
Hinit: composite_initial_state_prop
  (sub_IM (equivocator_IM IM) selection) is
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input := Some input;
      destination := s;
      output := output
    |}]): composite_state
  (sub_IM (equivocator_IM IM) selection)
final_descriptors: equivocator_descriptors IM'
Hproper: proper_equivocator_descriptors IM'
  final_descriptors s
initial_descriptors: equivocator_descriptors IM'
trX': list
  (composite_transition_item
     (λ i : sub_index selection, IM (`i)))
l0: composite_label IM'
final_descriptors': equivocator_descriptors IM'
Hpr_trX': equivocators_trace_project
  (λ i : sub_index selection, IM (`i))
  final_descriptors' tr' =
Some (trX', initial_descriptors)
HtrX': finite_valid_trace_from SeededX
  (equivocators_state_project IM'
     initial_descriptors is) trX'
HinitX: composite_initial_state_prop
  (sub_IM IM selection)
  (equivocators_state_project IM'
     initial_descriptors is)
Hproper': proper_equivocator_descriptors IM'
  final_descriptors'
  (finite_trace_last is tr')
Heq_final_descriptors': final_descriptors' =
equivocator_descriptors_update
  IM' final_descriptors
  (projT1 l)
  (final_descriptors'
     (projT1 l))
Hvx: let (i, li) := l0 in
valid li
  (finite_trace_last
     (equivocators_state_project IM'
        initial_descriptors is) trX' i,
   Some input)
Htx: (let (i, li) := l0 in
 let (si', om') :=
   transition li
     (finite_trace_last
        (equivocators_state_project IM'
           initial_descriptors is) trX' i,
      Some input) in
 (state_update IM'
    (finite_trace_last
       (equivocators_state_project IM'
          initial_descriptors is) trX') i si',
  om')) =
(equivocators_state_project IM'
   final_descriptors s, output)
Hl: ∃ Hex : existing_equivocator_label
          (IM' (projT1 l)) 
          (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM' (projT1 l)) 
       (projT2 l) Hex) = l0
Hprojectx: equivocators_transition_item_project IM'
  final_descriptors
  {|
    l := l;
    input := Some input;
    destination := s;
    output := output
  |} =
Some
  (Some
     {|
       l := l0;
       input := Some input;
       destination :=
         equivocators_state_project IM'
           final_descriptors s;
       output := output
     |}, final_descriptors')
Htl: finite_valid_trace_from SeededXE s []
Hs: valid_state_prop SeededXE
  (finite_trace_last is tr')
Hno_equiv: trace_has_message
  (field_selector VLSM.output) input tr'
Hiom: option_valid_message_prop SeededXE (Some input)
Hlst_trX': valid_state_prop SeededX
  (finite_trace_last
     (equivocators_state_project IM'
        initial_descriptors is) trX')
Htr_to: finite_valid_trace_init_to SeededXE is
  (finite_trace_last is tr') tr'
Hpre_tr_to: finite_constrained_trace_init_to SubFreeE
  is (finite_trace_last is tr') tr'
_Hpr_trX': Some (trX', initial_descriptors) =
Some (trX', initial_descriptors)
Heq_final_stateX': finite_trace_last
  (equivocators_state_project IM'
     initial_descriptors is) trX' =
equivocators_state_project IM'
  final_descriptors'
  (finite_trace_last is tr')
Hs_free: constrained_state_prop SubFreeE
  (finite_trace_last is tr')
Hpre_tr: finite_valid_trace
  (preloaded_with_all_messages_vlsm SubFreeE)
  is tr'
final_descriptors_m, initial_descriptors_m: equivocator_descriptors
  (λ ei : sub_index selection,
     IM (`ei))
trXm: list
  (composite_transition_item
     (λ ei : sub_index selection, IM (`ei)))
Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  (λ ei : sub_index selection,
     IM (`ei))
  final_descriptors_m
  (finite_trace_last is tr')
Hproject_trXm: equivocators_trace_project
  (λ ei : sub_index selection,
     IM (`ei)) final_descriptors_m tr' =
Some (trXm, initial_descriptors_m)
Hex: Exists (field_selector VLSM.output input) trXm
initial_descriptors_m': equivocator_descriptors
  (λ ei : sub_index selection,
     IM (`ei))
trXm': list
  (composite_transition_item
     (λ ei : sub_index selection, IM (`ei)))
H: finite_valid_trace SeededX
  (equivocators_state_project IM'
     initial_descriptors_m' is) trXm'
Hfinal_descriptors_m_proper: proper_equivocator_descriptors
  (λ ei : sub_index
            selection,
     IM (`ei))
  final_descriptors_m
  (finite_trace_last is
     tr')
Hproper_initial_m: proper_equivocator_descriptors
  (λ ei : sub_index selection,
     IM (`ei))
  initial_descriptors_m' is
Hproject_trXm': equivocators_trace_project
  (λ ei : sub_index selection,
     IM (`ei)) final_descriptors_m
  tr' =
Some (trXm', initial_descriptors_m')
HtrXm: finite_constrained_trace_init_to
  (free_composite_vlsm
     (λ ei : sub_index selection, IM (`ei)))
  (equivocators_state_project
     (λ ei : sub_index selection, IM (`ei))
     initial_descriptors_m' is)
  (equivocators_state_project
     (λ ei : sub_index selection, IM (`ei))
     final_descriptors_m
     (finite_trace_last is tr')) trXm'
option_valid_message_prop SeededX (Some input)
    simpl in *.message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
is: composite_state
  (sub_IM (equivocator_IM IM) selection)
IM':= λ i : sub_index selection, IM (`i): sub_index selection → VLSM message
tr': list
  (composite_transition_item
     (sub_IM (equivocator_IM IM) selection))
s: composite_state
  (sub_IM (equivocator_IM IM) selection)
l: composite_label
  (sub_IM (equivocator_IM IM) selection)
input: message
output: option message
Htr: finite_valid_trace_from SeededXE is tr'
Hlst: finite_valid_trace_from SeededXE
  (finite_trace_last is tr')
  [{|
     l := l;
     input := Some input;
     destination := s;
     output := output
   |}]
Hinit: composite_initial_state_prop
  (sub_IM (equivocator_IM IM) selection) is
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input := Some input;
      destination := s;
      output := output
    |}]): composite_state
  (sub_IM (equivocator_IM IM) selection)
final_descriptors: equivocator_descriptors IM'
Hproper: proper_equivocator_descriptors IM'
  final_descriptors s
initial_descriptors: equivocator_descriptors IM'
trX': list
  (composite_transition_item
     (λ i : sub_index selection, IM (`i)))
l0: composite_label IM'
final_descriptors': equivocator_descriptors IM'
Hpr_trX': equivocators_trace_project
  (λ i : sub_index selection, IM (`i))
  final_descriptors' tr' =
Some (trX', initial_descriptors)
HtrX': finite_valid_trace_from SeededX
  (equivocators_state_project IM'
     initial_descriptors is) trX'
HinitX: composite_initial_state_prop
  (sub_IM IM selection)
  (equivocators_state_project IM'
     initial_descriptors is)
Hproper': proper_equivocator_descriptors IM'
  final_descriptors'
  (finite_trace_last is tr')
Heq_final_descriptors': final_descriptors' =
equivocator_descriptors_update
  IM' final_descriptors
  (projT1 l)
  (final_descriptors'
     (projT1 l))
Hvx: let (i, li) := l0 in
valid li
  (finite_trace_last
     (equivocators_state_project IM'
        initial_descriptors is) trX' i,
   Some input)
Htx: (let (i, li) := l0 in
 let (si', om') :=
   transition li
     (finite_trace_last
        (equivocators_state_project IM'
           initial_descriptors is) trX' i,
      Some input) in
 (state_update IM'
    (finite_trace_last
       (equivocators_state_project IM'
          initial_descriptors is) trX') i si',
  om')) =
(equivocators_state_project IM'
   final_descriptors s, output)
Hl: ∃ Hex : existing_equivocator_label
          (IM' (projT1 l)) 
          (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM' (projT1 l)) 
       (projT2 l) Hex) = l0
Hprojectx: equivocators_transition_item_project IM'
  final_descriptors
  {|
    l := l;
    input := Some input;
    destination := s;
    output := output
  |} =
Some
  (Some
     {|
       l := l0;
       input := Some input;
       destination :=
         equivocators_state_project IM'
           final_descriptors s;
       output := output
     |}, final_descriptors')
Htl: finite_valid_trace_from SeededXE s []
Hs: valid_state_prop SeededXE
  (finite_trace_last is tr')
Hno_equiv: trace_has_message
  (field_selector VLSM.output) input tr'
Hiom: option_valid_message_prop SeededXE (Some input)
Hlst_trX': valid_state_prop SeededX
  (finite_trace_last
     (equivocators_state_project IM'
        initial_descriptors is) trX')
Htr_to: finite_valid_trace_init_to SeededXE is
  (finite_trace_last is tr') tr'
Hpre_tr_to: finite_constrained_trace_init_to SubFreeE
  is (finite_trace_last is tr') tr'
_Hpr_trX': Some (trX', initial_descriptors) =
Some (trX', initial_descriptors)
Heq_final_stateX': finite_trace_last
  (equivocators_state_project IM'
     initial_descriptors is) trX' =
equivocators_state_project IM'
  final_descriptors'
  (finite_trace_last is tr')
Hs_free: constrained_state_prop SubFreeE
  (finite_trace_last is tr')
Hpre_tr: finite_valid_trace
  (preloaded_with_all_messages_vlsm SubFreeE)
  is tr'
final_descriptors_m, initial_descriptors_m: equivocator_descriptors
  (λ ei : sub_index selection,
     IM (`ei))
trXm: list
  (composite_transition_item
     (λ ei : sub_index selection, IM (`ei)))
Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  (λ ei : sub_index selection,
     IM (`ei))
  final_descriptors_m
  (finite_trace_last is tr')
Hproject_trXm: equivocators_trace_project
  (λ ei : sub_index selection,
     IM (`ei)) final_descriptors_m tr' =
Some (trXm, initial_descriptors_m)
Hex: Exists (field_selector VLSM.output input) trXm
initial_descriptors_m': equivocator_descriptors
  (λ ei : sub_index selection,
     IM (`ei))
trXm': list
  (composite_transition_item
     (λ ei : sub_index selection, IM (`ei)))
H: finite_valid_trace SeededX
  (equivocators_state_project IM'
     initial_descriptors_m' is) trXm'
Hfinal_descriptors_m_proper: proper_equivocator_descriptors
  (λ ei : sub_index
            selection,
     IM (`ei))
  final_descriptors_m
  (finite_trace_last is
     tr')
Hproper_initial_m: proper_equivocator_descriptors
  (λ ei : sub_index selection,
     IM (`ei))
  initial_descriptors_m' is
Hproject_trXm': equivocators_trace_project
  (λ ei : sub_index selection,
     IM (`ei)) final_descriptors_m
  tr' =
Some (trXm', initial_descriptors_m')
HtrXm: finite_constrained_trace_init_to
  (free_composite_vlsm
     (λ ei : sub_index selection, IM (`ei)))
  (equivocators_state_project
     (λ ei : sub_index selection, IM (`ei))
     initial_descriptors_m' is)
  (equivocators_state_project
     (λ ei : sub_index selection, IM (`ei))
     final_descriptors_m
     (finite_trace_last is tr')) trXm'
option_valid_message_prop SeededX (Some input) rewrite Hproject_trXm in Hproject_trXm'.message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
is: composite_state
  (sub_IM (equivocator_IM IM) selection)
IM':= λ i : sub_index selection, IM (`i): sub_index selection → VLSM message
tr': list
  (composite_transition_item
     (sub_IM (equivocator_IM IM) selection))
s: composite_state
  (sub_IM (equivocator_IM IM) selection)
l: composite_label
  (sub_IM (equivocator_IM IM) selection)
input: message
output: option message
Htr: finite_valid_trace_from SeededXE is tr'
Hlst: finite_valid_trace_from SeededXE
  (finite_trace_last is tr')
  [{|
     l := l;
     input := Some input;
     destination := s;
     output := output
   |}]
Hinit: composite_initial_state_prop
  (sub_IM (equivocator_IM IM) selection) is
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input := Some input;
      destination := s;
      output := output
    |}]): composite_state
  (sub_IM (equivocator_IM IM) selection)
final_descriptors: equivocator_descriptors IM'
Hproper: proper_equivocator_descriptors IM'
  final_descriptors s
initial_descriptors: equivocator_descriptors IM'
trX': list
  (composite_transition_item
     (λ i : sub_index selection, IM (`i)))
l0: composite_label IM'
final_descriptors': equivocator_descriptors IM'
Hpr_trX': equivocators_trace_project
  (λ i : sub_index selection, IM (`i))
  final_descriptors' tr' =
Some (trX', initial_descriptors)
HtrX': finite_valid_trace_from SeededX
  (equivocators_state_project IM'
     initial_descriptors is) trX'
HinitX: composite_initial_state_prop
  (sub_IM IM selection)
  (equivocators_state_project IM'
     initial_descriptors is)
Hproper': proper_equivocator_descriptors IM'
  final_descriptors'
  (finite_trace_last is tr')
Heq_final_descriptors': final_descriptors' =
equivocator_descriptors_update
  IM' final_descriptors
  (projT1 l)
  (final_descriptors'
     (projT1 l))
Hvx: let (i, li) := l0 in
valid li
  (finite_trace_last
     (equivocators_state_project IM'
        initial_descriptors is) trX' i,
   Some input)
Htx: (let (i, li) := l0 in
 let (si', om') :=
   transition li
     (finite_trace_last
        (equivocators_state_project IM'
           initial_descriptors is) trX' i,
      Some input) in
 (state_update IM'
    (finite_trace_last
       (equivocators_state_project IM'
          initial_descriptors is) trX') i si',
  om')) =
(equivocators_state_project IM'
   final_descriptors s, output)
Hl: ∃ Hex : existing_equivocator_label
          (IM' (projT1 l)) 
          (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM' (projT1 l)) 
       (projT2 l) Hex) = l0
Hprojectx: equivocators_transition_item_project IM'
  final_descriptors
  {|
    l := l;
    input := Some input;
    destination := s;
    output := output
  |} =
Some
  (Some
     {|
       l := l0;
       input := Some input;
       destination :=
         equivocators_state_project IM'
           final_descriptors s;
       output := output
     |}, final_descriptors')
Htl: finite_valid_trace_from SeededXE s []
Hs: valid_state_prop SeededXE
  (finite_trace_last is tr')
Hno_equiv: trace_has_message
  (field_selector VLSM.output) input tr'
Hiom: option_valid_message_prop SeededXE (Some input)
Hlst_trX': valid_state_prop SeededX
  (finite_trace_last
     (equivocators_state_project IM'
        initial_descriptors is) trX')
Htr_to: finite_valid_trace_init_to SeededXE is
  (finite_trace_last is tr') tr'
Hpre_tr_to: finite_constrained_trace_init_to SubFreeE
  is (finite_trace_last is tr') tr'
_Hpr_trX': Some (trX', initial_descriptors) =
Some (trX', initial_descriptors)
Heq_final_stateX': finite_trace_last
  (equivocators_state_project IM'
     initial_descriptors is) trX' =
equivocators_state_project IM'
  final_descriptors'
  (finite_trace_last is tr')
Hs_free: constrained_state_prop SubFreeE
  (finite_trace_last is tr')
Hpre_tr: finite_valid_trace
  (preloaded_with_all_messages_vlsm SubFreeE)
  is tr'
final_descriptors_m, initial_descriptors_m: equivocator_descriptors
  (λ ei : sub_index selection,
     IM (`ei))
trXm: list
  (composite_transition_item
     (λ ei : sub_index selection, IM (`ei)))
Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  (λ ei : sub_index selection,
     IM (`ei))
  final_descriptors_m
  (finite_trace_last is tr')
Hproject_trXm: equivocators_trace_project
  (λ ei : sub_index selection,
     IM (`ei)) final_descriptors_m tr' =
Some (trXm, initial_descriptors_m)
Hex: Exists (field_selector VLSM.output input) trXm
initial_descriptors_m': equivocator_descriptors
  (λ ei : sub_index selection,
     IM (`ei))
trXm': list
  (composite_transition_item
     (λ ei : sub_index selection, IM (`ei)))
H: finite_valid_trace SeededX
  (equivocators_state_project IM'
     initial_descriptors_m' is) trXm'
Hfinal_descriptors_m_proper: proper_equivocator_descriptors
  (λ ei : sub_index
            selection,
     IM (`ei))
  final_descriptors_m
  (finite_trace_last is
     tr')
Hproper_initial_m: proper_equivocator_descriptors
  (λ ei : sub_index selection,
     IM (`ei))
  initial_descriptors_m' is
Hproject_trXm': Some (trXm, initial_descriptors_m) =
Some (trXm', initial_descriptors_m')
HtrXm: finite_constrained_trace_init_to
  (free_composite_vlsm
     (λ ei : sub_index selection, IM (`ei)))
  (equivocators_state_project
     (λ ei : sub_index selection, IM (`ei))
     initial_descriptors_m' is)
  (equivocators_state_project
     (λ ei : sub_index selection, IM (`ei))
     final_descriptors_m
     (finite_trace_last is tr')) trXm'
option_valid_message_prop SeededX (Some input)
    inversion Hproject_trXm'.message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
is: composite_state
  (sub_IM (equivocator_IM IM) selection)
IM':= λ i : sub_index selection, IM (`i): sub_index selection → VLSM message
tr': list
  (composite_transition_item
     (sub_IM (equivocator_IM IM) selection))
s: composite_state
  (sub_IM (equivocator_IM IM) selection)
l: composite_label
  (sub_IM (equivocator_IM IM) selection)
input: message
output: option message
Htr: finite_valid_trace_from SeededXE is tr'
Hlst: finite_valid_trace_from SeededXE
  (finite_trace_last is tr')
  [{|
     l := l;
     input := Some input;
     destination := s;
     output := output
   |}]
Hinit: composite_initial_state_prop
  (sub_IM (equivocator_IM IM) selection) is
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input := Some input;
      destination := s;
      output := output
    |}]): composite_state
  (sub_IM (equivocator_IM IM) selection)
final_descriptors: equivocator_descriptors IM'
Hproper: proper_equivocator_descriptors IM'
  final_descriptors s
initial_descriptors: equivocator_descriptors IM'
trX': list
  (composite_transition_item
     (λ i : sub_index selection, IM (`i)))
l0: composite_label IM'
final_descriptors': equivocator_descriptors IM'
Hpr_trX': equivocators_trace_project
  (λ i : sub_index selection, IM (`i))
  final_descriptors' tr' =
Some (trX', initial_descriptors)
HtrX': finite_valid_trace_from SeededX
  (equivocators_state_project IM'
     initial_descriptors is) trX'
HinitX: composite_initial_state_prop
  (sub_IM IM selection)
  (equivocators_state_project IM'
     initial_descriptors is)
Hproper': proper_equivocator_descriptors IM'
  final_descriptors'
  (finite_trace_last is tr')
Heq_final_descriptors': final_descriptors' =
equivocator_descriptors_update
  IM' final_descriptors
  (projT1 l)
  (final_descriptors'
     (projT1 l))
Hvx: let (i, li) := l0 in
valid li
  (finite_trace_last
     (equivocators_state_project IM'
        initial_descriptors is) trX' i,
   Some input)
Htx: (let (i, li) := l0 in
 let (si', om') :=
   transition li
     (finite_trace_last
        (equivocators_state_project IM'
           initial_descriptors is) trX' i,
      Some input) in
 (state_update IM'
    (finite_trace_last
       (equivocators_state_project IM'
          initial_descriptors is) trX') i si',
  om')) =
(equivocators_state_project IM'
   final_descriptors s, output)
Hl: ∃ Hex : existing_equivocator_label
          (IM' (projT1 l)) 
          (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM' (projT1 l)) 
       (projT2 l) Hex) = l0
Hprojectx: equivocators_transition_item_project IM'
  final_descriptors
  {|
    l := l;
    input := Some input;
    destination := s;
    output := output
  |} =
Some
  (Some
     {|
       l := l0;
       input := Some input;
       destination :=
         equivocators_state_project IM'
           final_descriptors s;
       output := output
     |}, final_descriptors')
Htl: finite_valid_trace_from SeededXE s []
Hs: valid_state_prop SeededXE
  (finite_trace_last is tr')
Hno_equiv: trace_has_message
  (field_selector VLSM.output) input tr'
Hiom: option_valid_message_prop SeededXE (Some input)
Hlst_trX': valid_state_prop SeededX
  (finite_trace_last
     (equivocators_state_project IM'
        initial_descriptors is) trX')
Htr_to: finite_valid_trace_init_to SeededXE is
  (finite_trace_last is tr') tr'
Hpre_tr_to: finite_constrained_trace_init_to SubFreeE
  is (finite_trace_last is tr') tr'
_Hpr_trX': Some (trX', initial_descriptors) =
Some (trX', initial_descriptors)
Heq_final_stateX': finite_trace_last
  (equivocators_state_project IM'
     initial_descriptors is) trX' =
equivocators_state_project IM'
  final_descriptors'
  (finite_trace_last is tr')
Hs_free: constrained_state_prop SubFreeE
  (finite_trace_last is tr')
Hpre_tr: finite_valid_trace
  (preloaded_with_all_messages_vlsm SubFreeE)
  is tr'
final_descriptors_m, initial_descriptors_m: equivocator_descriptors
  (λ ei : sub_index selection,
     IM (`ei))
trXm: list
  (composite_transition_item
     (λ ei : sub_index selection, IM (`ei)))
Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  (λ ei : sub_index selection,
     IM (`ei))
  final_descriptors_m
  (finite_trace_last is tr')
Hproject_trXm: equivocators_trace_project
  (λ ei : sub_index selection,
     IM (`ei)) final_descriptors_m tr' =
Some (trXm, initial_descriptors_m)
Hex: Exists (field_selector VLSM.output input) trXm
initial_descriptors_m': equivocator_descriptors
  (λ ei : sub_index selection,
     IM (`ei))
trXm': list
  (composite_transition_item
     (λ ei : sub_index selection, IM (`ei)))
H: finite_valid_trace SeededX
  (equivocators_state_project IM'
     initial_descriptors_m' is) trXm'
Hfinal_descriptors_m_proper: proper_equivocator_descriptors
  (λ ei : sub_index
            selection,
     IM (`ei))
  final_descriptors_m
  (finite_trace_last is
     tr')
Hproper_initial_m: proper_equivocator_descriptors
  (λ ei : sub_index selection,
     IM (`ei))
  initial_descriptors_m' is
Hproject_trXm': Some (trXm, initial_descriptors_m) =
Some (trXm', initial_descriptors_m')
HtrXm: finite_constrained_trace_init_to
  (free_composite_vlsm
     (λ ei : sub_index selection, IM (`ei)))
  (equivocators_state_project
     (λ ei : sub_index selection, IM (`ei))
     initial_descriptors_m' is)
  (equivocators_state_project
     (λ ei : sub_index selection, IM (`ei))
     final_descriptors_m
     (finite_trace_last is tr')) trXm'
H2: trXm = trXm'
H3: initial_descriptors_m = initial_descriptors_m'
option_valid_message_prop SeededX (Some input) subst trXm' initial_descriptors_m'.message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
is: composite_state
  (sub_IM (equivocator_IM IM) selection)
IM':= λ i : sub_index selection, IM (`i): sub_index selection → VLSM message
tr': list
  (composite_transition_item
     (sub_IM (equivocator_IM IM) selection))
s: composite_state
  (sub_IM (equivocator_IM IM) selection)
l: composite_label
  (sub_IM (equivocator_IM IM) selection)
input: message
output: option message
Htr: finite_valid_trace_from SeededXE is tr'
Hlst: finite_valid_trace_from SeededXE
  (finite_trace_last is tr')
  [{|
     l := l;
     input := Some input;
     destination := s;
     output := output
   |}]
Hinit: composite_initial_state_prop
  (sub_IM (equivocator_IM IM) selection) is
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input := Some input;
      destination := s;
      output := output
    |}]): composite_state
  (sub_IM (equivocator_IM IM) selection)
final_descriptors: equivocator_descriptors IM'
Hproper: proper_equivocator_descriptors IM'
  final_descriptors s
initial_descriptors: equivocator_descriptors IM'
trX': list
  (composite_transition_item
     (λ i : sub_index selection, IM (`i)))
l0: composite_label IM'
final_descriptors': equivocator_descriptors IM'
Hpr_trX': equivocators_trace_project
  (λ i : sub_index selection, IM (`i))
  final_descriptors' tr' =
Some (trX', initial_descriptors)
HtrX': finite_valid_trace_from SeededX
  (equivocators_state_project IM'
     initial_descriptors is) trX'
HinitX: composite_initial_state_prop
  (sub_IM IM selection)
  (equivocators_state_project IM'
     initial_descriptors is)
Hproper': proper_equivocator_descriptors IM'
  final_descriptors'
  (finite_trace_last is tr')
Heq_final_descriptors': final_descriptors' =
equivocator_descriptors_update
  IM' final_descriptors
  (projT1 l)
  (final_descriptors'
     (projT1 l))
Hvx: let (i, li) := l0 in
valid li
  (finite_trace_last
     (equivocators_state_project IM'
        initial_descriptors is) trX' i,
   Some input)
Htx: (let (i, li) := l0 in
 let (si', om') :=
   transition li
     (finite_trace_last
        (equivocators_state_project IM'
           initial_descriptors is) trX' i,
      Some input) in
 (state_update IM'
    (finite_trace_last
       (equivocators_state_project IM'
          initial_descriptors is) trX') i si',
  om')) =
(equivocators_state_project IM'
   final_descriptors s, output)
Hl: ∃ Hex : existing_equivocator_label
          (IM' (projT1 l)) 
          (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM' (projT1 l)) 
       (projT2 l) Hex) = l0
Hprojectx: equivocators_transition_item_project IM'
  final_descriptors
  {|
    l := l;
    input := Some input;
    destination := s;
    output := output
  |} =
Some
  (Some
     {|
       l := l0;
       input := Some input;
       destination :=
         equivocators_state_project IM'
           final_descriptors s;
       output := output
     |}, final_descriptors')
Htl: finite_valid_trace_from SeededXE s []
Hs: valid_state_prop SeededXE
  (finite_trace_last is tr')
Hno_equiv: trace_has_message
  (field_selector VLSM.output) input tr'
Hiom: option_valid_message_prop SeededXE (Some input)
Hlst_trX': valid_state_prop SeededX
  (finite_trace_last
     (equivocators_state_project IM'
        initial_descriptors is) trX')
Htr_to: finite_valid_trace_init_to SeededXE is
  (finite_trace_last is tr') tr'
Hpre_tr_to: finite_constrained_trace_init_to SubFreeE
  is (finite_trace_last is tr') tr'
_Hpr_trX': Some (trX', initial_descriptors) =
Some (trX', initial_descriptors)
Heq_final_stateX': finite_trace_last
  (equivocators_state_project IM'
     initial_descriptors is) trX' =
equivocators_state_project IM'
  final_descriptors'
  (finite_trace_last is tr')
Hs_free: constrained_state_prop SubFreeE
  (finite_trace_last is tr')
Hpre_tr: finite_valid_trace
  (preloaded_with_all_messages_vlsm SubFreeE)
  is tr'
final_descriptors_m, initial_descriptors_m: equivocator_descriptors
  (λ ei : sub_index selection,
     IM (`ei))
trXm: list
  (composite_transition_item
     (λ ei : sub_index selection, IM (`ei)))
Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  (λ ei : sub_index selection,
     IM (`ei))
  final_descriptors_m
  (finite_trace_last is tr')
Hproject_trXm: equivocators_trace_project
  (λ ei : sub_index selection,
     IM (`ei)) final_descriptors_m tr' =
Some (trXm, initial_descriptors_m)
Hex: Exists (field_selector VLSM.output input) trXm
H: finite_valid_trace SeededX
  (equivocators_state_project IM'
     initial_descriptors_m is) trXm
Hfinal_descriptors_m_proper: proper_equivocator_descriptors
  (λ ei : sub_index
            selection,
     IM (`ei))
  final_descriptors_m
  (finite_trace_last is
     tr')
Hproject_trXm': Some (trXm, initial_descriptors_m) =
Some (trXm, initial_descriptors_m)
HtrXm: finite_constrained_trace_init_to
  (free_composite_vlsm
     (λ ei : sub_index selection, IM (`ei)))
  (equivocators_state_project
     (λ ei : sub_index selection, IM (`ei))
     initial_descriptors_m is)
  (equivocators_state_project
     (λ ei : sub_index selection, IM (`ei))
     final_descriptors_m
     (finite_trace_last is tr')) trXm
Hproper_initial_m: proper_equivocator_descriptors
  (λ ei : sub_index selection,
     IM (`ei))
  initial_descriptors_m is
option_valid_message_prop SeededX (Some input) clear Hproject_trXm'.message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
is: composite_state
  (sub_IM (equivocator_IM IM) selection)
IM':= λ i : sub_index selection, IM (`i): sub_index selection → VLSM message
tr': list
  (composite_transition_item
     (sub_IM (equivocator_IM IM) selection))
s: composite_state
  (sub_IM (equivocator_IM IM) selection)
l: composite_label
  (sub_IM (equivocator_IM IM) selection)
input: message
output: option message
Htr: finite_valid_trace_from SeededXE is tr'
Hlst: finite_valid_trace_from SeededXE
  (finite_trace_last is tr')
  [{|
     l := l;
     input := Some input;
     destination := s;
     output := output
   |}]
Hinit: composite_initial_state_prop
  (sub_IM (equivocator_IM IM) selection) is
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input := Some input;
      destination := s;
      output := output
    |}]): composite_state
  (sub_IM (equivocator_IM IM) selection)
final_descriptors: equivocator_descriptors IM'
Hproper: proper_equivocator_descriptors IM'
  final_descriptors s
initial_descriptors: equivocator_descriptors IM'
trX': list
  (composite_transition_item
     (λ i : sub_index selection, IM (`i)))
l0: composite_label IM'
final_descriptors': equivocator_descriptors IM'
Hpr_trX': equivocators_trace_project
  (λ i : sub_index selection, IM (`i))
  final_descriptors' tr' =
Some (trX', initial_descriptors)
HtrX': finite_valid_trace_from SeededX
  (equivocators_state_project IM'
     initial_descriptors is) trX'
HinitX: composite_initial_state_prop
  (sub_IM IM selection)
  (equivocators_state_project IM'
     initial_descriptors is)
Hproper': proper_equivocator_descriptors IM'
  final_descriptors'
  (finite_trace_last is tr')
Heq_final_descriptors': final_descriptors' =
equivocator_descriptors_update
  IM' final_descriptors
  (projT1 l)
  (final_descriptors'
     (projT1 l))
Hvx: let (i, li) := l0 in
valid li
  (finite_trace_last
     (equivocators_state_project IM'
        initial_descriptors is) trX' i,
   Some input)
Htx: (let (i, li) := l0 in
 let (si', om') :=
   transition li
     (finite_trace_last
        (equivocators_state_project IM'
           initial_descriptors is) trX' i,
      Some input) in
 (state_update IM'
    (finite_trace_last
       (equivocators_state_project IM'
          initial_descriptors is) trX') i si',
  om')) =
(equivocators_state_project IM'
   final_descriptors s, output)
Hl: ∃ Hex : existing_equivocator_label
          (IM' (projT1 l)) 
          (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM' (projT1 l)) 
       (projT2 l) Hex) = l0
Hprojectx: equivocators_transition_item_project IM'
  final_descriptors
  {|
    l := l;
    input := Some input;
    destination := s;
    output := output
  |} =
Some
  (Some
     {|
       l := l0;
       input := Some input;
       destination :=
         equivocators_state_project IM'
           final_descriptors s;
       output := output
     |}, final_descriptors')
Htl: finite_valid_trace_from SeededXE s []
Hs: valid_state_prop SeededXE
  (finite_trace_last is tr')
Hno_equiv: trace_has_message
  (field_selector VLSM.output) input tr'
Hiom: option_valid_message_prop SeededXE (Some input)
Hlst_trX': valid_state_prop SeededX
  (finite_trace_last
     (equivocators_state_project IM'
        initial_descriptors is) trX')
Htr_to: finite_valid_trace_init_to SeededXE is
  (finite_trace_last is tr') tr'
Hpre_tr_to: finite_constrained_trace_init_to SubFreeE
  is (finite_trace_last is tr') tr'
_Hpr_trX': Some (trX', initial_descriptors) =
Some (trX', initial_descriptors)
Heq_final_stateX': finite_trace_last
  (equivocators_state_project IM'
     initial_descriptors is) trX' =
equivocators_state_project IM'
  final_descriptors'
  (finite_trace_last is tr')
Hs_free: constrained_state_prop SubFreeE
  (finite_trace_last is tr')
Hpre_tr: finite_valid_trace
  (preloaded_with_all_messages_vlsm SubFreeE)
  is tr'
final_descriptors_m, initial_descriptors_m: equivocator_descriptors
  (λ ei : sub_index selection,
     IM (`ei))
trXm: list
  (composite_transition_item
     (λ ei : sub_index selection, IM (`ei)))
Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  (λ ei : sub_index selection,
     IM (`ei))
  final_descriptors_m
  (finite_trace_last is tr')
Hproject_trXm: equivocators_trace_project
  (λ ei : sub_index selection,
     IM (`ei)) final_descriptors_m tr' =
Some (trXm, initial_descriptors_m)
Hex: Exists (field_selector VLSM.output input) trXm
H: finite_valid_trace SeededX
  (equivocators_state_project IM'
     initial_descriptors_m is) trXm
Hfinal_descriptors_m_proper: proper_equivocator_descriptors
  (λ ei : sub_index
            selection,
     IM (`ei))
  final_descriptors_m
  (finite_trace_last is
     tr')
HtrXm: finite_constrained_trace_init_to
  (free_composite_vlsm
     (λ ei : sub_index selection, IM (`ei)))
  (equivocators_state_project
     (λ ei : sub_index selection, IM (`ei))
     initial_descriptors_m is)
  (equivocators_state_project
     (λ ei : sub_index selection, IM (`ei))
     final_descriptors_m
     (finite_trace_last is tr')) trXm
Hproper_initial_m: proper_equivocator_descriptors
  (λ ei : sub_index selection,
     IM (`ei))
  initial_descriptors_m is
option_valid_message_prop SeededX (Some input)
    apply option_valid_message_Some.message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
is: composite_state
  (sub_IM (equivocator_IM IM) selection)
IM':= λ i : sub_index selection, IM (`i): sub_index selection → VLSM message
tr': list
  (composite_transition_item
     (sub_IM (equivocator_IM IM) selection))
s: composite_state
  (sub_IM (equivocator_IM IM) selection)
l: composite_label
  (sub_IM (equivocator_IM IM) selection)
input: message
output: option message
Htr: finite_valid_trace_from SeededXE is tr'
Hlst: finite_valid_trace_from SeededXE
  (finite_trace_last is tr')
  [{|
     l := l;
     input := Some input;
     destination := s;
     output := output
   |}]
Hinit: composite_initial_state_prop
  (sub_IM (equivocator_IM IM) selection) is
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input := Some input;
      destination := s;
      output := output
    |}]): composite_state
  (sub_IM (equivocator_IM IM) selection)
final_descriptors: equivocator_descriptors IM'
Hproper: proper_equivocator_descriptors IM'
  final_descriptors s
initial_descriptors: equivocator_descriptors IM'
trX': list
  (composite_transition_item
     (λ i : sub_index selection, IM (`i)))
l0: composite_label IM'
final_descriptors': equivocator_descriptors IM'
Hpr_trX': equivocators_trace_project
  (λ i : sub_index selection, IM (`i))
  final_descriptors' tr' =
Some (trX', initial_descriptors)
HtrX': finite_valid_trace_from SeededX
  (equivocators_state_project IM'
     initial_descriptors is) trX'
HinitX: composite_initial_state_prop
  (sub_IM IM selection)
  (equivocators_state_project IM'
     initial_descriptors is)
Hproper': proper_equivocator_descriptors IM'
  final_descriptors'
  (finite_trace_last is tr')
Heq_final_descriptors': final_descriptors' =
equivocator_descriptors_update
  IM' final_descriptors
  (projT1 l)
  (final_descriptors'
     (projT1 l))
Hvx: let (i, li) := l0 in
valid li
  (finite_trace_last
     (equivocators_state_project IM'
        initial_descriptors is) trX' i,
   Some input)
Htx: (let (i, li) := l0 in
 let (si', om') :=
   transition li
     (finite_trace_last
        (equivocators_state_project IM'
           initial_descriptors is) trX' i,
      Some input) in
 (state_update IM'
    (finite_trace_last
       (equivocators_state_project IM'
          initial_descriptors is) trX') i si',
  om')) =
(equivocators_state_project IM'
   final_descriptors s, output)
Hl: ∃ Hex : existing_equivocator_label
          (IM' (projT1 l)) 
          (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM' (projT1 l)) 
       (projT2 l) Hex) = l0
Hprojectx: equivocators_transition_item_project IM'
  final_descriptors
  {|
    l := l;
    input := Some input;
    destination := s;
    output := output
  |} =
Some
  (Some
     {|
       l := l0;
       input := Some input;
       destination :=
         equivocators_state_project IM'
           final_descriptors s;
       output := output
     |}, final_descriptors')
Htl: finite_valid_trace_from SeededXE s []
Hs: valid_state_prop SeededXE
  (finite_trace_last is tr')
Hno_equiv: trace_has_message
  (field_selector VLSM.output) input tr'
Hiom: option_valid_message_prop SeededXE (Some input)
Hlst_trX': valid_state_prop SeededX
  (finite_trace_last
     (equivocators_state_project IM'
        initial_descriptors is) trX')
Htr_to: finite_valid_trace_init_to SeededXE is
  (finite_trace_last is tr') tr'
Hpre_tr_to: finite_constrained_trace_init_to SubFreeE
  is (finite_trace_last is tr') tr'
_Hpr_trX': Some (trX', initial_descriptors) =
Some (trX', initial_descriptors)
Heq_final_stateX': finite_trace_last
  (equivocators_state_project IM'
     initial_descriptors is) trX' =
equivocators_state_project IM'
  final_descriptors'
  (finite_trace_last is tr')
Hs_free: constrained_state_prop SubFreeE
  (finite_trace_last is tr')
Hpre_tr: finite_valid_trace
  (preloaded_with_all_messages_vlsm SubFreeE)
  is tr'
final_descriptors_m, initial_descriptors_m: equivocator_descriptors
  (λ ei : sub_index selection,
     IM (`ei))
trXm: list
  (composite_transition_item
     (λ ei : sub_index selection, IM (`ei)))
Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  (λ ei : sub_index selection,
     IM (`ei))
  final_descriptors_m
  (finite_trace_last is tr')
Hproject_trXm: equivocators_trace_project
  (λ ei : sub_index selection,
     IM (`ei)) final_descriptors_m tr' =
Some (trXm, initial_descriptors_m)
Hex: Exists (field_selector VLSM.output input) trXm
H: finite_valid_trace SeededX
  (equivocators_state_project IM'
     initial_descriptors_m is) trXm
Hfinal_descriptors_m_proper: proper_equivocator_descriptors
  (λ ei : sub_index
            selection,
     IM (`ei))
  final_descriptors_m
  (finite_trace_last is
     tr')
HtrXm: finite_constrained_trace_init_to
  (free_composite_vlsm
     (λ ei : sub_index selection, IM (`ei)))
  (equivocators_state_project
     (λ ei : sub_index selection, IM (`ei))
     initial_descriptors_m is)
  (equivocators_state_project
     (λ ei : sub_index selection, IM (`ei))
     final_descriptors_m
     (finite_trace_last is tr')) trXm
Hproper_initial_m: proper_equivocator_descriptors
  (λ ei : sub_index selection,
     IM (`ei))
  initial_descriptors_m is
valid_message_prop SeededX input
    by apply (valid_trace_output_is_valid _ _ _ (proj1 H) _ Hex).
Qed.

Lemma SeededXE_incl_PreFreeE
  : VLSM_incl SeededXE SubPreFreeE.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
VLSM_incl SeededXE SubPreFreeE
Proof.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
VLSM_incl SeededXE SubPreFreeE
  apply basic_VLSM_strong_incl.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
strong_incl_initial_state_preservation
  (preloaded_vlsm_machine
     (composite_vlsm
        (sub_IM (equivocator_IM IM) selection)
        (no_equivocations_additional_constraint_with_preloaded
           (sub_IM (equivocator_IM IM) selection)
           (free_constraint
              (sub_IM (equivocator_IM IM) selection))
           seed)) seed)
  (preloaded_vlsm_machine SubFreeE
     (λ _ : message, True))
message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
strong_incl_initial_message_preservation
  (preloaded_vlsm_machine
     (composite_vlsm
        (sub_IM (equivocator_IM IM) selection)
        (no_equivocations_additional_constraint_with_preloaded
           (sub_IM (equivocator_IM IM) selection)
           (free_constraint
              (sub_IM (equivocator_IM IM) selection))
           seed)) seed)
  (preloaded_vlsm_machine SubFreeE
     (λ _ : message, True))
message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
strong_incl_valid_preservation
  (preloaded_vlsm_machine
     (composite_vlsm
        (sub_IM (equivocator_IM IM) selection)
        (no_equivocations_additional_constraint_with_preloaded
           (sub_IM (equivocator_IM IM) selection)
           (free_constraint
              (sub_IM (equivocator_IM IM) selection))
           seed)) seed)
  (preloaded_vlsm_machine SubFreeE
     (λ _ : message, True))
message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
strong_incl_transition_preservation
  (preloaded_vlsm_machine
     (composite_vlsm
        (sub_IM (equivocator_IM IM) selection)
        (no_equivocations_additional_constraint_with_preloaded
           (sub_IM (equivocator_IM IM) selection)
           (free_constraint
              (sub_IM (equivocator_IM IM) selection))
           seed)) seed)
  (preloaded_vlsm_machine SubFreeE
     (λ _ : message, True))
  -message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
strong_incl_initial_state_preservation
  (preloaded_vlsm_machine
     (composite_vlsm
        (sub_IM (equivocator_IM IM) selection)
        (no_equivocations_additional_constraint_with_preloaded
           (sub_IM (equivocator_IM IM) selection)
           (free_constraint
              (sub_IM (equivocator_IM IM) selection))
           seed)) seed)
  (preloaded_vlsm_machine SubFreeE
     (λ _ : message, True)) by intros s Hn n; itauto.
  -message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
strong_incl_initial_message_preservation
  (preloaded_vlsm_machine
     (composite_vlsm
        (sub_IM (equivocator_IM IM) selection)
        (no_equivocations_additional_constraint_with_preloaded
           (sub_IM (equivocator_IM IM) selection)
           (free_constraint
              (sub_IM (equivocator_IM IM) selection))
           seed)) seed)
  (preloaded_vlsm_machine SubFreeE
     (λ _ : message, True)) by cbv; itauto.
  -message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
strong_incl_valid_preservation
  (preloaded_vlsm_machine
     (composite_vlsm
        (sub_IM (equivocator_IM IM) selection)
        (no_equivocations_additional_constraint_with_preloaded
           (sub_IM (equivocator_IM IM) selection)
           (free_constraint
              (sub_IM (equivocator_IM IM) selection))
           seed)) seed)
  (preloaded_vlsm_machine SubFreeE
     (λ _ : message, True)) by destruct 1.
  -message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
strong_incl_transition_preservation
  (preloaded_vlsm_machine
     (composite_vlsm
        (sub_IM (equivocator_IM IM) selection)
        (no_equivocations_additional_constraint_with_preloaded
           (sub_IM (equivocator_IM IM) selection)
           (free_constraint
              (sub_IM (equivocator_IM IM) selection))
           seed)) seed)
  (preloaded_vlsm_machine SubFreeE
     (λ _ : message, True)) by cbv; itauto.
Qed.

Lemma PreSeededXE_incl_PreFreeE
  : VLSM_incl (preloaded_with_all_messages_vlsm SeededXE) SubPreFreeE.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
VLSM_incl (preloaded_with_all_messages_vlsm SeededXE)
  SubPreFreeE
Proof.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
VLSM_incl (preloaded_with_all_messages_vlsm SeededXE)
  SubPreFreeE
  by apply basic_VLSM_incl_preloaded; [intro | inversion 1 | intro].
Qed.

Lemma SeededXE_SeededX_vlsm_partial_projection
  (IM' := fun i : sub_index selection => IM (`i))
  (final_descriptors : equivocator_descriptors IM')
  : VLSM_partial_projection SeededXE SeededX
      (equivocators_partial_trace_project IM' final_descriptors).message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
IM':= λ i : sub_index selection, IM (`i): sub_index selection → VLSM message
final_descriptors: equivocator_descriptors IM'
VLSM_partial_projection SeededXE SeededX
  (equivocators_partial_trace_project IM'
     final_descriptors)
Proof.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
IM':= λ i : sub_index selection, IM (`i): sub_index selection → VLSM message
final_descriptors: equivocator_descriptors IM'
VLSM_partial_projection SeededXE SeededX
  (equivocators_partial_trace_project IM'
     final_descriptors)
  split; [split |].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
IM':= λ i : sub_index selection, IM (`i): sub_index selection → VLSM message
final_descriptors: equivocator_descriptors IM'
∀ (sX : state SeededXE) (trX : list transition_item) 
  (sY : state SeededX) (trY : list transition_item),
  equivocators_partial_trace_project IM'
    final_descriptors (sX, trX) = Some (sY, trY)
  → ∀ (s'X : state SeededXE) (preX : list
                                       transition_item),
      finite_trace_last s'X preX = sX
      → finite_valid_trace_from SeededXE s'X
          (preX ++ trX)
        → ∃ (s'Y : state SeededX) (preY : list
                                            transition_item),
            equivocators_partial_trace_project IM'
              final_descriptors (s'X, preX ++ trX) =
            Some (s'Y, preY ++ trY)
            ∧ finite_trace_last s'Y preY = sY
message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
IM':= λ i : sub_index selection, IM (`i): sub_index selection → VLSM message
final_descriptors: equivocator_descriptors IM'
∀ (sX : state SeededXE) 
  (trX : list transition_item) 
  (sY : state SeededX) 
  (trY : list transition_item),
  equivocators_partial_trace_project IM'
    final_descriptors (
    sX, trX) = Some (sY, trY)
  → finite_valid_trace SeededXE sX trX
    → finite_valid_trace SeededX sY trY
  -message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
IM':= λ i : sub_index selection, IM (`i): sub_index selection → VLSM message
final_descriptors: equivocator_descriptors IM'
∀ (sX : state SeededXE) (trX : list transition_item) 
  (sY : state SeededX) (trY : list transition_item),
  equivocators_partial_trace_project IM'
    final_descriptors (sX, trX) = Some (sY, trY)
  → ∀ (s'X : state SeededXE) (preX : list
                                       transition_item),
      finite_trace_last s'X preX = sX
      → finite_valid_trace_from SeededXE s'X
          (preX ++ trX)
        → ∃ (s'Y : state SeededX) (preY : list
                                            transition_item),
            equivocators_partial_trace_project IM'
              final_descriptors (s'X, preX ++ trX) =
            Some (s'Y, preY ++ trY)
            ∧ finite_trace_last s'Y preY = sY intros s tr sX trX Hpr_tr s_pre pre Hs_lst Hpre_tr.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
IM':= λ i : sub_index selection, IM (`i): sub_index selection → VLSM message
final_descriptors: equivocator_descriptors IM'
s: state SeededXE
tr: list transition_item
sX: state SeededX
trX: list transition_item
Hpr_tr: equivocators_partial_trace_project IM'
  final_descriptors (s, tr) = 
Some (sX, trX)
s_pre: state SeededXE
pre: list transition_item
Hs_lst: finite_trace_last s_pre pre = s
Hpre_tr: finite_valid_trace_from SeededXE s_pre
  (pre ++ tr)
∃ (s'Y : state SeededX) (preY : list transition_item),
  equivocators_partial_trace_project IM'
    final_descriptors (s_pre, pre ++ tr) =
  Some (s'Y, preY ++ trX)
  ∧ finite_trace_last s'Y preY = sX
    assert
      (HPreFree_pre_tr : finite_constrained_trace_from SubFreeE s_pre (pre ++ tr)).message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
IM':= λ i : sub_index selection, IM (`i): sub_index selection → VLSM message
final_descriptors: equivocator_descriptors IM'
s: state SeededXE
tr: list transition_item
sX: state SeededX
trX: list transition_item
Hpr_tr: equivocators_partial_trace_project IM'
  final_descriptors (s, tr) = 
Some (sX, trX)
s_pre: state SeededXE
pre: list transition_item
Hs_lst: finite_trace_last s_pre pre = s
Hpre_tr: finite_valid_trace_from SeededXE s_pre
  (pre ++ tr)
finite_constrained_trace_from SubFreeE s_pre
  (pre ++ tr)
message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
IM':= λ i : sub_index selection, IM (`i): sub_index selection → VLSM message
final_descriptors: equivocator_descriptors IM'
s: state SeededXE
tr: list transition_item
sX: state SeededX
trX: list transition_item
Hpr_tr: equivocators_partial_trace_project IM'
  final_descriptors (s, tr) = 
Some (sX, trX)
s_pre: state SeededXE
pre: list transition_item
Hs_lst: finite_trace_last s_pre pre = s
Hpre_tr: finite_valid_trace_from SeededXE s_pre
  (pre ++ tr)
HPreFree_pre_tr: finite_constrained_trace_from
  SubFreeE s_pre (pre ++ tr)
∃ (s'Y : state SeededX) (preY : list transition_item),
  equivocators_partial_trace_project IM'
    final_descriptors (
    s_pre, pre ++ tr) = 
  Some (s'Y, preY ++ trX)
  ∧ finite_trace_last s'Y preY = sX
    {message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
IM':= λ i : sub_index selection, IM (`i): sub_index selection → VLSM message
final_descriptors: equivocator_descriptors IM'
s: state SeededXE
tr: list transition_item
sX: state SeededX
trX: list transition_item
Hpr_tr: equivocators_partial_trace_project IM'
  final_descriptors (s, tr) = 
Some (sX, trX)
s_pre: state SeededXE
pre: list transition_item
Hs_lst: finite_trace_last s_pre pre = s
Hpre_tr: finite_valid_trace_from SeededXE s_pre
  (pre ++ tr)
finite_constrained_trace_from SubFreeE s_pre
  (pre ++ tr)
      revert Hpre_tr; apply VLSM_incl_finite_valid_trace_from.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
IM':= λ i : sub_index selection, IM (`i): sub_index selection → VLSM message
final_descriptors: equivocator_descriptors IM'
s: state SeededXE
tr: list transition_item
sX: state SeededX
trX: list transition_item
Hpr_tr: equivocators_partial_trace_project IM'
  final_descriptors (s, tr) = 
Some (sX, trX)
s_pre: state SeededXE
pre: list transition_item
Hs_lst: finite_trace_last s_pre pre = s
VLSM_incl_part
  (preloaded_vlsm_machine
     (composite_vlsm
        (sub_IM (equivocator_IM IM) selection)
        (no_equivocations_additional_constraint_with_preloaded
           (sub_IM (equivocator_IM IM) selection)
           (free_constraint
              (sub_IM (equivocator_IM IM) selection))
           seed)) seed)
  (preloaded_vlsm_machine SubFreeE
     (λ _ : message, True))
      by apply SeededXE_incl_PreFreeE.
    }message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
IM':= λ i : sub_index selection, IM (`i): sub_index selection → VLSM message
final_descriptors: equivocator_descriptors IM'
s: state SeededXE
tr: list transition_item
sX: state SeededX
trX: list transition_item
Hpr_tr: equivocators_partial_trace_project IM'
  final_descriptors (s, tr) = 
Some (sX, trX)
s_pre: state SeededXE
pre: list transition_item
Hs_lst: finite_trace_last s_pre pre = s
Hpre_tr: finite_valid_trace_from SeededXE s_pre
  (pre ++ tr)
HPreFree_pre_tr: finite_constrained_trace_from
  SubFreeE s_pre (pre ++ tr)
∃ (s'Y : state SeededX) (preY : list transition_item),
  equivocators_partial_trace_project IM'
    final_descriptors (s_pre, pre ++ tr) =
  Some (s'Y, preY ++ trX)
  ∧ finite_trace_last s'Y preY = sX
    clear Hpre_tr.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
IM':= λ i : sub_index selection, IM (`i): sub_index selection → VLSM message
final_descriptors: equivocator_descriptors IM'
s: state SeededXE
tr: list transition_item
sX: state SeededX
trX: list transition_item
Hpr_tr: equivocators_partial_trace_project IM'
  final_descriptors (s, tr) = 
Some (sX, trX)
s_pre: state SeededXE
pre: list transition_item
Hs_lst: finite_trace_last s_pre pre = s
HPreFree_pre_tr: finite_constrained_trace_from
  SubFreeE s_pre (pre ++ tr)
∃ (s'Y : state SeededX) (preY : list transition_item),
  equivocators_partial_trace_project IM'
    final_descriptors (s_pre, pre ++ tr) =
  Some (s'Y, preY ++ trX)
  ∧ finite_trace_last s'Y preY = sX revert s tr sX trX Hpr_tr s_pre pre Hs_lst HPreFree_pre_tr.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
IM':= λ i : sub_index selection, IM (`i): sub_index selection → VLSM message
final_descriptors: equivocator_descriptors IM'
∀ (s : state SeededXE) (tr : list transition_item) 
  (sX : state SeededX) (trX : list transition_item),
  equivocators_partial_trace_project IM'
    final_descriptors (s, tr) = Some (sX, trX)
  → ∀ (s_pre : state SeededXE) (pre : list
                                        transition_item),
      finite_trace_last s_pre pre = s
      → finite_constrained_trace_from SubFreeE s_pre
          (pre ++ tr)
        → ∃ (s'Y : state SeededX) (preY : list
                                            transition_item),
            equivocators_partial_trace_project IM'
              final_descriptors (s_pre, pre ++ tr) =
            Some (s'Y, preY ++ trX)
            ∧ finite_trace_last s'Y preY = sX
    by apply equivocators_partial_trace_project_extends_left.
  -message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
IM':= λ i : sub_index selection, IM (`i): sub_index selection → VLSM message
final_descriptors: equivocator_descriptors IM'
∀ (sX : state SeededXE) (trX : list transition_item) 
  (sY : state SeededX) (trY : list transition_item),
  equivocators_partial_trace_project IM'
    final_descriptors (sX, trX) = Some (sY, trY)
  → finite_valid_trace SeededXE sX trX
    → finite_valid_trace SeededX sY trY intros s tr sX trX Hpr_tr Htr.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
IM':= λ i : sub_index selection, IM (`i): sub_index selection → VLSM message
final_descriptors: equivocator_descriptors IM'
s: state SeededXE
tr: list transition_item
sX: state SeededX
trX: list transition_item
Hpr_tr: equivocators_partial_trace_project IM'
  final_descriptors (s, tr) = 
Some (sX, trX)
Htr: finite_valid_trace SeededXE s tr
finite_valid_trace SeededX sX trX
    destruct (destruct_equivocators_partial_trace_project _  Hpr_tr)
      as [Hnot_equiv [initial_descriptors [Htr_project Hs_project]]].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
IM':= λ i : sub_index selection, IM (`i): sub_index selection → VLSM message
final_descriptors: equivocator_descriptors IM'
s: state SeededXE
tr: list transition_item
sX: state SeededX
trX: list transition_item
Hpr_tr: equivocators_partial_trace_project IM'
  final_descriptors (s, tr) = 
Some (sX, trX)
Htr: finite_valid_trace SeededXE s tr
Hnot_equiv: not_equivocating_equivocator_descriptors
  (sub_IM IM selection) final_descriptors
  (finite_trace_last s tr)
initial_descriptors: equivocator_descriptors
  (sub_IM IM selection)
Htr_project: equivocators_trace_project
  (sub_IM IM selection)
  final_descriptors tr =
Some (trX, initial_descriptors)
Hs_project: equivocators_state_project
  (sub_IM IM selection)
  initial_descriptors s = sX
finite_valid_trace SeededX sX trX
    apply not_equivocating_equivocator_descriptors_proper in Hnot_equiv as Hproper.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
IM':= λ i : sub_index selection, IM (`i): sub_index selection → VLSM message
final_descriptors: equivocator_descriptors IM'
s: state SeededXE
tr: list transition_item
sX: state SeededX
trX: list transition_item
Hpr_tr: equivocators_partial_trace_project IM'
  final_descriptors (s, tr) = 
Some (sX, trX)
Htr: finite_valid_trace SeededXE s tr
Hnot_equiv: not_equivocating_equivocator_descriptors
  (sub_IM IM selection) final_descriptors
  (finite_trace_last s tr)
initial_descriptors: equivocator_descriptors
  (sub_IM IM selection)
Htr_project: equivocators_trace_project
  (sub_IM IM selection)
  final_descriptors tr =
Some (trX, initial_descriptors)
Hs_project: equivocators_state_project
  (sub_IM IM selection)
  initial_descriptors s = sX
Hproper: proper_equivocator_descriptors
  (sub_IM IM selection) final_descriptors
  (finite_trace_last s tr)
finite_valid_trace SeededX sX trX
    destruct (seeded_equivocators_valid_trace_project _ _ Htr _ Hproper)
      as [_trX [_initial_descriptors [_ [_Htr_project [_ HtrX]]]]].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
selection: list index
seed: message → Prop
SeededXE:= seeded_equivocators_no_equivocation_vlsm IM
  selection seed: VLSM message
SubFreeE:= free_composite_vlsm
  (sub_IM (equivocator_IM IM) selection): VLSM message
SubPreFreeE:= preloaded_with_all_messages_vlsm SubFreeE: VLSM message
SubFree:= free_composite_vlsm (sub_IM IM selection): VLSM message
SeededX:= preloaded_vlsm SubFree seed: VLSM message
IM':= λ i : sub_index selection, IM (`i): sub_index selection → VLSM message
final_descriptors: equivocator_descriptors IM'
s: state SeededXE
tr: list transition_item
sX: state SeededX
trX: list transition_item
Hpr_tr: equivocators_partial_trace_project IM'
  final_descriptors (s, tr) = 
Some (sX, trX)
Htr: finite_valid_trace SeededXE s tr
Hnot_equiv: not_equivocating_equivocator_descriptors
  (sub_IM IM selection) final_descriptors
  (finite_trace_last s tr)
initial_descriptors: equivocator_descriptors
  (sub_IM IM selection)
Htr_project: equivocators_trace_project
  (sub_IM IM selection)
  final_descriptors tr =
Some (trX, initial_descriptors)
Hs_project: equivocators_state_project
  (sub_IM IM selection)
  initial_descriptors s = sX
Hproper: proper_equivocator_descriptors
  (sub_IM IM selection) final_descriptors
  (finite_trace_last s tr)
_trX: list
  (composite_transition_item
     (λ i : sub_index selection, IM (`i)))
_initial_descriptors: equivocator_descriptors
  (λ i : sub_index selection,
     IM (`i))
_Htr_project: equivocators_trace_project
  (λ i : sub_index selection, IM (`i))
  final_descriptors tr =
Some (_trX, _initial_descriptors)
HtrX: finite_valid_trace SeededX
  (equivocators_state_project
     (λ i : sub_index selection, IM (`i))
     _initial_descriptors s) _trX
finite_valid_trace SeededX sX trX
    by unfold sub_IM in *; congruence.
Qed.

End sec_seeded_equivocators_valid_trace_project.

End sec_equivocators_composition_sub_projections.

Section sec_equivocators_composition_vlsm_projection.

Context
  {message : Type}
  `{finite.Finite index}
  (IM : index -> VLSM message)
  `{forall i : index, HasBeenSentCapability (IM i)}
  (FreeE := free_composite_vlsm (equivocator_IM IM))
  (PreFreeE := preloaded_with_all_messages_vlsm FreeE)
  (Free := free_composite_vlsm IM)
  (PreFree := preloaded_with_all_messages_vlsm Free)
  .

Definition free_sub_free_equivocator_descriptors
  (descriptors : equivocator_descriptors IM)
  : equivocator_descriptors (sub_IM IM (finite.enum index))
  := fun i => descriptors (proj1_sig i).

Lemma equivocators_no_equivocations_vlsm_X_vlsm_partial_projection
  (final_descriptors : equivocator_descriptors IM)
  : VLSM_partial_projection (equivocators_no_equivocations_vlsm IM) Free
      (equivocators_partial_trace_project IM final_descriptors).message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
final_descriptors: equivocator_descriptors IM
VLSM_partial_projection
  (equivocators_no_equivocations_vlsm IM) Free
  (equivocators_partial_trace_project IM
     final_descriptors)
Proof.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
final_descriptors: equivocator_descriptors IM
VLSM_partial_projection
  (equivocators_no_equivocations_vlsm IM) Free
  (equivocators_partial_trace_project IM
     final_descriptors)
  split; [split |].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
final_descriptors: equivocator_descriptors IM
∀ (sX : state (equivocators_no_equivocations_vlsm IM)) 
  (trX : list transition_item) (sY : state Free) (trY : 
                                                 list
                                                 transition_item),
  equivocators_partial_trace_project IM
    final_descriptors (sX, trX) = Some (sY, trY)
  → ∀ (s'X : state
               (equivocators_no_equivocations_vlsm IM)) 
      (preX : list transition_item),
      finite_trace_last s'X preX = sX
      → finite_valid_trace_from
          (equivocators_no_equivocations_vlsm IM) s'X
          (preX ++ trX)
        → ∃ (s'Y : state Free) (preY : list
                                         transition_item),
            equivocators_partial_trace_project IM
              final_descriptors (s'X, preX ++ trX) =
            Some (s'Y, preY ++ trY)
            ∧ finite_trace_last s'Y preY = sY
message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
final_descriptors: equivocator_descriptors IM
∀ (sX : state (equivocators_no_equivocations_vlsm IM)) 
  (trX : list transition_item) 
  (sY : state Free) (trY : list transition_item),
  equivocators_partial_trace_project IM
    final_descriptors (
    sX, trX) = Some (sY, trY)
  → finite_valid_trace
      (equivocators_no_equivocations_vlsm IM) sX trX
    → finite_valid_trace Free sY trY
  -message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
final_descriptors: equivocator_descriptors IM
∀ (sX : state (equivocators_no_equivocations_vlsm IM)) 
  (trX : list transition_item) (sY : state Free) (trY : 
                                                 list
                                                 transition_item),
  equivocators_partial_trace_project IM
    final_descriptors (sX, trX) = Some (sY, trY)
  → ∀ (s'X : state
               (equivocators_no_equivocations_vlsm IM)) 
      (preX : list transition_item),
      finite_trace_last s'X preX = sX
      → finite_valid_trace_from
          (equivocators_no_equivocations_vlsm IM) s'X
          (preX ++ trX)
        → ∃ (s'Y : state Free) (preY : list
                                         transition_item),
            equivocators_partial_trace_project IM
              final_descriptors (s'X, preX ++ trX) =
            Some (s'Y, preY ++ trY)
            ∧ finite_trace_last s'Y preY = sY intros s tr sX trX Hpr_tr s_pre pre Hs_lst Hpre_tr.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
final_descriptors: equivocator_descriptors IM
s: state (equivocators_no_equivocations_vlsm IM)
tr: list transition_item
sX: state Free
trX: list transition_item
Hpr_tr: equivocators_partial_trace_project IM
  final_descriptors (s, tr) = 
Some (sX, trX)
s_pre: state (equivocators_no_equivocations_vlsm IM)
pre: list transition_item
Hs_lst: finite_trace_last s_pre pre = s
Hpre_tr: finite_valid_trace_from
  (equivocators_no_equivocations_vlsm IM)
  s_pre (pre ++ tr)
∃ (s'Y : state Free) (preY : list transition_item),
  equivocators_partial_trace_project IM
    final_descriptors (s_pre, pre ++ tr) =
  Some (s'Y, preY ++ trX)
  ∧ finite_trace_last s'Y preY = sX
    assert
      (HPreFree_pre_tr : finite_constrained_trace_from FreeE s_pre (pre ++ tr)).message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
final_descriptors: equivocator_descriptors IM
s: state (equivocators_no_equivocations_vlsm IM)
tr: list transition_item
sX: state Free
trX: list transition_item
Hpr_tr: equivocators_partial_trace_project IM
  final_descriptors (s, tr) = 
Some (sX, trX)
s_pre: state (equivocators_no_equivocations_vlsm IM)
pre: list transition_item
Hs_lst: finite_trace_last s_pre pre = s
Hpre_tr: finite_valid_trace_from
  (equivocators_no_equivocations_vlsm IM)
  s_pre (pre ++ tr)
finite_constrained_trace_from FreeE s_pre (pre ++ tr)
message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
final_descriptors: equivocator_descriptors IM
s: state (equivocators_no_equivocations_vlsm IM)
tr: list transition_item
sX: state Free
trX: list transition_item
Hpr_tr: equivocators_partial_trace_project IM
  final_descriptors (s, tr) = 
Some (sX, trX)
s_pre: state (equivocators_no_equivocations_vlsm IM)
pre: list transition_item
Hs_lst: finite_trace_last s_pre pre = s
Hpre_tr: finite_valid_trace_from
  (equivocators_no_equivocations_vlsm IM)
  s_pre (pre ++ tr)
HPreFree_pre_tr: finite_constrained_trace_from FreeE
  s_pre (pre ++ tr)
∃ (s'Y : state Free) (preY : list transition_item),
  equivocators_partial_trace_project IM
    final_descriptors (
    s_pre, pre ++ tr) = 
  Some (s'Y, preY ++ trX)
  ∧ finite_trace_last s'Y preY = sX
    {message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
final_descriptors: equivocator_descriptors IM
s: state (equivocators_no_equivocations_vlsm IM)
tr: list transition_item
sX: state Free
trX: list transition_item
Hpr_tr: equivocators_partial_trace_project IM
  final_descriptors (s, tr) = 
Some (sX, trX)
s_pre: state (equivocators_no_equivocations_vlsm IM)
pre: list transition_item
Hs_lst: finite_trace_last s_pre pre = s
Hpre_tr: finite_valid_trace_from
  (equivocators_no_equivocations_vlsm IM)
  s_pre (pre ++ tr)
finite_constrained_trace_from FreeE s_pre (pre ++ tr)
      apply VLSM_incl_finite_valid_trace_from; [| done].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
final_descriptors: equivocator_descriptors IM
s: state (equivocators_no_equivocations_vlsm IM)
tr: list transition_item
sX: state Free
trX: list transition_item
Hpr_tr: equivocators_partial_trace_project IM
  final_descriptors (s, tr) = 
Some (sX, trX)
s_pre: state (equivocators_no_equivocations_vlsm IM)
pre: list transition_item
Hs_lst: finite_trace_last s_pre pre = s
Hpre_tr: finite_valid_trace_from
  (equivocators_no_equivocations_vlsm IM)
  s_pre (pre ++ tr)
VLSM_incl_part
  (constrained_vlsm_machine
     (free_composite_vlsm (equivocator_IM IM))
     (equivocators_no_equivocations_constraint IM))
  (preloaded_vlsm_machine FreeE (λ _ : message, True))
      by apply constrained_preloaded_incl.
    }message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
final_descriptors: equivocator_descriptors IM
s: state (equivocators_no_equivocations_vlsm IM)
tr: list transition_item
sX: state Free
trX: list transition_item
Hpr_tr: equivocators_partial_trace_project IM
  final_descriptors (s, tr) = 
Some (sX, trX)
s_pre: state (equivocators_no_equivocations_vlsm IM)
pre: list transition_item
Hs_lst: finite_trace_last s_pre pre = s
Hpre_tr: finite_valid_trace_from
  (equivocators_no_equivocations_vlsm IM)
  s_pre (pre ++ tr)
HPreFree_pre_tr: finite_constrained_trace_from FreeE
  s_pre (pre ++ tr)
∃ (s'Y : state Free) (preY : list transition_item),
  equivocators_partial_trace_project IM
    final_descriptors (s_pre, pre ++ tr) =
  Some (s'Y, preY ++ trX)
  ∧ finite_trace_last s'Y preY = sX
    clear Hpre_tr.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
final_descriptors: equivocator_descriptors IM
s: state (equivocators_no_equivocations_vlsm IM)
tr: list transition_item
sX: state Free
trX: list transition_item
Hpr_tr: equivocators_partial_trace_project IM
  final_descriptors (s, tr) = 
Some (sX, trX)
s_pre: state (equivocators_no_equivocations_vlsm IM)
pre: list transition_item
Hs_lst: finite_trace_last s_pre pre = s
HPreFree_pre_tr: finite_constrained_trace_from FreeE
  s_pre (pre ++ tr)
∃ (s'Y : state Free) (preY : list transition_item),
  equivocators_partial_trace_project IM
    final_descriptors (s_pre, pre ++ tr) =
  Some (s'Y, preY ++ trX)
  ∧ finite_trace_last s'Y preY = sX  revert s tr sX trX Hpr_tr s_pre pre Hs_lst HPreFree_pre_tr.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
final_descriptors: equivocator_descriptors IM
∀ (s : state (equivocators_no_equivocations_vlsm IM)) 
  (tr : list transition_item) (sX : state Free) (trX : 
                                                 list
                                                 transition_item),
  equivocators_partial_trace_project IM
    final_descriptors (s, tr) = Some (sX, trX)
  → ∀ (s_pre : state
                 (equivocators_no_equivocations_vlsm
                    IM)) (pre : list transition_item),
      finite_trace_last s_pre pre = s
      → finite_constrained_trace_from FreeE s_pre
          (pre ++ tr)
        → ∃ (s'Y : state Free) (preY : list
                                         transition_item),
            equivocators_partial_trace_project IM
              final_descriptors (s_pre, pre ++ tr) =
            Some (s'Y, preY ++ trX)
            ∧ finite_trace_last s'Y preY = sX
    by apply equivocators_partial_trace_project_extends_left.
  -message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
final_descriptors: equivocator_descriptors IM
∀ (sX : state (equivocators_no_equivocations_vlsm IM)) 
  (trX : list transition_item) (sY : state Free) (trY : 
                                                 list
                                                 transition_item),
  equivocators_partial_trace_project IM
    final_descriptors (sX, trX) = Some (sY, trY)
  → finite_valid_trace
      (equivocators_no_equivocations_vlsm IM) sX trX
    → finite_valid_trace Free sY trY intros s tr sX trX Hpr_tr Htr.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
final_descriptors: equivocator_descriptors IM
s: state (equivocators_no_equivocations_vlsm IM)
tr: list transition_item
sX: state Free
trX: list transition_item
Hpr_tr: equivocators_partial_trace_project IM
  final_descriptors (s, tr) = 
Some (sX, trX)
Htr: finite_valid_trace
  (equivocators_no_equivocations_vlsm IM) s tr
finite_valid_trace Free sX trX
    destruct (destruct_equivocators_partial_trace_project IM Hpr_tr)
      as [Hnot_equiv [initial_descriptors [Htr_project Hs_project]]].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
final_descriptors: equivocator_descriptors IM
s: state (equivocators_no_equivocations_vlsm IM)
tr: list transition_item
sX: state Free
trX: list transition_item
Hpr_tr: equivocators_partial_trace_project IM
  final_descriptors (s, tr) = 
Some (sX, trX)
Htr: finite_valid_trace
  (equivocators_no_equivocations_vlsm IM) s tr
Hnot_equiv: not_equivocating_equivocator_descriptors
  IM final_descriptors
  (finite_trace_last s tr)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors tr =
Some (trX, initial_descriptors)
Hs_project: equivocators_state_project IM
  initial_descriptors s = sX
finite_valid_trace Free sX trX
    apply not_equivocating_equivocator_descriptors_proper in Hnot_equiv as Hproper.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
final_descriptors: equivocator_descriptors IM
s: state (equivocators_no_equivocations_vlsm IM)
tr: list transition_item
sX: state Free
trX: list transition_item
Hpr_tr: equivocators_partial_trace_project IM
  final_descriptors (s, tr) = 
Some (sX, trX)
Htr: finite_valid_trace
  (equivocators_no_equivocations_vlsm IM) s tr
Hnot_equiv: not_equivocating_equivocator_descriptors
  IM final_descriptors
  (finite_trace_last s tr)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors tr =
Some (trX, initial_descriptors)
Hs_project: equivocators_state_project IM
  initial_descriptors s = sX
Hproper: proper_equivocator_descriptors IM
  final_descriptors (finite_trace_last s tr)
finite_valid_trace Free sX trX

    specialize (sub_composition_all_embedding (equivocator_IM IM)
      (equivocators_no_equivocations_constraint IM)) as Hproj.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
final_descriptors: equivocator_descriptors IM
s: state (equivocators_no_equivocations_vlsm IM)
tr: list transition_item
sX: state Free
trX: list transition_item
Hpr_tr: equivocators_partial_trace_project IM
  final_descriptors (s, tr) = 
Some (sX, trX)
Htr: finite_valid_trace
  (equivocators_no_equivocations_vlsm IM) s tr
Hnot_equiv: not_equivocating_equivocator_descriptors
  IM final_descriptors
  (finite_trace_last s tr)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors tr =
Some (trX, initial_descriptors)
Hs_project: equivocators_state_project IM
  initial_descriptors s = sX
Hproper: proper_equivocator_descriptors IM
  final_descriptors (finite_trace_last s tr)
Hproj: VLSM_embedding
  (composite_vlsm (equivocator_IM IM)
     (equivocators_no_equivocations_constraint
        IM))
  (composite_vlsm
     (sub_IM (equivocator_IM IM) (enum index))
     (free_sub_free_constraint
        (equivocator_IM IM)
        (equivocators_no_equivocations_constraint
           IM)))
  (free_sub_free_label (equivocator_IM IM))
  (composite_state_sub_projection
     (equivocator_IM IM) 
     (enum index))
finite_valid_trace Free sX trX
    apply (VLSM_embedding_finite_valid_trace Hproj) in Htr.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
final_descriptors: equivocator_descriptors IM
s: state (equivocators_no_equivocations_vlsm IM)
tr: list transition_item
sX: state Free
trX: list transition_item
Hpr_tr: equivocators_partial_trace_project IM
  final_descriptors (s, tr) = 
Some (sX, trX)
Hproj: VLSM_embedding
  (composite_vlsm (equivocator_IM IM)
     (equivocators_no_equivocations_constraint
        IM))
  (composite_vlsm
     (sub_IM (equivocator_IM IM) (enum index))
     (free_sub_free_constraint
        (equivocator_IM IM)
        (equivocators_no_equivocations_constraint
           IM)))
  (free_sub_free_label (equivocator_IM IM))
  (composite_state_sub_projection
     (equivocator_IM IM) 
     (enum index))
Htr: finite_valid_trace
  (composite_vlsm
     (sub_IM (equivocator_IM IM) (enum index))
     (free_sub_free_constraint
        (equivocator_IM IM)
        (equivocators_no_equivocations_constraint
           IM)))
  (composite_state_sub_projection
     (equivocator_IM IM) 
     (enum index) s)
  (VLSM_embedding_finite_trace_project Hproj tr)
Hnot_equiv: not_equivocating_equivocator_descriptors
  IM final_descriptors
  (finite_trace_last s tr)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors tr =
Some (trX, initial_descriptors)
Hs_project: equivocators_state_project IM
  initial_descriptors s = sX
Hproper: proper_equivocator_descriptors IM
  final_descriptors (finite_trace_last s tr)
finite_valid_trace Free sX trX
    specialize
      (false_composite_no_equivocation_vlsm_with_preloaded
        (SubProjectionTraces.sub_IM (equivocator_IM IM) (enum index))
        (free_constraint _))
      as Heq.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
final_descriptors: equivocator_descriptors IM
s: state (equivocators_no_equivocations_vlsm IM)
tr: list transition_item
sX: state Free
trX: list transition_item
Hpr_tr: equivocators_partial_trace_project IM
  final_descriptors (s, tr) = 
Some (sX, trX)
Hproj: VLSM_embedding
  (composite_vlsm (equivocator_IM IM)
     (equivocators_no_equivocations_constraint
        IM))
  (composite_vlsm
     (sub_IM (equivocator_IM IM) (enum index))
     (free_sub_free_constraint
        (equivocator_IM IM)
        (equivocators_no_equivocations_constraint
           IM)))
  (free_sub_free_label (equivocator_IM IM))
  (composite_state_sub_projection
     (equivocator_IM IM) 
     (enum index))
Htr: finite_valid_trace
  (composite_vlsm
     (sub_IM (equivocator_IM IM) (enum index))
     (free_sub_free_constraint
        (equivocator_IM IM)
        (equivocators_no_equivocations_constraint
           IM)))
  (composite_state_sub_projection
     (equivocator_IM IM) 
     (enum index) s)
  (VLSM_embedding_finite_trace_project Hproj tr)
Hnot_equiv: not_equivocating_equivocator_descriptors
  IM final_descriptors
  (finite_trace_last s tr)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors tr =
Some (trX, initial_descriptors)
Hs_project: equivocators_state_project IM
  initial_descriptors s = sX
Hproper: proper_equivocator_descriptors IM
  final_descriptors (finite_trace_last s tr)
Heq: VLSM_eq
  (composite_no_equivocation_vlsm_with_preloaded
     (sub_IM (equivocator_IM IM) (enum index))
     (free_constraint
        (sub_IM (equivocator_IM IM) (enum index)))
     (λ _ : message, False))
  (composite_vlsm
     (sub_IM (equivocator_IM IM) (enum index))
     (no_equivocations_additional_constraint
        (sub_IM (equivocator_IM IM) (enum index))
        (free_constraint
           (sub_IM (equivocator_IM IM)
              (enum index)))))
finite_valid_trace Free sX trX
    assert (Htr' :
      finite_valid_trace
        (composite_vlsm
          (SubProjectionTraces.sub_IM (equivocator_IM IM) (enum index))
          (no_equivocations_additional_constraint
            (SubProjectionTraces.sub_IM (equivocator_IM IM) (enum index))
            (free_constraint _)))
        (composite_state_sub_projection (equivocator_IM IM) (finite.enum index) s)
        (VLSM_embedding_finite_trace_project Hproj tr)).message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
final_descriptors: equivocator_descriptors IM
s: state (equivocators_no_equivocations_vlsm IM)
tr: list transition_item
sX: state Free
trX: list transition_item
Hpr_tr: equivocators_partial_trace_project IM
  final_descriptors (s, tr) = 
Some (sX, trX)
Hproj: VLSM_embedding
  (composite_vlsm (equivocator_IM IM)
     (equivocators_no_equivocations_constraint
        IM))
  (composite_vlsm
     (sub_IM (equivocator_IM IM) (enum index))
     (free_sub_free_constraint
        (equivocator_IM IM)
        (equivocators_no_equivocations_constraint
           IM)))
  (free_sub_free_label (equivocator_IM IM))
  (composite_state_sub_projection
     (equivocator_IM IM) 
     (enum index))
Htr: finite_valid_trace
  (composite_vlsm
     (sub_IM (equivocator_IM IM) (enum index))
     (free_sub_free_constraint
        (equivocator_IM IM)
        (equivocators_no_equivocations_constraint
           IM)))
  (composite_state_sub_projection
     (equivocator_IM IM) 
     (enum index) s)
  (VLSM_embedding_finite_trace_project Hproj tr)
Hnot_equiv: not_equivocating_equivocator_descriptors
  IM final_descriptors
  (finite_trace_last s tr)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors tr =
Some (trX, initial_descriptors)
Hs_project: equivocators_state_project IM
  initial_descriptors s = sX
Hproper: proper_equivocator_descriptors IM
  final_descriptors (finite_trace_last s tr)
Heq: VLSM_eq
  (composite_no_equivocation_vlsm_with_preloaded
     (sub_IM (equivocator_IM IM) (enum index))
     (free_constraint
        (sub_IM (equivocator_IM IM) (enum index)))
     (λ _ : message, False))
  (composite_vlsm
     (sub_IM (equivocator_IM IM) (enum index))
     (no_equivocations_additional_constraint
        (sub_IM (equivocator_IM IM) (enum index))
        (free_constraint
           (sub_IM (equivocator_IM IM)
              (enum index)))))
finite_valid_trace
  (composite_vlsm
     (sub_IM (equivocator_IM IM) (enum index))
     (no_equivocations_additional_constraint
        (sub_IM (equivocator_IM IM) (enum index))
        (free_constraint
           (sub_IM (equivocator_IM IM) (enum index)))))
  (composite_state_sub_projection (equivocator_IM IM)
     (enum index) s)
  (VLSM_embedding_finite_trace_project Hproj tr)
message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
final_descriptors: equivocator_descriptors IM
s: state (equivocators_no_equivocations_vlsm IM)
tr: list transition_item
sX: state Free
trX: list transition_item
Hpr_tr: equivocators_partial_trace_project IM
  final_descriptors (s, tr) = 
Some (sX, trX)
Hproj: VLSM_embedding
  (composite_vlsm (equivocator_IM IM)
     (equivocators_no_equivocations_constraint
        IM))
  (composite_vlsm
     (sub_IM (equivocator_IM IM) (enum index))
     (free_sub_free_constraint
        (equivocator_IM IM)
        (equivocators_no_equivocations_constraint
           IM)))
  (free_sub_free_label (equivocator_IM IM))
  (composite_state_sub_projection
     (equivocator_IM IM) 
     (enum index))
Htr: finite_valid_trace
  (composite_vlsm
     (sub_IM (equivocator_IM IM) (enum index))
     (free_sub_free_constraint
        (equivocator_IM IM)
        (equivocators_no_equivocations_constraint
           IM)))
  (composite_state_sub_projection
     (equivocator_IM IM) 
     (enum index) s)
  (VLSM_embedding_finite_trace_project Hproj tr)
Hnot_equiv: not_equivocating_equivocator_descriptors
  IM final_descriptors
  (finite_trace_last s tr)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors tr =
Some (trX, initial_descriptors)
Hs_project: equivocators_state_project IM
  initial_descriptors s = sX
Hproper: proper_equivocator_descriptors IM
  final_descriptors (finite_trace_last s tr)
Heq: VLSM_eq
  (composite_no_equivocation_vlsm_with_preloaded
     (sub_IM (equivocator_IM IM) (enum index))
     (free_constraint
        (sub_IM (equivocator_IM IM) (enum index)))
     (λ _ : message, False))
  (composite_vlsm
     (sub_IM (equivocator_IM IM) (enum index))
     (no_equivocations_additional_constraint
        (sub_IM (equivocator_IM IM) (enum index))
        (free_constraint
           (sub_IM (equivocator_IM IM)
              (enum index)))))
Htr': finite_valid_trace
  (composite_vlsm
     (sub_IM (equivocator_IM IM) (enum index))
     (no_equivocations_additional_constraint
        (sub_IM (equivocator_IM IM)
           (enum index))
        (free_constraint
           (sub_IM (equivocator_IM IM)
              (enum index)))))
  (composite_state_sub_projection
     (equivocator_IM IM) 
     (enum index) s)
  (VLSM_embedding_finite_trace_project Hproj tr)
finite_valid_trace Free sX trX
    {message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
final_descriptors: equivocator_descriptors IM
s: state (equivocators_no_equivocations_vlsm IM)
tr: list transition_item
sX: state Free
trX: list transition_item
Hpr_tr: equivocators_partial_trace_project IM
  final_descriptors (s, tr) = 
Some (sX, trX)
Hproj: VLSM_embedding
  (composite_vlsm (equivocator_IM IM)
     (equivocators_no_equivocations_constraint
        IM))
  (composite_vlsm
     (sub_IM (equivocator_IM IM) (enum index))
     (free_sub_free_constraint
        (equivocator_IM IM)
        (equivocators_no_equivocations_constraint
           IM)))
  (free_sub_free_label (equivocator_IM IM))
  (composite_state_sub_projection
     (equivocator_IM IM) 
     (enum index))
Htr: finite_valid_trace
  (composite_vlsm
     (sub_IM (equivocator_IM IM) (enum index))
     (free_sub_free_constraint
        (equivocator_IM IM)
        (equivocators_no_equivocations_constraint
           IM)))
  (composite_state_sub_projection
     (equivocator_IM IM) 
     (enum index) s)
  (VLSM_embedding_finite_trace_project Hproj tr)
Hnot_equiv: not_equivocating_equivocator_descriptors
  IM final_descriptors
  (finite_trace_last s tr)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors tr =
Some (trX, initial_descriptors)
Hs_project: equivocators_state_project IM
  initial_descriptors s = sX
Hproper: proper_equivocator_descriptors IM
  final_descriptors (finite_trace_last s tr)
Heq: VLSM_eq
  (composite_no_equivocation_vlsm_with_preloaded
     (sub_IM (equivocator_IM IM) (enum index))
     (free_constraint
        (sub_IM (equivocator_IM IM) (enum index)))
     (λ _ : message, False))
  (composite_vlsm
     (sub_IM (equivocator_IM IM) (enum index))
     (no_equivocations_additional_constraint
        (sub_IM (equivocator_IM IM) (enum index))
        (free_constraint
           (sub_IM (equivocator_IM IM)
              (enum index)))))
finite_valid_trace
  (composite_vlsm
     (sub_IM (equivocator_IM IM) (enum index))
     (no_equivocations_additional_constraint
        (sub_IM (equivocator_IM IM) (enum index))
        (free_constraint
           (sub_IM (equivocator_IM IM) (enum index)))))
  (composite_state_sub_projection (equivocator_IM IM)
     (enum index) s)
  (VLSM_embedding_finite_trace_project Hproj tr) revert Htr.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
final_descriptors: equivocator_descriptors IM
s: state (equivocators_no_equivocations_vlsm IM)
tr: list transition_item
sX: state Free
trX: list transition_item
Hpr_tr: equivocators_partial_trace_project IM
  final_descriptors (s, tr) = 
Some (sX, trX)
Hproj: VLSM_embedding
  (composite_vlsm (equivocator_IM IM)
     (equivocators_no_equivocations_constraint
        IM))
  (composite_vlsm
     (sub_IM (equivocator_IM IM) (enum index))
     (free_sub_free_constraint
        (equivocator_IM IM)
        (equivocators_no_equivocations_constraint
           IM)))
  (free_sub_free_label (equivocator_IM IM))
  (composite_state_sub_projection
     (equivocator_IM IM) 
     (enum index))
Hnot_equiv: not_equivocating_equivocator_descriptors
  IM final_descriptors
  (finite_trace_last s tr)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors tr =
Some (trX, initial_descriptors)
Hs_project: equivocators_state_project IM
  initial_descriptors s = sX
Hproper: proper_equivocator_descriptors IM
  final_descriptors (finite_trace_last s tr)
Heq: VLSM_eq
  (composite_no_equivocation_vlsm_with_preloaded
     (sub_IM (equivocator_IM IM) (enum index))
     (free_constraint
        (sub_IM (equivocator_IM IM) (enum index)))
     (λ _ : message, False))
  (composite_vlsm
     (sub_IM (equivocator_IM IM) (enum index))
     (no_equivocations_additional_constraint
        (sub_IM (equivocator_IM IM) (enum index))
        (free_constraint
           (sub_IM (equivocator_IM IM)
              (enum index)))))
finite_valid_trace
  (composite_vlsm
     (sub_IM (equivocator_IM IM) (enum index))
     (free_sub_free_constraint (equivocator_IM IM)
        (equivocators_no_equivocations_constraint IM)))
  (composite_state_sub_projection (equivocator_IM IM)
     (enum index) s)
  (VLSM_embedding_finite_trace_project Hproj tr)
→ finite_valid_trace
    (composite_vlsm
       (sub_IM (equivocator_IM IM) (enum index))
       (no_equivocations_additional_constraint
          (sub_IM (equivocator_IM IM) (enum index))
          (free_constraint
             (sub_IM (equivocator_IM IM) (enum index)))))
    (composite_state_sub_projection
       (equivocator_IM IM) (enum index) s)
    (VLSM_embedding_finite_trace_project Hproj tr)
      apply VLSM_incl_finite_valid_trace.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
final_descriptors: equivocator_descriptors IM
s: state (equivocators_no_equivocations_vlsm IM)
tr: list transition_item
sX: state Free
trX: list transition_item
Hpr_tr: equivocators_partial_trace_project IM
  final_descriptors (s, tr) = 
Some (sX, trX)
Hproj: VLSM_embedding
  (composite_vlsm (equivocator_IM IM)
     (equivocators_no_equivocations_constraint
        IM))
  (composite_vlsm
     (sub_IM (equivocator_IM IM) (enum index))
     (free_sub_free_constraint
        (equivocator_IM IM)
        (equivocators_no_equivocations_constraint
           IM)))
  (free_sub_free_label (equivocator_IM IM))
  (composite_state_sub_projection
     (equivocator_IM IM) 
     (enum index))
Hnot_equiv: not_equivocating_equivocator_descriptors
  IM final_descriptors
  (finite_trace_last s tr)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors tr =
Some (trX, initial_descriptors)
Hs_project: equivocators_state_project IM
  initial_descriptors s = sX
Hproper: proper_equivocator_descriptors IM
  final_descriptors (finite_trace_last s tr)
Heq: VLSM_eq
  (composite_no_equivocation_vlsm_with_preloaded
     (sub_IM (equivocator_IM IM) (enum index))
     (free_constraint
        (sub_IM (equivocator_IM IM) (enum index)))
     (λ _ : message, False))
  (composite_vlsm
     (sub_IM (equivocator_IM IM) (enum index))
     (no_equivocations_additional_constraint
        (sub_IM (equivocator_IM IM) (enum index))
        (free_constraint
           (sub_IM (equivocator_IM IM)
              (enum index)))))
VLSM_incl_part
  (constrained_vlsm_machine
     (free_composite_vlsm
        (sub_IM (equivocator_IM IM) (enum index)))
     (free_sub_free_constraint (equivocator_IM IM)
        (equivocators_no_equivocations_constraint IM)))
  (constrained_vlsm_machine
     (free_composite_vlsm
        (sub_IM (equivocator_IM IM) (enum index)))
     (no_equivocations_additional_constraint
        (sub_IM (equivocator_IM IM) (enum index))
        (free_constraint
           (sub_IM (equivocator_IM IM) (enum index)))))
      clear.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
VLSM_incl_part
  (constrained_vlsm_machine
     (free_composite_vlsm
        (sub_IM (equivocator_IM IM) (enum index)))
     (free_sub_free_constraint (equivocator_IM IM)
        (equivocators_no_equivocations_constraint IM)))
  (constrained_vlsm_machine
     (free_composite_vlsm
        (sub_IM (equivocator_IM IM) (enum index)))
     (no_equivocations_additional_constraint
        (sub_IM (equivocator_IM IM) (enum index))
        (free_constraint
           (sub_IM (equivocator_IM IM) (enum index)))))
      apply constraint_subsumption_incl.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
input_valid_constraint_subsumption
  (free_composite_vlsm
     (sub_IM (equivocator_IM IM) (enum index)))
  (free_sub_free_constraint (equivocator_IM IM)
     (equivocators_no_equivocations_constraint IM))
  (no_equivocations_additional_constraint
     (sub_IM (equivocator_IM IM) (enum index))
     (free_constraint
        (sub_IM (equivocator_IM IM) (enum index))))
      apply preloaded_constraint_subsumption_stronger.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
preloaded_constraint_subsumption
  (free_composite_vlsm
     (sub_IM (equivocator_IM IM) (enum index)))
  (free_sub_free_constraint (equivocator_IM IM)
     (equivocators_no_equivocations_constraint IM))
  (no_equivocations_additional_constraint
     (sub_IM (equivocator_IM IM) (enum index))
     (free_constraint
        (sub_IM (equivocator_IM IM) (enum index))))
      apply strong_constraint_subsumption_strongest.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
strong_constraint_subsumption
  (free_composite_vlsm
     (sub_IM (equivocator_IM IM) (enum index)))
  (free_sub_free_constraint (equivocator_IM IM)
     (equivocators_no_equivocations_constraint IM))
  (no_equivocations_additional_constraint
     (sub_IM (equivocator_IM IM) (enum index))
     (free_constraint
        (sub_IM (equivocator_IM IM) (enum index))))
      intros (i, li) (s, om).message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
i: sub_index (enum index)
li: label (sub_IM (equivocator_IM IM) (enum index) i)
s: state
  (free_composite_vlsm
     (sub_IM (equivocator_IM IM) (enum index)))
om: option message
free_sub_free_constraint (equivocator_IM IM)
  (equivocators_no_equivocations_constraint IM)
  (existT i li) (s, om)
→ no_equivocations_additional_constraint
    (sub_IM (equivocator_IM IM) (enum index))
    (free_constraint
       (sub_IM (equivocator_IM IM) (enum index)))
    (existT i li) (s, om)
      unfold free_sub_free_constraint, lift_sub_label, free_sub_free_state, free_sub_free_index.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
i: sub_index (enum index)
li: label (sub_IM (equivocator_IM IM) (enum index) i)
s: state
  (free_composite_vlsm
     (sub_IM (equivocator_IM IM) (enum index)))
om: option message
equivocators_no_equivocations_constraint IM
  (existT (`(projT1 (existT i li)))
     (projT2 (existT i li)))
  (λ i : index,
     s
       (dexist i
          (SubProjectionTraces.free_sub_free_index_obligation_1
             i)), om)
→ no_equivocations_additional_constraint
    (sub_IM (equivocator_IM IM) (enum index))
    (free_constraint
       (sub_IM (equivocator_IM IM) (enum index)))
    (existT i li) (s, om)
      unfold equivocators_no_equivocations_constraint.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
i: sub_index (enum index)
li: label (sub_IM (equivocator_IM IM) (enum index) i)
s: state
  (free_composite_vlsm
     (sub_IM (equivocator_IM IM) (enum index)))
om: option message
no_equivocations_additional_constraint
  (equivocator_IM IM)
  (free_constraint (equivocator_IM IM))
  (existT (`(projT1 (existT i li)))
     (projT2 (existT i li)))
  (λ i : index,
     s
       (dexist i
          (SubProjectionTraces.free_sub_free_index_obligation_1
             i)), om)
→ no_equivocations_additional_constraint
    (sub_IM (equivocator_IM IM) (enum index))
    (free_constraint
       (sub_IM (equivocator_IM IM) (enum index)))
    (existT i li) (s, om)
      intros [Hno_equiv _].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
i: sub_index (enum index)
li: label (sub_IM (equivocator_IM IM) (enum index) i)
s: state
  (free_composite_vlsm
     (sub_IM (equivocator_IM IM) (enum index)))
om: option message
Hno_equiv: composite_no_equivocations
  (equivocator_IM IM)
  (existT (`(projT1 (existT i li)))
     (projT2 (existT i li)))
  (λ i : index,
     s
       (dexist i
          (SubProjectionTraces.free_sub_free_index_obligation_1
             i)), om)
no_equivocations_additional_constraint
  (sub_IM (equivocator_IM IM) (enum index))
  (free_constraint
     (sub_IM (equivocator_IM IM) (enum index)))
  (existT i li) (s, om)
      split; [| done].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
i: sub_index (enum index)
li: label (sub_IM (equivocator_IM IM) (enum index) i)
s: state
  (free_composite_vlsm
     (sub_IM (equivocator_IM IM) (enum index)))
om: option message
Hno_equiv: composite_no_equivocations
  (equivocator_IM IM)
  (existT (`(projT1 (existT i li)))
     (projT2 (existT i li)))
  (λ i : index,
     s
       (dexist i
          (SubProjectionTraces.free_sub_free_index_obligation_1
             i)), om)
composite_no_equivocations
  (sub_IM (equivocator_IM IM) (enum index))
  (existT i li) (s, om)
      destruct om as [m |]; [| done].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
i: sub_index (enum index)
li: label (sub_IM (equivocator_IM IM) (enum index) i)
s: state
  (free_composite_vlsm
     (sub_IM (equivocator_IM IM) (enum index)))
m: message
Hno_equiv: composite_no_equivocations
  (equivocator_IM IM)
  (existT (`(projT1 (existT i li)))
     (projT2 (existT i li)))
  (λ i : index,
     s
       (dexist i
          (SubProjectionTraces.free_sub_free_index_obligation_1
             i)), Some m)
composite_no_equivocations
  (sub_IM (equivocator_IM IM) (enum index))
  (existT i li) (s, Some m)
      left.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
i: sub_index (enum index)
li: label (sub_IM (equivocator_IM IM) (enum index) i)
s: state
  (free_composite_vlsm
     (sub_IM (equivocator_IM IM) (enum index)))
m: message
Hno_equiv: composite_no_equivocations
  (equivocator_IM IM)
  (existT (`(projT1 (existT i li)))
     (projT2 (existT i li)))
  (λ i : index,
     s
       (dexist i
          (SubProjectionTraces.free_sub_free_index_obligation_1
             i)), Some m)
composite_has_been_sent
  (sub_IM (equivocator_IM IM) (enum index))
  (s, Some m).1 m destruct Hno_equiv as [Hno_equiv | Hfalse]; [| done].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
i: sub_index (enum index)
li: label (sub_IM (equivocator_IM IM) (enum index) i)
s: state
  (free_composite_vlsm
     (sub_IM (equivocator_IM IM) (enum index)))
m: message
Hno_equiv: composite_has_been_sent
  (equivocator_IM IM)
  (λ i : index,
     s
       (dexist i
          (SubProjectionTraces.free_sub_free_index_obligation_1
             i)), Some m).1 m
composite_has_been_sent
  (sub_IM (equivocator_IM IM) (enum index))
  (s, Some m).1 m
      destruct Hno_equiv as [eqv Hno_equiv].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
i: sub_index (enum index)
li: label (sub_IM (equivocator_IM IM) (enum index) i)
s: state
  (free_composite_vlsm
     (sub_IM (equivocator_IM IM) (enum index)))
m: message
eqv: index
Hno_equiv: has_been_sent (equivocator_IM IM eqv)
  ((λ i : index,
      s
        (dexist i
           (SubProjectionTraces.free_sub_free_index_obligation_1
              i)), Some m).1 eqv) m
composite_has_been_sent
  (sub_IM (equivocator_IM IM) (enum index))
  (s, Some m).1 m
      by exists (dexist eqv (SubProjectionTraces.free_sub_free_index_obligation_1 eqv)).
    }message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
final_descriptors: equivocator_descriptors IM
s: state (equivocators_no_equivocations_vlsm IM)
tr: list transition_item
sX: state Free
trX: list transition_item
Hpr_tr: equivocators_partial_trace_project IM
  final_descriptors (s, tr) = 
Some (sX, trX)
Hproj: VLSM_embedding
  (composite_vlsm (equivocator_IM IM)
     (equivocators_no_equivocations_constraint
        IM))
  (composite_vlsm
     (sub_IM (equivocator_IM IM) (enum index))
     (free_sub_free_constraint
        (equivocator_IM IM)
        (equivocators_no_equivocations_constraint
           IM)))
  (free_sub_free_label (equivocator_IM IM))
  (composite_state_sub_projection
     (equivocator_IM IM) 
     (enum index))
Htr: finite_valid_trace
  (composite_vlsm
     (sub_IM (equivocator_IM IM) (enum index))
     (free_sub_free_constraint
        (equivocator_IM IM)
        (equivocators_no_equivocations_constraint
           IM)))
  (composite_state_sub_projection
     (equivocator_IM IM) 
     (enum index) s)
  (VLSM_embedding_finite_trace_project Hproj tr)
Hnot_equiv: not_equivocating_equivocator_descriptors
  IM final_descriptors
  (finite_trace_last s tr)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors tr =
Some (trX, initial_descriptors)
Hs_project: equivocators_state_project IM
  initial_descriptors s = sX
Hproper: proper_equivocator_descriptors IM
  final_descriptors (finite_trace_last s tr)
Heq: VLSM_eq
  (composite_no_equivocation_vlsm_with_preloaded
     (sub_IM (equivocator_IM IM) (enum index))
     (free_constraint
        (sub_IM (equivocator_IM IM) (enum index)))
     (λ _ : message, False))
  (composite_vlsm
     (sub_IM (equivocator_IM IM) (enum index))
     (no_equivocations_additional_constraint
        (sub_IM (equivocator_IM IM) (enum index))
        (free_constraint
           (sub_IM (equivocator_IM IM)
              (enum index)))))
Htr': finite_valid_trace
  (composite_vlsm
     (sub_IM (equivocator_IM IM) (enum index))
     (no_equivocations_additional_constraint
        (sub_IM (equivocator_IM IM)
           (enum index))
        (free_constraint
           (sub_IM (equivocator_IM IM)
              (enum index)))))
  (composite_state_sub_projection
     (equivocator_IM IM) 
     (enum index) s)
  (VLSM_embedding_finite_trace_project Hproj tr)
finite_valid_trace Free sX trX
    apply (VLSM_eq_finite_valid_trace Heq) in Htr'.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
final_descriptors: equivocator_descriptors IM
s: state (equivocators_no_equivocations_vlsm IM)
tr: list transition_item
sX: state Free
trX: list transition_item
Hpr_tr: equivocators_partial_trace_project IM
  final_descriptors (s, tr) = 
Some (sX, trX)
Hproj: VLSM_embedding
  (composite_vlsm (equivocator_IM IM)
     (equivocators_no_equivocations_constraint
        IM))
  (composite_vlsm
     (sub_IM (equivocator_IM IM) (enum index))
     (free_sub_free_constraint
        (equivocator_IM IM)
        (equivocators_no_equivocations_constraint
           IM)))
  (free_sub_free_label (equivocator_IM IM))
  (composite_state_sub_projection
     (equivocator_IM IM) 
     (enum index))
Htr: finite_valid_trace
  (composite_vlsm
     (sub_IM (equivocator_IM IM) (enum index))
     (free_sub_free_constraint
        (equivocator_IM IM)
        (equivocators_no_equivocations_constraint
           IM)))
  (composite_state_sub_projection
     (equivocator_IM IM) 
     (enum index) s)
  (VLSM_embedding_finite_trace_project Hproj tr)
Hnot_equiv: not_equivocating_equivocator_descriptors
  IM final_descriptors
  (finite_trace_last s tr)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors tr =
Some (trX, initial_descriptors)
Hs_project: equivocators_state_project IM
  initial_descriptors s = sX
Hproper: proper_equivocator_descriptors IM
  final_descriptors (finite_trace_last s tr)
Heq: VLSM_eq
  (composite_no_equivocation_vlsm_with_preloaded
     (sub_IM (equivocator_IM IM) (enum index))
     (free_constraint
        (sub_IM (equivocator_IM IM) (enum index)))
     (λ _ : message, False))
  (composite_vlsm
     (sub_IM (equivocator_IM IM) (enum index))
     (no_equivocations_additional_constraint
        (sub_IM (equivocator_IM IM) (enum index))
        (free_constraint
           (sub_IM (equivocator_IM IM)
              (enum index)))))
Htr': finite_valid_trace
  {|
    vlsm_type :=
      composite_no_equivocation_vlsm_with_preloaded
        (sub_IM (equivocator_IM IM)
           (enum index))
        (free_constraint
           (sub_IM (equivocator_IM IM)
              (enum index)))
        (λ _ : message, False);
    vlsm_machine :=
      composite_no_equivocation_vlsm_with_preloaded
        (sub_IM (equivocator_IM IM)
           (enum index))
        (free_constraint
           (sub_IM (equivocator_IM IM)
              (enum index)))
        (λ _ : message, False)
  |}
  (composite_state_sub_projection
     (equivocator_IM IM) 
     (enum index) s)
  (VLSM_embedding_finite_trace_project Hproj tr)
finite_valid_trace Free sX trX

    specialize
      (seeded_equivocators_valid_trace_project IM
        (enum index)
        (fun m => False)
        _ _ Htr'
        (free_sub_free_equivocator_descriptors final_descriptors))
      as Hproject.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
final_descriptors: equivocator_descriptors IM
s: state (equivocators_no_equivocations_vlsm IM)
tr: list transition_item
sX: state Free
trX: list transition_item
Hpr_tr: equivocators_partial_trace_project IM
  final_descriptors (s, tr) = 
Some (sX, trX)
Hproj: VLSM_embedding
  (composite_vlsm (equivocator_IM IM)
     (equivocators_no_equivocations_constraint
        IM))
  (composite_vlsm
     (sub_IM (equivocator_IM IM) (enum index))
     (free_sub_free_constraint
        (equivocator_IM IM)
        (equivocators_no_equivocations_constraint
           IM)))
  (free_sub_free_label (equivocator_IM IM))
  (composite_state_sub_projection
     (equivocator_IM IM) 
     (enum index))
Htr: finite_valid_trace
  (composite_vlsm
     (sub_IM (equivocator_IM IM) (enum index))
     (free_sub_free_constraint
        (equivocator_IM IM)
        (equivocators_no_equivocations_constraint
           IM)))
  (composite_state_sub_projection
     (equivocator_IM IM) 
     (enum index) s)
  (VLSM_embedding_finite_trace_project Hproj tr)
Hnot_equiv: not_equivocating_equivocator_descriptors
  IM final_descriptors
  (finite_trace_last s tr)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors tr =
Some (trX, initial_descriptors)
Hs_project: equivocators_state_project IM
  initial_descriptors s = sX
Hproper: proper_equivocator_descriptors IM
  final_descriptors (finite_trace_last s tr)
Heq: VLSM_eq
  (composite_no_equivocation_vlsm_with_preloaded
     (sub_IM (equivocator_IM IM) (enum index))
     (free_constraint
        (sub_IM (equivocator_IM IM) (enum index)))
     (λ _ : message, False))
  (composite_vlsm
     (sub_IM (equivocator_IM IM) (enum index))
     (no_equivocations_additional_constraint
        (sub_IM (equivocator_IM IM) (enum index))
        (free_constraint
           (sub_IM (equivocator_IM IM)
              (enum index)))))
Htr': finite_valid_trace
  {|
    vlsm_type :=
      composite_no_equivocation_vlsm_with_preloaded
        (sub_IM (equivocator_IM IM)
           (enum index))
        (free_constraint
           (sub_IM (equivocator_IM IM)
              (enum index)))
        (λ _ : message, False);
    vlsm_machine :=
      composite_no_equivocation_vlsm_with_preloaded
        (sub_IM (equivocator_IM IM)
           (enum index))
        (free_constraint
           (sub_IM (equivocator_IM IM)
              (enum index)))
        (λ _ : message, False)
  |}
  (composite_state_sub_projection
     (equivocator_IM IM) 
     (enum index) s)
  (VLSM_embedding_finite_trace_project Hproj tr)
Hproject: proper_equivocator_descriptors
  (λ i : sub_index (enum index), IM (`i))
  (free_sub_free_equivocator_descriptors
     final_descriptors)
  (finite_trace_last
     (composite_state_sub_projection
        (equivocator_IM IM) 
        (enum index) s)
     (VLSM_embedding_finite_trace_project
        Hproj tr))
→ ∃ (trX : list
             (composite_transition_item
                (λ i : 
                    sub_index 
                      (enum index), 
                   IM (`i)))) 
    (initial_descriptors : 
     equivocator_descriptors
       (λ i : sub_index (enum index),
          IM (`i))),
    let isX :=
      equivocators_state_project
        (λ i : sub_index (enum index),
           IM (`i)) initial_descriptors
        (composite_state_sub_projection
           (equivocator_IM IM) 
           (enum index) s) in
    let final_stateX :=
      finite_trace_last isX trX in
    proper_equivocator_descriptors
      (λ i : sub_index (enum index),
         IM (`i)) initial_descriptors
      (composite_state_sub_projection
         (equivocator_IM IM) 
         (enum index) s)
    ∧ equivocators_trace_project
        (λ i : sub_index (enum index),
           IM (`i))
        (free_sub_free_equivocator_descriptors
           final_descriptors)
        (VLSM_embedding_finite_trace_project
           Hproj tr) =
      Some (trX, initial_descriptors)
      ∧ equivocators_state_project
          (λ i : sub_index (enum index),
             IM (`i))
          (free_sub_free_equivocator_descriptors
             final_descriptors)
          (finite_trace_last
             (composite_state_sub_projection
                (equivocator_IM IM)
                (enum index) s)
             (VLSM_embedding_finite_trace_project
                Hproj tr)) = final_stateX
        ∧ finite_valid_trace
            (preloaded_vlsm
               (free_composite_vlsm
                  (sub_IM IM (enum index)))
               (λ _ : message, False)) isX
            trX
finite_valid_trace Free sX trX
    spec Hproject.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
final_descriptors: equivocator_descriptors IM
s: state (equivocators_no_equivocations_vlsm IM)
tr: list transition_item
sX: state Free
trX: list transition_item
Hpr_tr: equivocators_partial_trace_project IM
  final_descriptors (s, tr) = 
Some (sX, trX)
Hproj: VLSM_embedding
  (composite_vlsm (equivocator_IM IM)
     (equivocators_no_equivocations_constraint
        IM))
  (composite_vlsm
     (sub_IM (equivocator_IM IM) (enum index))
     (free_sub_free_constraint
        (equivocator_IM IM)
        (equivocators_no_equivocations_constraint
           IM)))
  (free_sub_free_label (equivocator_IM IM))
  (composite_state_sub_projection
     (equivocator_IM IM) 
     (enum index))
Htr: finite_valid_trace
  (composite_vlsm
     (sub_IM (equivocator_IM IM) (enum index))
     (free_sub_free_constraint
        (equivocator_IM IM)
        (equivocators_no_equivocations_constraint
           IM)))
  (composite_state_sub_projection
     (equivocator_IM IM) 
     (enum index) s)
  (VLSM_embedding_finite_trace_project Hproj tr)
Hnot_equiv: not_equivocating_equivocator_descriptors
  IM final_descriptors
  (finite_trace_last s tr)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors tr =
Some (trX, initial_descriptors)
Hs_project: equivocators_state_project IM
  initial_descriptors s = sX
Hproper: proper_equivocator_descriptors IM
  final_descriptors (finite_trace_last s tr)
Heq: VLSM_eq
  (composite_no_equivocation_vlsm_with_preloaded
     (sub_IM (equivocator_IM IM) (enum index))
     (free_constraint
        (sub_IM (equivocator_IM IM) (enum index)))
     (λ _ : message, False))
  (composite_vlsm
     (sub_IM (equivocator_IM IM) (enum index))
     (no_equivocations_additional_constraint
        (sub_IM (equivocator_IM IM) (enum index))
        (free_constraint
           (sub_IM (equivocator_IM IM)
              (enum index)))))
Htr': finite_valid_trace
  {|
    vlsm_type :=
      composite_no_equivocation_vlsm_with_preloaded
        (sub_IM (equivocator_IM IM)
           (enum index))
        (free_constraint
           (sub_IM (equivocator_IM IM)
              (enum index)))
        (λ _ : message, False);
    vlsm_machine :=
      composite_no_equivocation_vlsm_with_preloaded
        (sub_IM (equivocator_IM IM)
           (enum index))
        (free_constraint
           (sub_IM (equivocator_IM IM)
              (enum index)))
        (λ _ : message, False)
  |}
  (composite_state_sub_projection
     (equivocator_IM IM) 
     (enum index) s)
  (VLSM_embedding_finite_trace_project Hproj tr)
Hproject: proper_equivocator_descriptors
  (λ i : sub_index (enum index), IM (`i))
  (free_sub_free_equivocator_descriptors
     final_descriptors)
  (finite_trace_last
     (composite_state_sub_projection
        (equivocator_IM IM) 
        (enum index) s)
     (VLSM_embedding_finite_trace_project
        Hproj tr))
→ ∃ (trX : list
             (composite_transition_item
                (λ i : 
                    sub_index 
                      (enum index), 
                   IM (`i)))) 
    (initial_descriptors : 
     equivocator_descriptors
       (λ i : sub_index (enum index),
          IM (`i))),
    let isX :=
      equivocators_state_project
        (λ i : sub_index (enum index),
           IM (`i)) initial_descriptors
        (composite_state_sub_projection
           (equivocator_IM IM) 
           (enum index) s) in
    let final_stateX :=
      finite_trace_last isX trX in
    proper_equivocator_descriptors
      (λ i : sub_index (enum index),
         IM (`i)) initial_descriptors
      (composite_state_sub_projection
         (equivocator_IM IM) 
         (enum index) s)
    ∧ equivocators_trace_project
        (λ i : sub_index (enum index),
           IM (`i))
        (free_sub_free_equivocator_descriptors
           final_descriptors)
        (VLSM_embedding_finite_trace_project
           Hproj tr) =
      Some (trX, initial_descriptors)
      ∧ equivocators_state_project
          (λ i : sub_index (enum index),
             IM (`i))
          (free_sub_free_equivocator_descriptors
             final_descriptors)
          (finite_trace_last
             (composite_state_sub_projection
                (equivocator_IM IM)
                (enum index) s)
             (VLSM_embedding_finite_trace_project
                Hproj tr)) = final_stateX
        ∧ finite_valid_trace
            (preloaded_vlsm
               (free_composite_vlsm
                  (sub_IM IM (enum index)))
               (λ _ : message, False)) isX
            trX
proper_equivocator_descriptors
  (λ i : sub_index (enum index), IM (`i))
  (free_sub_free_equivocator_descriptors
     final_descriptors)
  (finite_trace_last
     (composite_state_sub_projection
        (equivocator_IM IM) (enum index) s)
     (VLSM_embedding_finite_trace_project Hproj tr))
message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
final_descriptors: equivocator_descriptors IM
s: state (equivocators_no_equivocations_vlsm IM)
tr: list transition_item
sX: state Free
trX: list transition_item
Hpr_tr: equivocators_partial_trace_project IM
  final_descriptors (s, tr) = 
Some (sX, trX)
Hproj: VLSM_embedding
  (composite_vlsm (equivocator_IM IM)
     (equivocators_no_equivocations_constraint
        IM))
  (composite_vlsm
     (sub_IM (equivocator_IM IM) (enum index))
     (free_sub_free_constraint
        (equivocator_IM IM)
        (equivocators_no_equivocations_constraint
           IM)))
  (free_sub_free_label (equivocator_IM IM))
  (composite_state_sub_projection
     (equivocator_IM IM) 
     (enum index))
Htr: finite_valid_trace
  (composite_vlsm
     (sub_IM (equivocator_IM IM) (enum index))
     (free_sub_free_constraint
        (equivocator_IM IM)
        (equivocators_no_equivocations_constraint
           IM)))
  (composite_state_sub_projection
     (equivocator_IM IM) 
     (enum index) s)
  (VLSM_embedding_finite_trace_project Hproj tr)
Hnot_equiv: not_equivocating_equivocator_descriptors
  IM final_descriptors
  (finite_trace_last s tr)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors tr =
Some (trX, initial_descriptors)
Hs_project: equivocators_state_project IM
  initial_descriptors s = sX
Hproper: proper_equivocator_descriptors IM
  final_descriptors (finite_trace_last s tr)
Heq: VLSM_eq
  (composite_no_equivocation_vlsm_with_preloaded
     (sub_IM (equivocator_IM IM) (enum index))
     (free_constraint
        (sub_IM (equivocator_IM IM) (enum index)))
     (λ _ : message, False))
  (composite_vlsm
     (sub_IM (equivocator_IM IM) (enum index))
     (no_equivocations_additional_constraint
        (sub_IM (equivocator_IM IM) (enum index))
        (free_constraint
           (sub_IM (equivocator_IM IM)
              (enum index)))))
Htr': finite_valid_trace
  {|
    vlsm_type :=
      composite_no_equivocation_vlsm_with_preloaded
        (sub_IM (equivocator_IM IM)
           (enum index))
        (free_constraint
           (sub_IM (equivocator_IM IM)
              (enum index)))
        (λ _ : message, False);
    vlsm_machine :=
      composite_no_equivocation_vlsm_with_preloaded
        (sub_IM (equivocator_IM IM)
           (enum index))
        (free_constraint
           (sub_IM (equivocator_IM IM)
              (enum index)))
        (λ _ : message, False)
  |}
  (composite_state_sub_projection
     (equivocator_IM IM) 
     (enum index) s)
  (VLSM_embedding_finite_trace_project Hproj tr)
Hproject: ∃ (trX : list
           (composite_transition_item
              (λ i : sub_index (enum index),
                 IM (`i)))) 
  (initial_descriptors : 
   equivocator_descriptors
     (λ i : sub_index (enum index), IM (`i))),
  let isX :=
    equivocators_state_project
      (λ i : sub_index (enum index),
         IM (`i)) initial_descriptors
      (composite_state_sub_projection
         (equivocator_IM IM) 
         (enum index) s) in
  let final_stateX :=
    finite_trace_last isX trX in
  proper_equivocator_descriptors
    (λ i : sub_index (enum index), IM (`i))
    initial_descriptors
    (composite_state_sub_projection
       (equivocator_IM IM) 
       (enum index) s)
  ∧ equivocators_trace_project
      (λ i : sub_index (enum index),
         IM (`i))
      (free_sub_free_equivocator_descriptors
         final_descriptors)
      (VLSM_embedding_finite_trace_project
         Hproj tr) =
    Some (trX, initial_descriptors)
    ∧ equivocators_state_project
        (λ i : sub_index (enum index),
           IM (`i))
        (free_sub_free_equivocator_descriptors
           final_descriptors)
        (finite_trace_last
           (composite_state_sub_projection
              (equivocator_IM IM)
              (enum index) s)
           (VLSM_embedding_finite_trace_project
              Hproj tr)) = final_stateX
      ∧ finite_valid_trace
          (preloaded_vlsm
             (free_composite_vlsm
                (sub_IM IM (enum index)))
             (λ _ : message, False)) isX
          trX
finite_valid_trace Free sX trX
    {message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
final_descriptors: equivocator_descriptors IM
s: state (equivocators_no_equivocations_vlsm IM)
tr: list transition_item
sX: state Free
trX: list transition_item
Hpr_tr: equivocators_partial_trace_project IM
  final_descriptors (s, tr) = 
Some (sX, trX)
Hproj: VLSM_embedding
  (composite_vlsm (equivocator_IM IM)
     (equivocators_no_equivocations_constraint
        IM))
  (composite_vlsm
     (sub_IM (equivocator_IM IM) (enum index))
     (free_sub_free_constraint
        (equivocator_IM IM)
        (equivocators_no_equivocations_constraint
           IM)))
  (free_sub_free_label (equivocator_IM IM))
  (composite_state_sub_projection
     (equivocator_IM IM) 
     (enum index))
Htr: finite_valid_trace
  (composite_vlsm
     (sub_IM (equivocator_IM IM) (enum index))
     (free_sub_free_constraint
        (equivocator_IM IM)
        (equivocators_no_equivocations_constraint
           IM)))
  (composite_state_sub_projection
     (equivocator_IM IM) 
     (enum index) s)
  (VLSM_embedding_finite_trace_project Hproj tr)
Hnot_equiv: not_equivocating_equivocator_descriptors
  IM final_descriptors
  (finite_trace_last s tr)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors tr =
Some (trX, initial_descriptors)
Hs_project: equivocators_state_project IM
  initial_descriptors s = sX
Hproper: proper_equivocator_descriptors IM
  final_descriptors (finite_trace_last s tr)
Heq: VLSM_eq
  (composite_no_equivocation_vlsm_with_preloaded
     (sub_IM (equivocator_IM IM) (enum index))
     (free_constraint
        (sub_IM (equivocator_IM IM) (enum index)))
     (λ _ : message, False))
  (composite_vlsm
     (sub_IM (equivocator_IM IM) (enum index))
     (no_equivocations_additional_constraint
        (sub_IM (equivocator_IM IM) (enum index))
        (free_constraint
           (sub_IM (equivocator_IM IM)
              (enum index)))))
Htr': finite_valid_trace
  {|
    vlsm_type :=
      composite_no_equivocation_vlsm_with_preloaded
        (sub_IM (equivocator_IM IM)
           (enum index))
        (free_constraint
           (sub_IM (equivocator_IM IM)
              (enum index)))
        (λ _ : message, False);
    vlsm_machine :=
      composite_no_equivocation_vlsm_with_preloaded
        (sub_IM (equivocator_IM IM)
           (enum index))
        (free_constraint
           (sub_IM (equivocator_IM IM)
              (enum index)))
        (λ _ : message, False)
  |}
  (composite_state_sub_projection
     (equivocator_IM IM) 
     (enum index) s)
  (VLSM_embedding_finite_trace_project Hproj tr)
Hproject: proper_equivocator_descriptors
  (λ i : sub_index (enum index), IM (`i))
  (free_sub_free_equivocator_descriptors
     final_descriptors)
  (finite_trace_last
     (composite_state_sub_projection
        (equivocator_IM IM) 
        (enum index) s)
     (VLSM_embedding_finite_trace_project
        Hproj tr))
→ ∃ (trX : list
             (composite_transition_item
                (λ i : 
                    sub_index 
                      (enum index), 
                   IM (`i)))) 
    (initial_descriptors : 
     equivocator_descriptors
       (λ i : sub_index (enum index),
          IM (`i))),
    let isX :=
      equivocators_state_project
        (λ i : sub_index (enum index),
           IM (`i)) initial_descriptors
        (composite_state_sub_projection
           (equivocator_IM IM) 
           (enum index) s) in
    let final_stateX :=
      finite_trace_last isX trX in
    proper_equivocator_descriptors
      (λ i : sub_index (enum index),
         IM (`i)) initial_descriptors
      (composite_state_sub_projection
         (equivocator_IM IM) 
         (enum index) s)
    ∧ equivocators_trace_project
        (λ i : sub_index (enum index),
           IM (`i))
        (free_sub_free_equivocator_descriptors
           final_descriptors)
        (VLSM_embedding_finite_trace_project
           Hproj tr) =
      Some (trX, initial_descriptors)
      ∧ equivocators_state_project
          (λ i : sub_index (enum index),
             IM (`i))
          (free_sub_free_equivocator_descriptors
             final_descriptors)
          (finite_trace_last
             (composite_state_sub_projection
                (equivocator_IM IM)
                (enum index) s)
             (VLSM_embedding_finite_trace_project
                Hproj tr)) = final_stateX
        ∧ finite_valid_trace
            (preloaded_vlsm
               (free_composite_vlsm
                  (sub_IM IM (enum index)))
               (λ _ : message, False)) isX
            trX
proper_equivocator_descriptors
  (λ i : sub_index (enum index), IM (`i))
  (free_sub_free_equivocator_descriptors
     final_descriptors)
  (finite_trace_last
     (composite_state_sub_projection
        (equivocator_IM IM) (enum index) s)
     (VLSM_embedding_finite_trace_project Hproj tr))
      clear -Hproper.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
final_descriptors: equivocator_descriptors IM
s: state (equivocators_no_equivocations_vlsm IM)
tr: list transition_item
Hproj: VLSM_embedding
  (composite_vlsm (equivocator_IM IM)
     (equivocators_no_equivocations_constraint
        IM))
  (composite_vlsm
     (sub_IM (equivocator_IM IM) (enum index))
     (free_sub_free_constraint
        (equivocator_IM IM)
        (equivocators_no_equivocations_constraint
           IM)))
  (free_sub_free_label (equivocator_IM IM))
  (composite_state_sub_projection
     (equivocator_IM IM) 
     (enum index))
Hproper: proper_equivocator_descriptors IM
  final_descriptors (finite_trace_last s tr)
proper_equivocator_descriptors
  (λ i : sub_index (enum index), IM (`i))
  (free_sub_free_equivocator_descriptors
     final_descriptors)
  (finite_trace_last
     (composite_state_sub_projection
        (equivocator_IM IM) (enum index) s)
     (VLSM_embedding_finite_trace_project Hproj tr)) intro sub_i.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
final_descriptors: equivocator_descriptors IM
s: state (equivocators_no_equivocations_vlsm IM)
tr: list transition_item
Hproj: VLSM_embedding
  (composite_vlsm (equivocator_IM IM)
     (equivocators_no_equivocations_constraint
        IM))
  (composite_vlsm
     (sub_IM (equivocator_IM IM) (enum index))
     (free_sub_free_constraint
        (equivocator_IM IM)
        (equivocators_no_equivocations_constraint
           IM)))
  (free_sub_free_label (equivocator_IM IM))
  (composite_state_sub_projection
     (equivocator_IM IM) 
     (enum index))
Hproper: proper_equivocator_descriptors IM
  final_descriptors (finite_trace_last s tr)
sub_i: sub_index (enum index)
proper_descriptor (IM (`sub_i))
  (free_sub_free_equivocator_descriptors
     final_descriptors sub_i)
  (finite_trace_last
     (composite_state_sub_projection
        (equivocator_IM IM) (enum index) s)
     (VLSM_embedding_finite_trace_project Hproj tr)
     sub_i)
      destruct_dec_sig sub_i i Hi Heqsub_i.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
final_descriptors: equivocator_descriptors IM
s: state (equivocators_no_equivocations_vlsm IM)
tr: list transition_item
Hproj: VLSM_embedding
  (composite_vlsm (equivocator_IM IM)
     (equivocators_no_equivocations_constraint
        IM))
  (composite_vlsm
     (sub_IM (equivocator_IM IM) (enum index))
     (free_sub_free_constraint
        (equivocator_IM IM)
        (equivocators_no_equivocations_constraint
           IM)))
  (free_sub_free_label (equivocator_IM IM))
  (composite_state_sub_projection
     (equivocator_IM IM) 
     (enum index))
Hproper: proper_equivocator_descriptors IM
  final_descriptors (finite_trace_last s tr)
sub_i: sub_index (enum index)
i: index
Hi: sub_index_prop (enum index) i
Heqsub_i: sub_i = dexist i Hi
proper_descriptor (IM (`sub_i))
  (free_sub_free_equivocator_descriptors
     final_descriptors sub_i)
  (finite_trace_last
     (composite_state_sub_projection
        (equivocator_IM IM) (enum index) s)
     (VLSM_embedding_finite_trace_project Hproj tr)
     sub_i) subst.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
final_descriptors: equivocator_descriptors IM
s: state (equivocators_no_equivocations_vlsm IM)
tr: list transition_item
Hproj: VLSM_embedding
  (composite_vlsm (equivocator_IM IM)
     (equivocators_no_equivocations_constraint
        IM))
  (composite_vlsm
     (sub_IM (equivocator_IM IM) (enum index))
     (free_sub_free_constraint
        (equivocator_IM IM)
        (equivocators_no_equivocations_constraint
           IM)))
  (free_sub_free_label (equivocator_IM IM))
  (composite_state_sub_projection
     (equivocator_IM IM) 
     (enum index))
Hproper: proper_equivocator_descriptors IM
  final_descriptors (finite_trace_last s tr)
i: index
Hi: sub_index_prop (enum index) i
proper_descriptor (IM (`(dexist i Hi)))
  (free_sub_free_equivocator_descriptors
     final_descriptors (dexist i Hi))
  (finite_trace_last
     (composite_state_sub_projection
        (equivocator_IM IM) (enum index) s)
     (VLSM_embedding_finite_trace_project Hproj tr)
     (dexist i Hi))
      rewrite <- (VLSM_embedding_finite_trace_last Hproj).message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
final_descriptors: equivocator_descriptors IM
s: state (equivocators_no_equivocations_vlsm IM)
tr: list transition_item
Hproj: VLSM_embedding
  (composite_vlsm (equivocator_IM IM)
     (equivocators_no_equivocations_constraint
        IM))
  (composite_vlsm
     (sub_IM (equivocator_IM IM) (enum index))
     (free_sub_free_constraint
        (equivocator_IM IM)
        (equivocators_no_equivocations_constraint
           IM)))
  (free_sub_free_label (equivocator_IM IM))
  (composite_state_sub_projection
     (equivocator_IM IM) 
     (enum index))
Hproper: proper_equivocator_descriptors IM
  final_descriptors (finite_trace_last s tr)
i: index
Hi: sub_index_prop (enum index) i
proper_descriptor (IM (`(dexist i Hi)))
  (free_sub_free_equivocator_descriptors
     final_descriptors (dexist i Hi))
  (composite_state_sub_projection (equivocator_IM IM)
     (enum index) (finite_trace_last s tr)
     (dexist i Hi))
      by apply Hproper.
    }message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
final_descriptors: equivocator_descriptors IM
s: state (equivocators_no_equivocations_vlsm IM)
tr: list transition_item
sX: state Free
trX: list transition_item
Hpr_tr: equivocators_partial_trace_project IM
  final_descriptors (s, tr) = 
Some (sX, trX)
Hproj: VLSM_embedding
  (composite_vlsm (equivocator_IM IM)
     (equivocators_no_equivocations_constraint
        IM))
  (composite_vlsm
     (sub_IM (equivocator_IM IM) (enum index))
     (free_sub_free_constraint
        (equivocator_IM IM)
        (equivocators_no_equivocations_constraint
           IM)))
  (free_sub_free_label (equivocator_IM IM))
  (composite_state_sub_projection
     (equivocator_IM IM) 
     (enum index))
Htr: finite_valid_trace
  (composite_vlsm
     (sub_IM (equivocator_IM IM) (enum index))
     (free_sub_free_constraint
        (equivocator_IM IM)
        (equivocators_no_equivocations_constraint
           IM)))
  (composite_state_sub_projection
     (equivocator_IM IM) 
     (enum index) s)
  (VLSM_embedding_finite_trace_project Hproj tr)
Hnot_equiv: not_equivocating_equivocator_descriptors
  IM final_descriptors
  (finite_trace_last s tr)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors tr =
Some (trX, initial_descriptors)
Hs_project: equivocators_state_project IM
  initial_descriptors s = sX
Hproper: proper_equivocator_descriptors IM
  final_descriptors (finite_trace_last s tr)
Heq: VLSM_eq
  (composite_no_equivocation_vlsm_with_preloaded
     (sub_IM (equivocator_IM IM) (enum index))
     (free_constraint
        (sub_IM (equivocator_IM IM) (enum index)))
     (λ _ : message, False))
  (composite_vlsm
     (sub_IM (equivocator_IM IM) (enum index))
     (no_equivocations_additional_constraint
        (sub_IM (equivocator_IM IM) (enum index))
        (free_constraint
           (sub_IM (equivocator_IM IM)
              (enum index)))))
Htr': finite_valid_trace
  {|
    vlsm_type :=
      composite_no_equivocation_vlsm_with_preloaded
        (sub_IM (equivocator_IM IM)
           (enum index))
        (free_constraint
           (sub_IM (equivocator_IM IM)
              (enum index)))
        (λ _ : message, False);
    vlsm_machine :=
      composite_no_equivocation_vlsm_with_preloaded
        (sub_IM (equivocator_IM IM)
           (enum index))
        (free_constraint
           (sub_IM (equivocator_IM IM)
              (enum index)))
        (λ _ : message, False)
  |}
  (composite_state_sub_projection
     (equivocator_IM IM) 
     (enum index) s)
  (VLSM_embedding_finite_trace_project Hproj tr)
Hproject: ∃ (trX : list
           (composite_transition_item
              (λ i : sub_index (enum index),
                 IM (`i)))) 
  (initial_descriptors : 
   equivocator_descriptors
     (λ i : sub_index (enum index), IM (`i))),
  let isX :=
    equivocators_state_project
      (λ i : sub_index (enum index),
         IM (`i)) initial_descriptors
      (composite_state_sub_projection
         (equivocator_IM IM) 
         (enum index) s) in
  let final_stateX :=
    finite_trace_last isX trX in
  proper_equivocator_descriptors
    (λ i : sub_index (enum index), IM (`i))
    initial_descriptors
    (composite_state_sub_projection
       (equivocator_IM IM) 
       (enum index) s)
  ∧ equivocators_trace_project
      (λ i : sub_index (enum index),
         IM (`i))
      (free_sub_free_equivocator_descriptors
         final_descriptors)
      (VLSM_embedding_finite_trace_project
         Hproj tr) =
    Some (trX, initial_descriptors)
    ∧ equivocators_state_project
        (λ i : sub_index (enum index),
           IM (`i))
        (free_sub_free_equivocator_descriptors
           final_descriptors)
        (finite_trace_last
           (composite_state_sub_projection
              (equivocator_IM IM)
              (enum index) s)
           (VLSM_embedding_finite_trace_project
              Hproj tr)) = final_stateX
      ∧ finite_valid_trace
          (preloaded_vlsm
             (free_composite_vlsm
                (sub_IM IM (enum index)))
             (λ _ : message, False)) isX
          trX
finite_valid_trace Free sX trX
    destruct Hproject  as [_trX [_initial_descriptors [_ [_Htr_project [_ HtrX]]]]].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
final_descriptors: equivocator_descriptors IM
s: state (equivocators_no_equivocations_vlsm IM)
tr: list transition_item
sX: state Free
trX: list transition_item
Hpr_tr: equivocators_partial_trace_project IM
  final_descriptors (s, tr) = 
Some (sX, trX)
Hproj: VLSM_embedding
  (composite_vlsm (equivocator_IM IM)
     (equivocators_no_equivocations_constraint
        IM))
  (composite_vlsm
     (sub_IM (equivocator_IM IM) (enum index))
     (free_sub_free_constraint
        (equivocator_IM IM)
        (equivocators_no_equivocations_constraint
           IM)))
  (free_sub_free_label (equivocator_IM IM))
  (composite_state_sub_projection
     (equivocator_IM IM) 
     (enum index))
Htr: finite_valid_trace
  (composite_vlsm
     (sub_IM (equivocator_IM IM) (enum index))
     (free_sub_free_constraint
        (equivocator_IM IM)
        (equivocators_no_equivocations_constraint
           IM)))
  (composite_state_sub_projection
     (equivocator_IM IM) 
     (enum index) s)
  (VLSM_embedding_finite_trace_project Hproj tr)
Hnot_equiv: not_equivocating_equivocator_descriptors
  IM final_descriptors
  (finite_trace_last s tr)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors tr =
Some (trX, initial_descriptors)
Hs_project: equivocators_state_project IM
  initial_descriptors s = sX
Hproper: proper_equivocator_descriptors IM
  final_descriptors (finite_trace_last s tr)
Heq: VLSM_eq
  (composite_no_equivocation_vlsm_with_preloaded
     (sub_IM (equivocator_IM IM) (enum index))
     (free_constraint
        (sub_IM (equivocator_IM IM) (enum index)))
     (λ _ : message, False))
  (composite_vlsm
     (sub_IM (equivocator_IM IM) (enum index))
     (no_equivocations_additional_constraint
        (sub_IM (equivocator_IM IM) (enum index))
        (free_constraint
           (sub_IM (equivocator_IM IM)
              (enum index)))))
Htr': finite_valid_trace
  {|
    vlsm_type :=
      composite_no_equivocation_vlsm_with_preloaded
        (sub_IM (equivocator_IM IM)
           (enum index))
        (free_constraint
           (sub_IM (equivocator_IM IM)
              (enum index)))
        (λ _ : message, False);
    vlsm_machine :=
      composite_no_equivocation_vlsm_with_preloaded
        (sub_IM (equivocator_IM IM)
           (enum index))
        (free_constraint
           (sub_IM (equivocator_IM IM)
              (enum index)))
        (λ _ : message, False)
  |}
  (composite_state_sub_projection
     (equivocator_IM IM) 
     (enum index) s)
  (VLSM_embedding_finite_trace_project Hproj tr)
_trX: list
  (composite_transition_item
     (λ i : sub_index (enum index), IM (`i)))
_initial_descriptors: equivocator_descriptors
  (λ i : sub_index (enum index),
     IM (`i))
_Htr_project: equivocators_trace_project
  (λ i : sub_index (enum index),
     IM (`i))
  (free_sub_free_equivocator_descriptors
     final_descriptors)
  (VLSM_embedding_finite_trace_project
     Hproj tr) =
Some (_trX, _initial_descriptors)
HtrX: finite_valid_trace
  (preloaded_vlsm
     (free_composite_vlsm
        (sub_IM IM (enum index)))
     (λ _ : message, False))
  (equivocators_state_project
     (λ i : sub_index (enum index), IM (`i))
     _initial_descriptors
     (composite_state_sub_projection
        (equivocator_IM IM) 
        (enum index) s)) _trX
finite_valid_trace Free sX trX

    specialize
      (equivocators_trace_project_finite_trace_sub_projection_commute IM (enum index)
        final_descriptors initial_descriptors _initial_descriptors tr trX _trX
        Htr_project)
      as Hcommute.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
final_descriptors: equivocator_descriptors IM
s: state (equivocators_no_equivocations_vlsm IM)
tr: list transition_item
sX: state Free
trX: list transition_item
Hpr_tr: equivocators_partial_trace_project IM
  final_descriptors (s, tr) = 
Some (sX, trX)
Hproj: VLSM_embedding
  (composite_vlsm (equivocator_IM IM)
     (equivocators_no_equivocations_constraint
        IM))
  (composite_vlsm
     (sub_IM (equivocator_IM IM) (enum index))
     (free_sub_free_constraint
        (equivocator_IM IM)
        (equivocators_no_equivocations_constraint
           IM)))
  (free_sub_free_label (equivocator_IM IM))
  (composite_state_sub_projection
     (equivocator_IM IM) 
     (enum index))
Htr: finite_valid_trace
  (composite_vlsm
     (sub_IM (equivocator_IM IM) (enum index))
     (free_sub_free_constraint
        (equivocator_IM IM)
        (equivocators_no_equivocations_constraint
           IM)))
  (composite_state_sub_projection
     (equivocator_IM IM) 
     (enum index) s)
  (VLSM_embedding_finite_trace_project Hproj tr)
Hnot_equiv: not_equivocating_equivocator_descriptors
  IM final_descriptors
  (finite_trace_last s tr)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors tr =
Some (trX, initial_descriptors)
Hs_project: equivocators_state_project IM
  initial_descriptors s = sX
Hproper: proper_equivocator_descriptors IM
  final_descriptors (finite_trace_last s tr)
Heq: VLSM_eq
  (composite_no_equivocation_vlsm_with_preloaded
     (sub_IM (equivocator_IM IM) (enum index))
     (free_constraint
        (sub_IM (equivocator_IM IM) (enum index)))
     (λ _ : message, False))
  (composite_vlsm
     (sub_IM (equivocator_IM IM) (enum index))
     (no_equivocations_additional_constraint
        (sub_IM (equivocator_IM IM) (enum index))
        (free_constraint
           (sub_IM (equivocator_IM IM)
              (enum index)))))
Htr': finite_valid_trace
  {|
    vlsm_type :=
      composite_no_equivocation_vlsm_with_preloaded
        (sub_IM (equivocator_IM IM)
           (enum index))
        (free_constraint
           (sub_IM (equivocator_IM IM)
              (enum index)))
        (λ _ : message, False);
    vlsm_machine :=
      composite_no_equivocation_vlsm_with_preloaded
        (sub_IM (equivocator_IM IM)
           (enum index))
        (free_constraint
           (sub_IM (equivocator_IM IM)
              (enum index)))
        (λ _ : message, False)
  |}
  (composite_state_sub_projection
     (equivocator_IM IM) 
     (enum index) s)
  (VLSM_embedding_finite_trace_project Hproj tr)
_trX: list
  (composite_transition_item
     (λ i : sub_index (enum index), IM (`i)))
_initial_descriptors: equivocator_descriptors
  (λ i : sub_index (enum index),
     IM (`i))
_Htr_project: equivocators_trace_project
  (λ i : sub_index (enum index),
     IM (`i))
  (free_sub_free_equivocator_descriptors
     final_descriptors)
  (VLSM_embedding_finite_trace_project
     Hproj tr) =
Some (_trX, _initial_descriptors)
HtrX: finite_valid_trace
  (preloaded_vlsm
     (free_composite_vlsm
        (sub_IM IM (enum index)))
     (λ _ : message, False))
  (equivocators_state_project
     (λ i : sub_index (enum index), IM (`i))
     _initial_descriptors
     (composite_state_sub_projection
        (equivocator_IM IM) 
        (enum index) s)) _trX
Hcommute: equivocators_trace_project
  (sub_IM IM (enum index))
  (λ i : sub_index (enum index),
     final_descriptors (`i))
  (finite_trace_sub_projection
     (equivocator_IM IM) 
     (enum index) tr) =
Some (_trX, _initial_descriptors)
→ _initial_descriptors =
  (λ i : sub_index (enum index),
     initial_descriptors (`i))
  ∧ finite_trace_sub_projection IM
      (enum index) trX = _trX
finite_valid_trace Free sX trX
    spec Hcommute.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
final_descriptors: equivocator_descriptors IM
s: state (equivocators_no_equivocations_vlsm IM)
tr: list transition_item
sX: state Free
trX: list transition_item
Hpr_tr: equivocators_partial_trace_project IM
  final_descriptors (s, tr) = 
Some (sX, trX)
Hproj: VLSM_embedding
  (composite_vlsm (equivocator_IM IM)
     (equivocators_no_equivocations_constraint
        IM))
  (composite_vlsm
     (sub_IM (equivocator_IM IM) (enum index))
     (free_sub_free_constraint
        (equivocator_IM IM)
        (equivocators_no_equivocations_constraint
           IM)))
  (free_sub_free_label (equivocator_IM IM))
  (composite_state_sub_projection
     (equivocator_IM IM) 
     (enum index))
Htr: finite_valid_trace
  (composite_vlsm
     (sub_IM (equivocator_IM IM) (enum index))
     (free_sub_free_constraint
        (equivocator_IM IM)
        (equivocators_no_equivocations_constraint
           IM)))
  (composite_state_sub_projection
     (equivocator_IM IM) 
     (enum index) s)
  (VLSM_embedding_finite_trace_project Hproj tr)
Hnot_equiv: not_equivocating_equivocator_descriptors
  IM final_descriptors
  (finite_trace_last s tr)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors tr =
Some (trX, initial_descriptors)
Hs_project: equivocators_state_project IM
  initial_descriptors s = sX
Hproper: proper_equivocator_descriptors IM
  final_descriptors (finite_trace_last s tr)
Heq: VLSM_eq
  (composite_no_equivocation_vlsm_with_preloaded
     (sub_IM (equivocator_IM IM) (enum index))
     (free_constraint
        (sub_IM (equivocator_IM IM) (enum index)))
     (λ _ : message, False))
  (composite_vlsm
     (sub_IM (equivocator_IM IM) (enum index))
     (no_equivocations_additional_constraint
        (sub_IM (equivocator_IM IM) (enum index))
        (free_constraint
           (sub_IM (equivocator_IM IM)
              (enum index)))))
Htr': finite_valid_trace
  {|
    vlsm_type :=
      composite_no_equivocation_vlsm_with_preloaded
        (sub_IM (equivocator_IM IM)
           (enum index))
        (free_constraint
           (sub_IM (equivocator_IM IM)
              (enum index)))
        (λ _ : message, False);
    vlsm_machine :=
      composite_no_equivocation_vlsm_with_preloaded
        (sub_IM (equivocator_IM IM)
           (enum index))
        (free_constraint
           (sub_IM (equivocator_IM IM)
              (enum index)))
        (λ _ : message, False)
  |}
  (composite_state_sub_projection
     (equivocator_IM IM) 
     (enum index) s)
  (VLSM_embedding_finite_trace_project Hproj tr)
_trX: list
  (composite_transition_item
     (λ i : sub_index (enum index), IM (`i)))
_initial_descriptors: equivocator_descriptors
  (λ i : sub_index (enum index),
     IM (`i))
_Htr_project: equivocators_trace_project
  (λ i : sub_index (enum index),
     IM (`i))
  (free_sub_free_equivocator_descriptors
     final_descriptors)
  (VLSM_embedding_finite_trace_project
     Hproj tr) =
Some (_trX, _initial_descriptors)
HtrX: finite_valid_trace
  (preloaded_vlsm
     (free_composite_vlsm
        (sub_IM IM (enum index)))
     (λ _ : message, False))
  (equivocators_state_project
     (λ i : sub_index (enum index), IM (`i))
     _initial_descriptors
     (composite_state_sub_projection
        (equivocator_IM IM) 
        (enum index) s)) _trX
Hcommute: equivocators_trace_project
  (sub_IM IM (enum index))
  (λ i : sub_index (enum index),
     final_descriptors (`i))
  (finite_trace_sub_projection
     (equivocator_IM IM) 
     (enum index) tr) =
Some (_trX, _initial_descriptors)
→ _initial_descriptors =
  (λ i : sub_index (enum index),
     initial_descriptors (`i))
  ∧ finite_trace_sub_projection IM
      (enum index) trX = _trX
equivocators_trace_project (sub_IM IM (enum index))
  (λ i : sub_index (enum index),
     final_descriptors (`i))
  (finite_trace_sub_projection (equivocator_IM IM)
     (enum index) tr) =
Some (_trX, _initial_descriptors)
message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
final_descriptors: equivocator_descriptors IM
s: state (equivocators_no_equivocations_vlsm IM)
tr: list transition_item
sX: state Free
trX: list transition_item
Hpr_tr: equivocators_partial_trace_project IM
  final_descriptors (s, tr) = 
Some (sX, trX)
Hproj: VLSM_embedding
  (composite_vlsm (equivocator_IM IM)
     (equivocators_no_equivocations_constraint
        IM))
  (composite_vlsm
     (sub_IM (equivocator_IM IM) (enum index))
     (free_sub_free_constraint
        (equivocator_IM IM)
        (equivocators_no_equivocations_constraint
           IM)))
  (free_sub_free_label (equivocator_IM IM))
  (composite_state_sub_projection
     (equivocator_IM IM) 
     (enum index))
Htr: finite_valid_trace
  (composite_vlsm
     (sub_IM (equivocator_IM IM) (enum index))
     (free_sub_free_constraint
        (equivocator_IM IM)
        (equivocators_no_equivocations_constraint
           IM)))
  (composite_state_sub_projection
     (equivocator_IM IM) 
     (enum index) s)
  (VLSM_embedding_finite_trace_project Hproj tr)
Hnot_equiv: not_equivocating_equivocator_descriptors
  IM final_descriptors
  (finite_trace_last s tr)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors tr =
Some (trX, initial_descriptors)
Hs_project: equivocators_state_project IM
  initial_descriptors s = sX
Hproper: proper_equivocator_descriptors IM
  final_descriptors (finite_trace_last s tr)
Heq: VLSM_eq
  (composite_no_equivocation_vlsm_with_preloaded
     (sub_IM (equivocator_IM IM) (enum index))
     (free_constraint
        (sub_IM (equivocator_IM IM) (enum index)))
     (λ _ : message, False))
  (composite_vlsm
     (sub_IM (equivocator_IM IM) (enum index))
     (no_equivocations_additional_constraint
        (sub_IM (equivocator_IM IM) (enum index))
        (free_constraint
           (sub_IM (equivocator_IM IM)
              (enum index)))))
Htr': finite_valid_trace
  {|
    vlsm_type :=
      composite_no_equivocation_vlsm_with_preloaded
        (sub_IM (equivocator_IM IM)
           (enum index))
        (free_constraint
           (sub_IM (equivocator_IM IM)
              (enum index)))
        (λ _ : message, False);
    vlsm_machine :=
      composite_no_equivocation_vlsm_with_preloaded
        (sub_IM (equivocator_IM IM)
           (enum index))
        (free_constraint
           (sub_IM (equivocator_IM IM)
              (enum index)))
        (λ _ : message, False)
  |}
  (composite_state_sub_projection
     (equivocator_IM IM) 
     (enum index) s)
  (VLSM_embedding_finite_trace_project Hproj tr)
_trX: list
  (composite_transition_item
     (λ i : sub_index (enum index), IM (`i)))
_initial_descriptors: equivocator_descriptors
  (λ i : sub_index (enum index),
     IM (`i))
_Htr_project: equivocators_trace_project
  (λ i : sub_index (enum index),
     IM (`i))
  (free_sub_free_equivocator_descriptors
     final_descriptors)
  (VLSM_embedding_finite_trace_project
     Hproj tr) =
Some (_trX, _initial_descriptors)
HtrX: finite_valid_trace
  (preloaded_vlsm
     (free_composite_vlsm
        (sub_IM IM (enum index)))
     (λ _ : message, False))
  (equivocators_state_project
     (λ i : sub_index (enum index), IM (`i))
     _initial_descriptors
     (composite_state_sub_projection
        (equivocator_IM IM) 
        (enum index) s)) _trX
Hcommute: _initial_descriptors =
(λ i : sub_index (enum index),
   initial_descriptors (`i))
∧ finite_trace_sub_projection IM
    (enum index) trX = _trX
finite_valid_trace Free sX trX
    {message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
final_descriptors: equivocator_descriptors IM
s: state (equivocators_no_equivocations_vlsm IM)
tr: list transition_item
sX: state Free
trX: list transition_item
Hpr_tr: equivocators_partial_trace_project IM
  final_descriptors (s, tr) = 
Some (sX, trX)
Hproj: VLSM_embedding
  (composite_vlsm (equivocator_IM IM)
     (equivocators_no_equivocations_constraint
        IM))
  (composite_vlsm
     (sub_IM (equivocator_IM IM) (enum index))
     (free_sub_free_constraint
        (equivocator_IM IM)
        (equivocators_no_equivocations_constraint
           IM)))
  (free_sub_free_label (equivocator_IM IM))
  (composite_state_sub_projection
     (equivocator_IM IM) 
     (enum index))
Htr: finite_valid_trace
  (composite_vlsm
     (sub_IM (equivocator_IM IM) (enum index))
     (free_sub_free_constraint
        (equivocator_IM IM)
        (equivocators_no_equivocations_constraint
           IM)))
  (composite_state_sub_projection
     (equivocator_IM IM) 
     (enum index) s)
  (VLSM_embedding_finite_trace_project Hproj tr)
Hnot_equiv: not_equivocating_equivocator_descriptors
  IM final_descriptors
  (finite_trace_last s tr)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors tr =
Some (trX, initial_descriptors)
Hs_project: equivocators_state_project IM
  initial_descriptors s = sX
Hproper: proper_equivocator_descriptors IM
  final_descriptors (finite_trace_last s tr)
Heq: VLSM_eq
  (composite_no_equivocation_vlsm_with_preloaded
     (sub_IM (equivocator_IM IM) (enum index))
     (free_constraint
        (sub_IM (equivocator_IM IM) (enum index)))
     (λ _ : message, False))
  (composite_vlsm
     (sub_IM (equivocator_IM IM) (enum index))
     (no_equivocations_additional_constraint
        (sub_IM (equivocator_IM IM) (enum index))
        (free_constraint
           (sub_IM (equivocator_IM IM)
              (enum index)))))
Htr': finite_valid_trace
  {|
    vlsm_type :=
      composite_no_equivocation_vlsm_with_preloaded
        (sub_IM (equivocator_IM IM)
           (enum index))
        (free_constraint
           (sub_IM (equivocator_IM IM)
              (enum index)))
        (λ _ : message, False);
    vlsm_machine :=
      composite_no_equivocation_vlsm_with_preloaded
        (sub_IM (equivocator_IM IM)
           (enum index))
        (free_constraint
           (sub_IM (equivocator_IM IM)
              (enum index)))
        (λ _ : message, False)
  |}
  (composite_state_sub_projection
     (equivocator_IM IM) 
     (enum index) s)
  (VLSM_embedding_finite_trace_project Hproj tr)
_trX: list
  (composite_transition_item
     (λ i : sub_index (enum index), IM (`i)))
_initial_descriptors: equivocator_descriptors
  (λ i : sub_index (enum index),
     IM (`i))
_Htr_project: equivocators_trace_project
  (λ i : sub_index (enum index),
     IM (`i))
  (free_sub_free_equivocator_descriptors
     final_descriptors)
  (VLSM_embedding_finite_trace_project
     Hproj tr) =
Some (_trX, _initial_descriptors)
HtrX: finite_valid_trace
  (preloaded_vlsm
     (free_composite_vlsm
        (sub_IM IM (enum index)))
     (λ _ : message, False))
  (equivocators_state_project
     (λ i : sub_index (enum index), IM (`i))
     _initial_descriptors
     (composite_state_sub_projection
        (equivocator_IM IM) 
        (enum index) s)) _trX
Hcommute: equivocators_trace_project
  (sub_IM IM (enum index))
  (λ i : sub_index (enum index),
     final_descriptors (`i))
  (finite_trace_sub_projection
     (equivocator_IM IM) 
     (enum index) tr) =
Some (_trX, _initial_descriptors)
→ _initial_descriptors =
  (λ i : sub_index (enum index),
     initial_descriptors (`i))
  ∧ finite_trace_sub_projection IM
      (enum index) trX = _trX
equivocators_trace_project (sub_IM IM (enum index))
  (λ i : sub_index (enum index),
     final_descriptors (`i))
  (finite_trace_sub_projection (equivocator_IM IM)
     (enum index) tr) =
Some (_trX, _initial_descriptors) replace (finite_trace_sub_projection _ _ _)
        with (VLSM_embedding_finite_trace_project Hproj tr)
      ; [done |].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
final_descriptors: equivocator_descriptors IM
s: state (equivocators_no_equivocations_vlsm IM)
tr: list transition_item
sX: state Free
trX: list transition_item
Hpr_tr: equivocators_partial_trace_project IM
  final_descriptors (s, tr) = 
Some (sX, trX)
Hproj: VLSM_embedding
  (composite_vlsm (equivocator_IM IM)
     (equivocators_no_equivocations_constraint
        IM))
  (composite_vlsm
     (sub_IM (equivocator_IM IM) (enum index))
     (free_sub_free_constraint
        (equivocator_IM IM)
        (equivocators_no_equivocations_constraint
           IM)))
  (free_sub_free_label (equivocator_IM IM))
  (composite_state_sub_projection
     (equivocator_IM IM) 
     (enum index))
Htr: finite_valid_trace
  (composite_vlsm
     (sub_IM (equivocator_IM IM) (enum index))
     (free_sub_free_constraint
        (equivocator_IM IM)
        (equivocators_no_equivocations_constraint
           IM)))
  (composite_state_sub_projection
     (equivocator_IM IM) 
     (enum index) s)
  (VLSM_embedding_finite_trace_project Hproj tr)
Hnot_equiv: not_equivocating_equivocator_descriptors
  IM final_descriptors
  (finite_trace_last s tr)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors tr =
Some (trX, initial_descriptors)
Hs_project: equivocators_state_project IM
  initial_descriptors s = sX
Hproper: proper_equivocator_descriptors IM
  final_descriptors (finite_trace_last s tr)
Heq: VLSM_eq
  (composite_no_equivocation_vlsm_with_preloaded
     (sub_IM (equivocator_IM IM) (enum index))
     (free_constraint
        (sub_IM (equivocator_IM IM) (enum index)))
     (λ _ : message, False))
  (composite_vlsm
     (sub_IM (equivocator_IM IM) (enum index))
     (no_equivocations_additional_constraint
        (sub_IM (equivocator_IM IM) (enum index))
        (free_constraint
           (sub_IM (equivocator_IM IM)
              (enum index)))))
Htr': finite_valid_trace
  {|
    vlsm_type :=
      composite_no_equivocation_vlsm_with_preloaded
        (sub_IM (equivocator_IM IM)
           (enum index))
        (free_constraint
           (sub_IM (equivocator_IM IM)
              (enum index)))
        (λ _ : message, False);
    vlsm_machine :=
      composite_no_equivocation_vlsm_with_preloaded
        (sub_IM (equivocator_IM IM)
           (enum index))
        (free_constraint
           (sub_IM (equivocator_IM IM)
              (enum index)))
        (λ _ : message, False)
  |}
  (composite_state_sub_projection
     (equivocator_IM IM) 
     (enum index) s)
  (VLSM_embedding_finite_trace_project Hproj tr)
_trX: list
  (composite_transition_item
     (λ i : sub_index (enum index), IM (`i)))
_initial_descriptors: equivocator_descriptors
  (λ i : sub_index (enum index),
     IM (`i))
_Htr_project: equivocators_trace_project
  (λ i : sub_index (enum index),
     IM (`i))
  (free_sub_free_equivocator_descriptors
     final_descriptors)
  (VLSM_embedding_finite_trace_project
     Hproj tr) =
Some (_trX, _initial_descriptors)
HtrX: finite_valid_trace
  (preloaded_vlsm
     (free_composite_vlsm
        (sub_IM IM (enum index)))
     (λ _ : message, False))
  (equivocators_state_project
     (λ i : sub_index (enum index), IM (`i))
     _initial_descriptors
     (composite_state_sub_projection
        (equivocator_IM IM) 
        (enum index) s)) _trX
Hcommute: equivocators_trace_project
  (sub_IM IM (enum index))
  (λ i : sub_index (enum index),
     final_descriptors (`i))
  (finite_trace_sub_projection
     (equivocator_IM IM) 
     (enum index) tr) =
Some (_trX, _initial_descriptors)
→ _initial_descriptors =
  (λ i : sub_index (enum index),
     initial_descriptors (`i))
  ∧ finite_trace_sub_projection IM
      (enum index) trX = _trX
VLSM_embedding_finite_trace_project Hproj tr =
finite_trace_sub_projection (equivocator_IM IM)
  (enum index) tr
      clear.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
tr: list transition_item
Hproj: VLSM_embedding
  (composite_vlsm (equivocator_IM IM)
     (equivocators_no_equivocations_constraint
        IM))
  (composite_vlsm
     (sub_IM (equivocator_IM IM) (enum index))
     (free_sub_free_constraint
        (equivocator_IM IM)
        (equivocators_no_equivocations_constraint
           IM)))
  (free_sub_free_label (equivocator_IM IM))
  (composite_state_sub_projection
     (equivocator_IM IM) 
     (enum index))
VLSM_embedding_finite_trace_project Hproj tr =
finite_trace_sub_projection (equivocator_IM IM)
  (enum index) tr
      induction tr; [done |].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
a: transition_item
tr: list transition_item
Hproj: VLSM_embedding
  (composite_vlsm (equivocator_IM IM)
     (equivocators_no_equivocations_constraint
        IM))
  (composite_vlsm
     (sub_IM (equivocator_IM IM) (enum index))
     (free_sub_free_constraint
        (equivocator_IM IM)
        (equivocators_no_equivocations_constraint
           IM)))
  (free_sub_free_label (equivocator_IM IM))
  (composite_state_sub_projection
     (equivocator_IM IM) 
     (enum index))
IHtr: VLSM_embedding_finite_trace_project Hproj tr =
finite_trace_sub_projection 
  (equivocator_IM IM) (enum index) tr
VLSM_embedding_finite_trace_project Hproj (a :: tr) =
finite_trace_sub_projection (equivocator_IM IM)
  (enum index) (a :: tr)
      simpl.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
a: transition_item
tr: list transition_item
Hproj: VLSM_embedding
  (composite_vlsm (equivocator_IM IM)
     (equivocators_no_equivocations_constraint
        IM))
  (composite_vlsm
     (sub_IM (equivocator_IM IM) (enum index))
     (free_sub_free_constraint
        (equivocator_IM IM)
        (equivocators_no_equivocations_constraint
           IM)))
  (free_sub_free_label (equivocator_IM IM))
  (composite_state_sub_projection
     (equivocator_IM IM) 
     (enum index))
IHtr: VLSM_embedding_finite_trace_project Hproj tr =
finite_trace_sub_projection 
  (equivocator_IM IM) (enum index) tr
pre_VLSM_embedding_transition_item_project
  (composite_type (equivocator_IM IM))
  (composite_type
     (sub_IM (equivocator_IM IM) (enum index)))
  (free_sub_free_label (equivocator_IM IM))
  (composite_state_sub_projection (equivocator_IM IM)
     (enum index)) a
:: VLSM_embedding_finite_trace_project Hproj tr =
match
  pre_VLSM_projection_transition_item_project
    (composite_type (equivocator_IM IM))
    (composite_type
       (sub_IM (equivocator_IM IM) (enum index)))
    (composite_label_sub_projection_option
       (equivocator_IM IM) (enum index))
    (composite_state_sub_projection
       (equivocator_IM IM) (enum index)) a
with
| Some y =>
    y
    :: finite_trace_sub_projection (equivocator_IM IM)
         (enum index) tr
| None =>
    finite_trace_sub_projection (equivocator_IM IM)
      (enum index) tr
end
      unfold pre_VLSM_projection_transition_item_project,
        composite_label_sub_projection_option,
        pre_VLSM_embedding_transition_item_project.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
a: transition_item
tr: list transition_item
Hproj: VLSM_embedding
  (composite_vlsm (equivocator_IM IM)
     (equivocators_no_equivocations_constraint
        IM))
  (composite_vlsm
     (sub_IM (equivocator_IM IM) (enum index))
     (free_sub_free_constraint
        (equivocator_IM IM)
        (equivocators_no_equivocations_constraint
           IM)))
  (free_sub_free_label (equivocator_IM IM))
  (composite_state_sub_projection
     (equivocator_IM IM) 
     (enum index))
IHtr: VLSM_embedding_finite_trace_project Hproj tr =
finite_trace_sub_projection 
  (equivocator_IM IM) (enum index) tr
{|
  l := free_sub_free_label (equivocator_IM IM) (l a);
  input := input a;
  destination :=
    composite_state_sub_projection (equivocator_IM IM)
      (enum index) (destination a);
  output := output a
|} :: VLSM_embedding_finite_trace_project Hproj tr =
match
  match
    match decide (projT1 (l a) ∈ enum index) with
    | left i_in =>
        Some
          (composite_label_sub_projection
             (equivocator_IM IM) (enum index) (l a)
             i_in)
    | right _ => None
    end
  with
  | Some lY =>
      Some
        {|
          l := lY;
          input := input a;
          destination :=
            composite_state_sub_projection
              (equivocator_IM IM) (enum index)
              (destination a);
          output := output a
        |}
  | None => None
  end
with
| Some y =>
    y
    :: finite_trace_sub_projection (equivocator_IM IM)
         (enum index) tr
| None =>
    finite_trace_sub_projection (equivocator_IM IM)
      (enum index) tr
end
      simpl.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
a: transition_item
tr: list transition_item
Hproj: VLSM_embedding
  (composite_vlsm (equivocator_IM IM)
     (equivocators_no_equivocations_constraint
        IM))
  (composite_vlsm
     (sub_IM (equivocator_IM IM) (enum index))
     (free_sub_free_constraint
        (equivocator_IM IM)
        (equivocators_no_equivocations_constraint
           IM)))
  (free_sub_free_label (equivocator_IM IM))
  (composite_state_sub_projection
     (equivocator_IM IM) 
     (enum index))
IHtr: VLSM_embedding_finite_trace_project Hproj tr =
finite_trace_sub_projection 
  (equivocator_IM IM) (enum index) tr
{|
  l := free_sub_free_label (equivocator_IM IM) (l a);
  input := input a;
  destination :=
    composite_state_sub_projection (equivocator_IM IM)
      (enum index) (destination a);
  output := output a
|} :: VLSM_embedding_finite_trace_project Hproj tr =
match
  match
    match decide (projT1 (l a) ∈ enum index) with
    | left i_in =>
        Some
          (composite_label_sub_projection
             (equivocator_IM IM) (enum index) (l a)
             i_in)
    | right _ => None
    end
  with
  | Some lY =>
      Some
        {|
          l := lY;
          input := input a;
          destination :=
            composite_state_sub_projection
              (equivocator_IM IM) (enum index)
              (destination a);
          output := output a
        |}
  | None => None
  end
with
| Some y =>
    y
    :: finite_trace_sub_projection (equivocator_IM IM)
         (enum index) tr
| None =>
    finite_trace_sub_projection (equivocator_IM IM)
      (enum index) tr
end
      case_decide as Hla; [| contradict Hla; apply elem_of_enum].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
a: transition_item
tr: list transition_item
Hproj: VLSM_embedding
  (composite_vlsm (equivocator_IM IM)
     (equivocators_no_equivocations_constraint
        IM))
  (composite_vlsm
     (sub_IM (equivocator_IM IM) (enum index))
     (free_sub_free_constraint
        (equivocator_IM IM)
        (equivocators_no_equivocations_constraint
           IM)))
  (free_sub_free_label (equivocator_IM IM))
  (composite_state_sub_projection
     (equivocator_IM IM) 
     (enum index))
IHtr: VLSM_embedding_finite_trace_project Hproj tr =
finite_trace_sub_projection 
  (equivocator_IM IM) (enum index) tr
Hla: projT1 (l a) ∈ enum index
{|
  l := free_sub_free_label (equivocator_IM IM) (l a);
  input := input a;
  destination :=
    composite_state_sub_projection (equivocator_IM IM)
      (enum index) (destination a);
  output := output a
|} :: VLSM_embedding_finite_trace_project Hproj tr =
{|
  l :=
    composite_label_sub_projection (equivocator_IM IM)
      (enum index) (l a) Hla;
  input := input a;
  destination :=
    composite_state_sub_projection (equivocator_IM IM)
      (enum index) (destination a);
  output := output a
|}
:: finite_trace_sub_projection (equivocator_IM IM)
     (enum index) tr
      f_equal; [| done].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
a: transition_item
tr: list transition_item
Hproj: VLSM_embedding
  (composite_vlsm (equivocator_IM IM)
     (equivocators_no_equivocations_constraint
        IM))
  (composite_vlsm
     (sub_IM (equivocator_IM IM) (enum index))
     (free_sub_free_constraint
        (equivocator_IM IM)
        (equivocators_no_equivocations_constraint
           IM)))
  (free_sub_free_label (equivocator_IM IM))
  (composite_state_sub_projection
     (equivocator_IM IM) 
     (enum index))
IHtr: VLSM_embedding_finite_trace_project Hproj tr =
finite_trace_sub_projection 
  (equivocator_IM IM) (enum index) tr
Hla: projT1 (l a) ∈ enum index
{|
  l := free_sub_free_label (equivocator_IM IM) (l a);
  input := input a;
  destination :=
    composite_state_sub_projection (equivocator_IM IM)
      (enum index) (destination a);
  output := output a
|} =
{|
  l :=
    composite_label_sub_projection (equivocator_IM IM)
      (enum index) (l a) Hla;
  input := input a;
  destination :=
    composite_state_sub_projection (equivocator_IM IM)
      (enum index) (destination a);
  output := output a
|}
      destruct a, l as (i, li); cbn; f_equal.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
i: index
li: label (equivocator_IM IM i)
input: option message
destination: state
  (equivocators_no_equivocations_vlsm IM)
output: option message
tr: list transition_item
Hproj: VLSM_embedding
  (composite_vlsm (equivocator_IM IM)
     (equivocators_no_equivocations_constraint
        IM))
  (composite_vlsm
     (sub_IM (equivocator_IM IM) (enum index))
     (free_sub_free_constraint
        (equivocator_IM IM)
        (equivocators_no_equivocations_constraint
           IM)))
  (free_sub_free_label (equivocator_IM IM))
  (composite_state_sub_projection
     (equivocator_IM IM) 
     (enum index))
IHtr: VLSM_embedding_finite_trace_project Hproj tr =
finite_trace_sub_projection 
  (equivocator_IM IM) (enum index) tr
Hla: projT1
  (l
     {|
       l := existT i li;
       input := input;
       destination := destination;
       output := output
     |}) ∈ enum index
existT (free_sub_free_index i) li =
composite_label_sub_projection (equivocator_IM IM)
  (enum index) (existT i li) Hla
      unfold composite_label_sub_projection;
      cbn; unfold free_sub_free_index.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
i: index
li: label (equivocator_IM IM i)
input: option message
destination: state
  (equivocators_no_equivocations_vlsm IM)
output: option message
tr: list transition_item
Hproj: VLSM_embedding
  (composite_vlsm (equivocator_IM IM)
     (equivocators_no_equivocations_constraint
        IM))
  (composite_vlsm
     (sub_IM (equivocator_IM IM) (enum index))
     (free_sub_free_constraint
        (equivocator_IM IM)
        (equivocators_no_equivocations_constraint
           IM)))
  (free_sub_free_label (equivocator_IM IM))
  (composite_state_sub_projection
     (equivocator_IM IM) 
     (enum index))
IHtr: VLSM_embedding_finite_trace_project Hproj tr =
finite_trace_sub_projection 
  (equivocator_IM IM) (enum index) tr
Hla: projT1
  (l
     {|
       l := existT i li;
       input := input;
       destination := destination;
       output := output
     |}) ∈ enum index
existT
  (dexist i
     (SubProjectionTraces.free_sub_free_index_obligation_1
        i)) li = existT (dexist i Hla) li
      by apply
        (@dec_sig_sigT_eq _ _
          (sub_index_prop_dec (enum index))
          (fun n => label (EquivocatorsComposition.equivocator_IM IM n))
          i li li).
    }message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
final_descriptors: equivocator_descriptors IM
s: state (equivocators_no_equivocations_vlsm IM)
tr: list transition_item
sX: state Free
trX: list transition_item
Hpr_tr: equivocators_partial_trace_project IM
  final_descriptors (s, tr) = 
Some (sX, trX)
Hproj: VLSM_embedding
  (composite_vlsm (equivocator_IM IM)
     (equivocators_no_equivocations_constraint
        IM))
  (composite_vlsm
     (sub_IM (equivocator_IM IM) (enum index))
     (free_sub_free_constraint
        (equivocator_IM IM)
        (equivocators_no_equivocations_constraint
           IM)))
  (free_sub_free_label (equivocator_IM IM))
  (composite_state_sub_projection
     (equivocator_IM IM) 
     (enum index))
Htr: finite_valid_trace
  (composite_vlsm
     (sub_IM (equivocator_IM IM) (enum index))
     (free_sub_free_constraint
        (equivocator_IM IM)
        (equivocators_no_equivocations_constraint
           IM)))
  (composite_state_sub_projection
     (equivocator_IM IM) 
     (enum index) s)
  (VLSM_embedding_finite_trace_project Hproj tr)
Hnot_equiv: not_equivocating_equivocator_descriptors
  IM final_descriptors
  (finite_trace_last s tr)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors tr =
Some (trX, initial_descriptors)
Hs_project: equivocators_state_project IM
  initial_descriptors s = sX
Hproper: proper_equivocator_descriptors IM
  final_descriptors (finite_trace_last s tr)
Heq: VLSM_eq
  (composite_no_equivocation_vlsm_with_preloaded
     (sub_IM (equivocator_IM IM) (enum index))
     (free_constraint
        (sub_IM (equivocator_IM IM) (enum index)))
     (λ _ : message, False))
  (composite_vlsm
     (sub_IM (equivocator_IM IM) (enum index))
     (no_equivocations_additional_constraint
        (sub_IM (equivocator_IM IM) (enum index))
        (free_constraint
           (sub_IM (equivocator_IM IM)
              (enum index)))))
Htr': finite_valid_trace
  {|
    vlsm_type :=
      composite_no_equivocation_vlsm_with_preloaded
        (sub_IM (equivocator_IM IM)
           (enum index))
        (free_constraint
           (sub_IM (equivocator_IM IM)
              (enum index)))
        (λ _ : message, False);
    vlsm_machine :=
      composite_no_equivocation_vlsm_with_preloaded
        (sub_IM (equivocator_IM IM)
           (enum index))
        (free_constraint
           (sub_IM (equivocator_IM IM)
              (enum index)))
        (λ _ : message, False)
  |}
  (composite_state_sub_projection
     (equivocator_IM IM) 
     (enum index) s)
  (VLSM_embedding_finite_trace_project Hproj tr)
_trX: list
  (composite_transition_item
     (λ i : sub_index (enum index), IM (`i)))
_initial_descriptors: equivocator_descriptors
  (λ i : sub_index (enum index),
     IM (`i))
_Htr_project: equivocators_trace_project
  (λ i : sub_index (enum index),
     IM (`i))
  (free_sub_free_equivocator_descriptors
     final_descriptors)
  (VLSM_embedding_finite_trace_project
     Hproj tr) =
Some (_trX, _initial_descriptors)
HtrX: finite_valid_trace
  (preloaded_vlsm
     (free_composite_vlsm
        (sub_IM IM (enum index)))
     (λ _ : message, False))
  (equivocators_state_project
     (λ i : sub_index (enum index), IM (`i))
     _initial_descriptors
     (composite_state_sub_projection
        (equivocator_IM IM) 
        (enum index) s)) _trX
Hcommute: _initial_descriptors =
(λ i : sub_index (enum index),
   initial_descriptors (`i))
∧ finite_trace_sub_projection IM
    (enum index) trX = _trX
finite_valid_trace Free sX trX
    destruct Hcommute as [Heq_initial Heq_trX].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
final_descriptors: equivocator_descriptors IM
s: state (equivocators_no_equivocations_vlsm IM)
tr: list transition_item
sX: state Free
trX: list transition_item
Hpr_tr: equivocators_partial_trace_project IM
  final_descriptors (s, tr) = 
Some (sX, trX)
Hproj: VLSM_embedding
  (composite_vlsm (equivocator_IM IM)
     (equivocators_no_equivocations_constraint
        IM))
  (composite_vlsm
     (sub_IM (equivocator_IM IM) (enum index))
     (free_sub_free_constraint
        (equivocator_IM IM)
        (equivocators_no_equivocations_constraint
           IM)))
  (free_sub_free_label (equivocator_IM IM))
  (composite_state_sub_projection
     (equivocator_IM IM) 
     (enum index))
Htr: finite_valid_trace
  (composite_vlsm
     (sub_IM (equivocator_IM IM) (enum index))
     (free_sub_free_constraint
        (equivocator_IM IM)
        (equivocators_no_equivocations_constraint
           IM)))
  (composite_state_sub_projection
     (equivocator_IM IM) 
     (enum index) s)
  (VLSM_embedding_finite_trace_project Hproj tr)
Hnot_equiv: not_equivocating_equivocator_descriptors
  IM final_descriptors
  (finite_trace_last s tr)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors tr =
Some (trX, initial_descriptors)
Hs_project: equivocators_state_project IM
  initial_descriptors s = sX
Hproper: proper_equivocator_descriptors IM
  final_descriptors (finite_trace_last s tr)
Heq: VLSM_eq
  (composite_no_equivocation_vlsm_with_preloaded
     (sub_IM (equivocator_IM IM) (enum index))
     (free_constraint
        (sub_IM (equivocator_IM IM) (enum index)))
     (λ _ : message, False))
  (composite_vlsm
     (sub_IM (equivocator_IM IM) (enum index))
     (no_equivocations_additional_constraint
        (sub_IM (equivocator_IM IM) (enum index))
        (free_constraint
           (sub_IM (equivocator_IM IM)
              (enum index)))))
Htr': finite_valid_trace
  {|
    vlsm_type :=
      composite_no_equivocation_vlsm_with_preloaded
        (sub_IM (equivocator_IM IM)
           (enum index))
        (free_constraint
           (sub_IM (equivocator_IM IM)
              (enum index)))
        (λ _ : message, False);
    vlsm_machine :=
      composite_no_equivocation_vlsm_with_preloaded
        (sub_IM (equivocator_IM IM)
           (enum index))
        (free_constraint
           (sub_IM (equivocator_IM IM)
              (enum index)))
        (λ _ : message, False)
  |}
  (composite_state_sub_projection
     (equivocator_IM IM) 
     (enum index) s)
  (VLSM_embedding_finite_trace_project Hproj tr)
_trX: list
  (composite_transition_item
     (λ i : sub_index (enum index), IM (`i)))
_initial_descriptors: equivocator_descriptors
  (λ i : sub_index (enum index),
     IM (`i))
_Htr_project: equivocators_trace_project
  (λ i : sub_index (enum index),
     IM (`i))
  (free_sub_free_equivocator_descriptors
     final_descriptors)
  (VLSM_embedding_finite_trace_project
     Hproj tr) =
Some (_trX, _initial_descriptors)
HtrX: finite_valid_trace
  (preloaded_vlsm
     (free_composite_vlsm
        (sub_IM IM (enum index)))
     (λ _ : message, False))
  (equivocators_state_project
     (λ i : sub_index (enum index), IM (`i))
     _initial_descriptors
     (composite_state_sub_projection
        (equivocator_IM IM) 
        (enum index) s)) _trX
Heq_initial: _initial_descriptors =
(λ i : sub_index (enum index),
   initial_descriptors (`i))
Heq_trX: finite_trace_sub_projection IM 
  (enum index) trX = _trX
finite_valid_trace Free sX trX
    subst.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
final_descriptors: equivocator_descriptors IM
s: state (equivocators_no_equivocations_vlsm IM)
tr, trX: list transition_item
initial_descriptors: equivocator_descriptors IM
Hpr_tr: equivocators_partial_trace_project IM
  final_descriptors (s, tr) =
Some
  (equivocators_state_project IM
     initial_descriptors s, trX)
Hproj: VLSM_embedding
  (composite_vlsm (equivocator_IM IM)
     (equivocators_no_equivocations_constraint
        IM))
  (composite_vlsm
     (sub_IM (equivocator_IM IM) (enum index))
     (free_sub_free_constraint
        (equivocator_IM IM)
        (equivocators_no_equivocations_constraint
           IM)))
  (free_sub_free_label (equivocator_IM IM))
  (composite_state_sub_projection
     (equivocator_IM IM) 
     (enum index))
Htr: finite_valid_trace
  (composite_vlsm
     (sub_IM (equivocator_IM IM) (enum index))
     (free_sub_free_constraint
        (equivocator_IM IM)
        (equivocators_no_equivocations_constraint
           IM)))
  (composite_state_sub_projection
     (equivocator_IM IM) 
     (enum index) s)
  (VLSM_embedding_finite_trace_project Hproj tr)
Hnot_equiv: not_equivocating_equivocator_descriptors
  IM final_descriptors
  (finite_trace_last s tr)
Htr_project: equivocators_trace_project IM
  final_descriptors tr =
Some (trX, initial_descriptors)
Hproper: proper_equivocator_descriptors IM
  final_descriptors (finite_trace_last s tr)
Heq: VLSM_eq
  (composite_no_equivocation_vlsm_with_preloaded
     (sub_IM (equivocator_IM IM) (enum index))
     (free_constraint
        (sub_IM (equivocator_IM IM) (enum index)))
     (λ _ : message, False))
  (composite_vlsm
     (sub_IM (equivocator_IM IM) (enum index))
     (no_equivocations_additional_constraint
        (sub_IM (equivocator_IM IM) (enum index))
        (free_constraint
           (sub_IM (equivocator_IM IM)
              (enum index)))))
Htr': finite_valid_trace
  {|
    vlsm_type :=
      composite_no_equivocation_vlsm_with_preloaded
        (sub_IM (equivocator_IM IM)
           (enum index))
        (free_constraint
           (sub_IM (equivocator_IM IM)
              (enum index)))
        (λ _ : message, False);
    vlsm_machine :=
      composite_no_equivocation_vlsm_with_preloaded
        (sub_IM (equivocator_IM IM)
           (enum index))
        (free_constraint
           (sub_IM (equivocator_IM IM)
              (enum index)))
        (λ _ : message, False)
  |}
  (composite_state_sub_projection
     (equivocator_IM IM) 
     (enum index) s)
  (VLSM_embedding_finite_trace_project Hproj tr)
_Htr_project: equivocators_trace_project
  (λ i : sub_index (enum index),
     IM (`i))
  (free_sub_free_equivocator_descriptors
     final_descriptors)
  (VLSM_embedding_finite_trace_project
     Hproj tr) =
Some
  (finite_trace_sub_projection IM
     (enum index) trX,
   λ i : sub_index (enum index),
     initial_descriptors (`i))
HtrX: finite_valid_trace
  (preloaded_vlsm
     (free_composite_vlsm
        (sub_IM IM (enum index)))
     (λ _ : message, False))
  (equivocators_state_project
     (λ i : sub_index (enum index), IM (`i))
     (λ i : sub_index (enum index),
        initial_descriptors (`i))
     (composite_state_sub_projection
        (equivocator_IM IM) 
        (enum index) s))
  (finite_trace_sub_projection IM 
     (enum index) trX)
finite_valid_trace Free
  (equivocators_state_project IM initial_descriptors s)
  trX
    clear -HtrX.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
Free:= free_composite_vlsm IM: VLSM message
s: state (equivocators_no_equivocations_vlsm IM)
trX: list transition_item
initial_descriptors: equivocator_descriptors IM
HtrX: finite_valid_trace
  (preloaded_vlsm
     (free_composite_vlsm
        (sub_IM IM (enum index)))
     (λ _ : message, False))
  (equivocators_state_project
     (λ i : sub_index (enum index), IM (`i))
     (λ i : sub_index (enum index),
        initial_descriptors (`i))
     (composite_state_sub_projection
        (equivocator_IM IM) 
        (enum index) s))
  (finite_trace_sub_projection IM 
     (enum index) trX)
finite_valid_trace Free
  (equivocators_state_project IM initial_descriptors s)
  trX
    specialize
      (vlsm_is_preloaded_with_False
        (free_composite_vlsm (SubProjectionTraces.sub_IM IM (finite.enum index))))
      as Heq.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
Free:= free_composite_vlsm IM: VLSM message
s: state (equivocators_no_equivocations_vlsm IM)
trX: list transition_item
initial_descriptors: equivocator_descriptors IM
HtrX: finite_valid_trace
  (preloaded_vlsm
     (free_composite_vlsm
        (sub_IM IM (enum index)))
     (λ _ : message, False))
  (equivocators_state_project
     (λ i : sub_index (enum index), IM (`i))
     (λ i : sub_index (enum index),
        initial_descriptors (`i))
     (composite_state_sub_projection
        (equivocator_IM IM) 
        (enum index) s))
  (finite_trace_sub_projection IM 
     (enum index) trX)
Heq: VLSM_eq
  (free_composite_vlsm (sub_IM IM (enum index)))
  (preloaded_vlsm
     (free_composite_vlsm
        (sub_IM IM (enum index)))
     (λ _ : message, False))
finite_valid_trace Free
  (equivocators_state_project IM initial_descriptors s)
  trX
    apply (VLSM_eq_finite_valid_trace Heq) in HtrX.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
Free:= free_composite_vlsm IM: VLSM message
s: state (equivocators_no_equivocations_vlsm IM)
trX: list transition_item
initial_descriptors: equivocator_descriptors IM
HtrX: finite_valid_trace
  {|
    vlsm_type :=
      free_composite_vlsm
        (sub_IM IM (enum index));
    vlsm_machine :=
      free_composite_vlsm
        (sub_IM IM (enum index))
  |}
  (equivocators_state_project
     (λ i : sub_index (enum index), IM (`i))
     (λ i : sub_index (enum index),
        initial_descriptors (`i))
     (composite_state_sub_projection
        (equivocator_IM IM) 
        (enum index) s))
  (finite_trace_sub_projection IM 
     (enum index) trX)
Heq: VLSM_eq
  (free_composite_vlsm (sub_IM IM (enum index)))
  (preloaded_vlsm
     (free_composite_vlsm
        (sub_IM IM (enum index)))
     (λ _ : message, False))
finite_valid_trace Free
  (equivocators_state_project IM initial_descriptors s)
  trX
    specialize (sub_composition_all_embedding_rev IM (free_constraint IM)) as Hproj.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
Free:= free_composite_vlsm IM: VLSM message
s: state (equivocators_no_equivocations_vlsm IM)
trX: list transition_item
initial_descriptors: equivocator_descriptors IM
HtrX: finite_valid_trace
  {|
    vlsm_type :=
      free_composite_vlsm
        (sub_IM IM (enum index));
    vlsm_machine :=
      free_composite_vlsm
        (sub_IM IM (enum index))
  |}
  (equivocators_state_project
     (λ i : sub_index (enum index), IM (`i))
     (λ i : sub_index (enum index),
        initial_descriptors (`i))
     (composite_state_sub_projection
        (equivocator_IM IM) 
        (enum index) s))
  (finite_trace_sub_projection IM 
     (enum index) trX)
Heq: VLSM_eq
  (free_composite_vlsm (sub_IM IM (enum index)))
  (preloaded_vlsm
     (free_composite_vlsm
        (sub_IM IM (enum index)))
     (λ _ : message, False))
Hproj: VLSM_embedding
  (composite_vlsm (sub_IM IM (enum index))
     (free_sub_free_constraint IM
        (free_constraint IM)))
  (composite_vlsm IM (free_constraint IM))
  (lift_sub_label IM (enum index))
  (free_sub_free_state IM)
finite_valid_trace Free
  (equivocators_state_project IM initial_descriptors s)
  trX
    assert (HtrX' : finite_valid_trace
      (composite_vlsm (SubProjectionTraces.sub_IM IM (finite.enum index))
      (free_sub_free_constraint IM (free_constraint IM)))
      (EquivocatorsComposition.equivocators_state_project
        (SubProjectionTraces.sub_IM IM (finite.enum index))
        (fun i : sub_index (finite.enum index) => initial_descriptors (` i))
        (composite_state_sub_projection (equivocator_IM IM) (finite.enum index) s))
      (finite_trace_sub_projection IM (finite.enum index) trX)).message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
Free:= free_composite_vlsm IM: VLSM message
s: state (equivocators_no_equivocations_vlsm IM)
trX: list transition_item
initial_descriptors: equivocator_descriptors IM
HtrX: finite_valid_trace
  {|
    vlsm_type :=
      free_composite_vlsm
        (sub_IM IM (enum index));
    vlsm_machine :=
      free_composite_vlsm
        (sub_IM IM (enum index))
  |}
  (equivocators_state_project
     (λ i : sub_index (enum index), IM (`i))
     (λ i : sub_index (enum index),
        initial_descriptors (`i))
     (composite_state_sub_projection
        (equivocator_IM IM) 
        (enum index) s))
  (finite_trace_sub_projection IM 
     (enum index) trX)
Heq: VLSM_eq
  (free_composite_vlsm (sub_IM IM (enum index)))
  (preloaded_vlsm
     (free_composite_vlsm
        (sub_IM IM (enum index)))
     (λ _ : message, False))
Hproj: VLSM_embedding
  (composite_vlsm (sub_IM IM (enum index))
     (free_sub_free_constraint IM
        (free_constraint IM)))
  (composite_vlsm IM (free_constraint IM))
  (lift_sub_label IM (enum index))
  (free_sub_free_state IM)
finite_valid_trace
  (composite_vlsm (sub_IM IM (enum index))
     (free_sub_free_constraint IM (free_constraint IM)))
  (equivocators_state_project (sub_IM IM (enum index))
     (λ i : sub_index (enum index),
        initial_descriptors (`i))
     (composite_state_sub_projection
        (equivocator_IM IM) (enum index) s))
  (finite_trace_sub_projection IM (enum index) trX)
message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
Free:= free_composite_vlsm IM: VLSM message
s: state (equivocators_no_equivocations_vlsm IM)
trX: list transition_item
initial_descriptors: equivocator_descriptors IM
HtrX: finite_valid_trace
  {|
    vlsm_type :=
      free_composite_vlsm
        (sub_IM IM (enum index));
    vlsm_machine :=
      free_composite_vlsm
        (sub_IM IM (enum index))
  |}
  (equivocators_state_project
     (λ i : sub_index (enum index), IM (`i))
     (λ i : sub_index (enum index),
        initial_descriptors (`i))
     (composite_state_sub_projection
        (equivocator_IM IM) 
        (enum index) s))
  (finite_trace_sub_projection IM 
     (enum index) trX)
Heq: VLSM_eq
  (free_composite_vlsm (sub_IM IM (enum index)))
  (preloaded_vlsm
     (free_composite_vlsm
        (sub_IM IM (enum index)))
     (λ _ : message, False))
Hproj: VLSM_embedding
  (composite_vlsm (sub_IM IM (enum index))
     (free_sub_free_constraint IM
        (free_constraint IM)))
  (composite_vlsm IM (free_constraint IM))
  (lift_sub_label IM (enum index))
  (free_sub_free_state IM)
HtrX': finite_valid_trace
  (composite_vlsm (sub_IM IM (enum index))
     (free_sub_free_constraint IM
        (free_constraint IM)))
  (equivocators_state_project
     (sub_IM IM (enum index))
     (λ i : sub_index (enum index),
        initial_descriptors (`i))
     (composite_state_sub_projection
        (equivocator_IM IM) 
        (enum index) s))
  (finite_trace_sub_projection IM 
     (enum index) trX)
finite_valid_trace Free
  (equivocators_state_project IM initial_descriptors s)
  trX
    {message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
Free:= free_composite_vlsm IM: VLSM message
s: state (equivocators_no_equivocations_vlsm IM)
trX: list transition_item
initial_descriptors: equivocator_descriptors IM
HtrX: finite_valid_trace
  {|
    vlsm_type :=
      free_composite_vlsm
        (sub_IM IM (enum index));
    vlsm_machine :=
      free_composite_vlsm
        (sub_IM IM (enum index))
  |}
  (equivocators_state_project
     (λ i : sub_index (enum index), IM (`i))
     (λ i : sub_index (enum index),
        initial_descriptors (`i))
     (composite_state_sub_projection
        (equivocator_IM IM) 
        (enum index) s))
  (finite_trace_sub_projection IM 
     (enum index) trX)
Heq: VLSM_eq
  (free_composite_vlsm (sub_IM IM (enum index)))
  (preloaded_vlsm
     (free_composite_vlsm
        (sub_IM IM (enum index)))
     (λ _ : message, False))
Hproj: VLSM_embedding
  (composite_vlsm (sub_IM IM (enum index))
     (free_sub_free_constraint IM
        (free_constraint IM)))
  (composite_vlsm IM (free_constraint IM))
  (lift_sub_label IM (enum index))
  (free_sub_free_state IM)
finite_valid_trace
  (composite_vlsm (sub_IM IM (enum index))
     (free_sub_free_constraint IM (free_constraint IM)))
  (equivocators_state_project (sub_IM IM (enum index))
     (λ i : sub_index (enum index),
        initial_descriptors (`i))
     (composite_state_sub_projection
        (equivocator_IM IM) (enum index) s))
  (finite_trace_sub_projection IM (enum index) trX)
      revert HtrX.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
Free:= free_composite_vlsm IM: VLSM message
s: state (equivocators_no_equivocations_vlsm IM)
trX: list transition_item
initial_descriptors: equivocator_descriptors IM
Heq: VLSM_eq
  (free_composite_vlsm (sub_IM IM (enum index)))
  (preloaded_vlsm
     (free_composite_vlsm
        (sub_IM IM (enum index)))
     (λ _ : message, False))
Hproj: VLSM_embedding
  (composite_vlsm (sub_IM IM (enum index))
     (free_sub_free_constraint IM
        (free_constraint IM)))
  (composite_vlsm IM (free_constraint IM))
  (lift_sub_label IM (enum index))
  (free_sub_free_state IM)
finite_valid_trace
  {|
    vlsm_type :=
      free_composite_vlsm (sub_IM IM (enum index));
    vlsm_machine :=
      free_composite_vlsm (sub_IM IM (enum index))
  |}
  (equivocators_state_project
     (λ i : sub_index (enum index), IM (`i))
     (λ i : sub_index (enum index),
        initial_descriptors (`i))
     (composite_state_sub_projection
        (equivocator_IM IM) (enum index) s))
  (finite_trace_sub_projection IM (enum index) trX)
→ finite_valid_trace
    (composite_vlsm (sub_IM IM (enum index))
       (free_sub_free_constraint IM
          (free_constraint IM)))
    (equivocators_state_project
       (sub_IM IM (enum index))
       (λ i : sub_index (enum index),
          initial_descriptors (`i))
       (composite_state_sub_projection
          (equivocator_IM IM) (enum index) s))
    (finite_trace_sub_projection IM (enum index) trX)
      apply VLSM_incl_finite_valid_trace.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
Free:= free_composite_vlsm IM: VLSM message
s: state (equivocators_no_equivocations_vlsm IM)
trX: list transition_item
initial_descriptors: equivocator_descriptors IM
Heq: VLSM_eq
  (free_composite_vlsm (sub_IM IM (enum index)))
  (preloaded_vlsm
     (free_composite_vlsm
        (sub_IM IM (enum index)))
     (λ _ : message, False))
Hproj: VLSM_embedding
  (composite_vlsm (sub_IM IM (enum index))
     (free_sub_free_constraint IM
        (free_constraint IM)))
  (composite_vlsm IM (free_constraint IM))
  (lift_sub_label IM (enum index))
  (free_sub_free_state IM)
VLSM_incl_part
  (free_composite_vlsm (sub_IM IM (enum index)))
  (constrained_vlsm_machine
     (free_composite_vlsm (sub_IM IM (enum index)))
     (free_sub_free_constraint IM (free_constraint IM)))
      eapply VLSM_incl_trans.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
Free:= free_composite_vlsm IM: VLSM message
s: state (equivocators_no_equivocations_vlsm IM)
trX: list transition_item
initial_descriptors: equivocator_descriptors IM
Heq: VLSM_eq
  (free_composite_vlsm (sub_IM IM (enum index)))
  (preloaded_vlsm
     (free_composite_vlsm
        (sub_IM IM (enum index)))
     (λ _ : message, False))
Hproj: VLSM_embedding
  (composite_vlsm (sub_IM IM (enum index))
     (free_sub_free_constraint IM
        (free_constraint IM)))
  (composite_vlsm IM (free_constraint IM))
  (lift_sub_label IM (enum index))
  (free_sub_free_state IM)
VLSM_incl
  {|
    vlsm_type :=
      free_composite_vlsm (sub_IM IM (enum index));
    vlsm_machine :=
      free_composite_vlsm_machine
        (sub_IM IM (enum index))
  |}
  {|
    vlsm_type :=
      free_composite_vlsm (sub_IM IM (enum index));
    vlsm_machine := ?MY
  |}
message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
Free:= free_composite_vlsm IM: VLSM message
s: state (equivocators_no_equivocations_vlsm IM)
trX: list transition_item
initial_descriptors: equivocator_descriptors IM
Heq: VLSM_eq
  (free_composite_vlsm (sub_IM IM (enum index)))
  (preloaded_vlsm
     (free_composite_vlsm
        (sub_IM IM (enum index)))
     (λ _ : message, False))
Hproj: VLSM_embedding
  (composite_vlsm (sub_IM IM (enum index))
     (free_sub_free_constraint IM
        (free_constraint IM)))
  (composite_vlsm IM (free_constraint IM))
  (lift_sub_label IM (enum index))
  (free_sub_free_state IM)
VLSM_incl
  {|
    vlsm_type :=
      free_composite_vlsm (sub_IM IM (enum index));
    vlsm_machine := ?MY
  |}
  {|
    vlsm_type :=
      free_composite_vlsm (sub_IM IM (enum index));
    vlsm_machine :=
      {|
        initial_state_prop := initial_state_prop;
        s0 :=
          s0
            (free_composite_vlsm
               (sub_IM IM (enum index)))
            (free_composite_vlsm
               (sub_IM IM (enum index)));
        initial_message_prop := initial_message_prop;
        transition := transition;
        valid :=
          λ (l : label
                   (free_composite_vlsm
                      (sub_IM IM (enum index)))) 
            (som : state
                     (free_composite_vlsm
                      (sub_IM IM (enum index))) *
                   option message),
            valid l som
            ∧ free_sub_free_constraint IM
                (free_constraint IM) l som
      |}
  |}
      -message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
Free:= free_composite_vlsm IM: VLSM message
s: state (equivocators_no_equivocations_vlsm IM)
trX: list transition_item
initial_descriptors: equivocator_descriptors IM
Heq: VLSM_eq
  (free_composite_vlsm (sub_IM IM (enum index)))
  (preloaded_vlsm
     (free_composite_vlsm
        (sub_IM IM (enum index)))
     (λ _ : message, False))
Hproj: VLSM_embedding
  (composite_vlsm (sub_IM IM (enum index))
     (free_sub_free_constraint IM
        (free_constraint IM)))
  (composite_vlsm IM (free_constraint IM))
  (lift_sub_label IM (enum index))
  (free_sub_free_state IM)
VLSM_incl
  {|
    vlsm_type :=
      free_composite_vlsm (sub_IM IM (enum index));
    vlsm_machine :=
      free_composite_vlsm_machine
        (sub_IM IM (enum index))
  |}
  {|
    vlsm_type :=
      free_composite_vlsm (sub_IM IM (enum index));
    vlsm_machine := ?MY
  |} by apply free_composite_vlsm_spec.
      -message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
Free:= free_composite_vlsm IM: VLSM message
s: state (equivocators_no_equivocations_vlsm IM)
trX: list transition_item
initial_descriptors: equivocator_descriptors IM
Heq: VLSM_eq
  (free_composite_vlsm (sub_IM IM (enum index)))
  (preloaded_vlsm
     (free_composite_vlsm
        (sub_IM IM (enum index)))
     (λ _ : message, False))
Hproj: VLSM_embedding
  (composite_vlsm (sub_IM IM (enum index))
     (free_sub_free_constraint IM
        (free_constraint IM)))
  (composite_vlsm IM (free_constraint IM))
  (lift_sub_label IM (enum index))
  (free_sub_free_state IM)
VLSM_incl
  {|
    vlsm_type :=
      free_composite_vlsm (sub_IM IM (enum index));
    vlsm_machine :=
      composite_vlsm (sub_IM IM (enum index))
        (free_constraint (sub_IM IM (enum index)))
  |}
  {|
    vlsm_type :=
      free_composite_vlsm (sub_IM IM (enum index));
    vlsm_machine :=
      {|
        initial_state_prop := initial_state_prop;
        s0 :=
          s0
            (free_composite_vlsm
               (sub_IM IM (enum index)))
            (free_composite_vlsm
               (sub_IM IM (enum index)));
        initial_message_prop := initial_message_prop;
        transition := transition;
        valid :=
          λ (l : label
                   (free_composite_vlsm
                      (sub_IM IM (enum index)))) (som : 
                                                 state
                                                 (free_composite_vlsm
                                                 (sub_IM
                                                 IM
                                                 (enum
                                                 index))) *
                                                 option
                                                 message),
            valid l som
            ∧ free_sub_free_constraint IM
                (free_constraint IM) l som
      |}
  |} by apply constraint_subsumption_incl; intros [] [].
    }message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
Free:= free_composite_vlsm IM: VLSM message
s: state (equivocators_no_equivocations_vlsm IM)
trX: list transition_item
initial_descriptors: equivocator_descriptors IM
HtrX: finite_valid_trace
  {|
    vlsm_type :=
      free_composite_vlsm
        (sub_IM IM (enum index));
    vlsm_machine :=
      free_composite_vlsm
        (sub_IM IM (enum index))
  |}
  (equivocators_state_project
     (λ i : sub_index (enum index), IM (`i))
     (λ i : sub_index (enum index),
        initial_descriptors (`i))
     (composite_state_sub_projection
        (equivocator_IM IM) 
        (enum index) s))
  (finite_trace_sub_projection IM 
     (enum index) trX)
Heq: VLSM_eq
  (free_composite_vlsm (sub_IM IM (enum index)))
  (preloaded_vlsm
     (free_composite_vlsm
        (sub_IM IM (enum index)))
     (λ _ : message, False))
Hproj: VLSM_embedding
  (composite_vlsm (sub_IM IM (enum index))
     (free_sub_free_constraint IM
        (free_constraint IM)))
  (composite_vlsm IM (free_constraint IM))
  (lift_sub_label IM (enum index))
  (free_sub_free_state IM)
HtrX': finite_valid_trace
  (composite_vlsm (sub_IM IM (enum index))
     (free_sub_free_constraint IM
        (free_constraint IM)))
  (equivocators_state_project
     (sub_IM IM (enum index))
     (λ i : sub_index (enum index),
        initial_descriptors (`i))
     (composite_state_sub_projection
        (equivocator_IM IM) 
        (enum index) s))
  (finite_trace_sub_projection IM 
     (enum index) trX)
finite_valid_trace Free
  (equivocators_state_project IM initial_descriptors s)
  trX
    apply (VLSM_embedding_finite_valid_trace Hproj) in HtrX'.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
Free:= free_composite_vlsm IM: VLSM message
s: state (equivocators_no_equivocations_vlsm IM)
trX: list transition_item
initial_descriptors: equivocator_descriptors IM
HtrX: finite_valid_trace
  {|
    vlsm_type :=
      free_composite_vlsm
        (sub_IM IM (enum index));
    vlsm_machine :=
      free_composite_vlsm
        (sub_IM IM (enum index))
  |}
  (equivocators_state_project
     (λ i : sub_index (enum index), IM (`i))
     (λ i : sub_index (enum index),
        initial_descriptors (`i))
     (composite_state_sub_projection
        (equivocator_IM IM) 
        (enum index) s))
  (finite_trace_sub_projection IM 
     (enum index) trX)
Heq: VLSM_eq
  (free_composite_vlsm (sub_IM IM (enum index)))
  (preloaded_vlsm
     (free_composite_vlsm
        (sub_IM IM (enum index)))
     (λ _ : message, False))
Hproj: VLSM_embedding
  (composite_vlsm (sub_IM IM (enum index))
     (free_sub_free_constraint IM
        (free_constraint IM)))
  (composite_vlsm IM (free_constraint IM))
  (lift_sub_label IM (enum index))
  (free_sub_free_state IM)
HtrX': finite_valid_trace
  (composite_vlsm IM (free_constraint IM))
  (free_sub_free_state IM
     (equivocators_state_project
        (sub_IM IM (enum index))
        (λ i : sub_index (enum index),
           initial_descriptors (`i))
        (composite_state_sub_projection
           (equivocator_IM IM) 
           (enum index) s)))
  (VLSM_embedding_finite_trace_project Hproj
     (finite_trace_sub_projection IM
        (enum index) trX))
finite_valid_trace Free
  (equivocators_state_project IM initial_descriptors s)
  trX
    replace (free_sub_free_state _ _)
      with (EquivocatorsComposition.equivocators_state_project IM initial_descriptors s)
      in HtrX'
    ; [replace (VLSM_embedding_finite_trace_project Hproj _) with trX
      in HtrX' |]; only 3: done.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
Free:= free_composite_vlsm IM: VLSM message
s: state (equivocators_no_equivocations_vlsm IM)
trX: list transition_item
initial_descriptors: equivocator_descriptors IM
HtrX: finite_valid_trace
  {|
    vlsm_type :=
      free_composite_vlsm
        (sub_IM IM (enum index));
    vlsm_machine :=
      free_composite_vlsm
        (sub_IM IM (enum index))
  |}
  (equivocators_state_project
     (λ i : sub_index (enum index), IM (`i))
     (λ i : sub_index (enum index),
        initial_descriptors (`i))
     (composite_state_sub_projection
        (equivocator_IM IM) 
        (enum index) s))
  (finite_trace_sub_projection IM 
     (enum index) trX)
Heq: VLSM_eq
  (free_composite_vlsm (sub_IM IM (enum index)))
  (preloaded_vlsm
     (free_composite_vlsm
        (sub_IM IM (enum index)))
     (λ _ : message, False))
Hproj: VLSM_embedding
  (composite_vlsm (sub_IM IM (enum index))
     (free_sub_free_constraint IM
        (free_constraint IM)))
  (composite_vlsm IM (free_constraint IM))
  (lift_sub_label IM (enum index))
  (free_sub_free_state IM)
HtrX': finite_valid_trace
  (composite_vlsm IM (free_constraint IM))
  (equivocators_state_project IM
     initial_descriptors s) trX
finite_valid_trace Free
  (equivocators_state_project IM initial_descriptors s)
  trX
message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
Free:= free_composite_vlsm IM: VLSM message
s: state (equivocators_no_equivocations_vlsm IM)
trX: list transition_item
initial_descriptors: equivocator_descriptors IM
HtrX: finite_valid_trace
  {|
    vlsm_type :=
      free_composite_vlsm
        (sub_IM IM (enum index));
    vlsm_machine :=
      free_composite_vlsm
        (sub_IM IM (enum index))
  |}
  (equivocators_state_project
     (λ i : sub_index (enum index), IM (`i))
     (λ i : sub_index (enum index),
        initial_descriptors (`i))
     (composite_state_sub_projection
        (equivocator_IM IM) 
        (enum index) s))
  (finite_trace_sub_projection IM 
     (enum index) trX)
Heq: VLSM_eq
  (free_composite_vlsm (sub_IM IM (enum index)))
  (preloaded_vlsm
     (free_composite_vlsm
        (sub_IM IM (enum index)))
     (λ _ : message, False))
Hproj: VLSM_embedding
  (composite_vlsm (sub_IM IM (enum index))
     (free_sub_free_constraint IM
        (free_constraint IM)))
  (composite_vlsm IM (free_constraint IM))
  (lift_sub_label IM (enum index))
  (free_sub_free_state IM)
HtrX': finite_valid_trace
  (composite_vlsm IM (free_constraint IM))
  (equivocators_state_project IM
     initial_descriptors s)
  (VLSM_embedding_finite_trace_project Hproj
     (finite_trace_sub_projection IM
        (enum index) trX))
trX =
VLSM_embedding_finite_trace_project Hproj
  (finite_trace_sub_projection IM (enum index) trX)
    +message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
Free:= free_composite_vlsm IM: VLSM message
s: state (equivocators_no_equivocations_vlsm IM)
trX: list transition_item
initial_descriptors: equivocator_descriptors IM
HtrX: finite_valid_trace
  {|
    vlsm_type :=
      free_composite_vlsm
        (sub_IM IM (enum index));
    vlsm_machine :=
      free_composite_vlsm
        (sub_IM IM (enum index))
  |}
  (equivocators_state_project
     (λ i : sub_index (enum index), IM (`i))
     (λ i : sub_index (enum index),
        initial_descriptors (`i))
     (composite_state_sub_projection
        (equivocator_IM IM) 
        (enum index) s))
  (finite_trace_sub_projection IM 
     (enum index) trX)
Heq: VLSM_eq
  (free_composite_vlsm (sub_IM IM (enum index)))
  (preloaded_vlsm
     (free_composite_vlsm
        (sub_IM IM (enum index)))
     (λ _ : message, False))
Hproj: VLSM_embedding
  (composite_vlsm (sub_IM IM (enum index))
     (free_sub_free_constraint IM
        (free_constraint IM)))
  (composite_vlsm IM (free_constraint IM))
  (lift_sub_label IM (enum index))
  (free_sub_free_state IM)
HtrX': finite_valid_trace
  (composite_vlsm IM (free_constraint IM))
  (equivocators_state_project IM
     initial_descriptors s) trX
finite_valid_trace Free
  (equivocators_state_project IM initial_descriptors s)
  trX apply (@VLSM_incl_finite_valid_trace _ _ (composite_vlsm IM (free_constraint IM))); [| done].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
Free:= free_composite_vlsm IM: VLSM message
s: state (equivocators_no_equivocations_vlsm IM)
trX: list transition_item
initial_descriptors: equivocator_descriptors IM
HtrX: finite_valid_trace
  {|
    vlsm_type :=
      free_composite_vlsm
        (sub_IM IM (enum index));
    vlsm_machine :=
      free_composite_vlsm
        (sub_IM IM (enum index))
  |}
  (equivocators_state_project
     (λ i : sub_index (enum index), IM (`i))
     (λ i : sub_index (enum index),
        initial_descriptors (`i))
     (composite_state_sub_projection
        (equivocator_IM IM) 
        (enum index) s))
  (finite_trace_sub_projection IM 
     (enum index) trX)
Heq: VLSM_eq
  (free_composite_vlsm (sub_IM IM (enum index)))
  (preloaded_vlsm
     (free_composite_vlsm
        (sub_IM IM (enum index)))
     (λ _ : message, False))
Hproj: VLSM_embedding
  (composite_vlsm (sub_IM IM (enum index))
     (free_sub_free_constraint IM
        (free_constraint IM)))
  (composite_vlsm IM (free_constraint IM))
  (lift_sub_label IM (enum index))
  (free_sub_free_state IM)
HtrX': finite_valid_trace
  (composite_vlsm IM (free_constraint IM))
  (equivocators_state_project IM
     initial_descriptors s) trX
VLSM_incl_part
  (composite_vlsm IM (free_constraint IM))
  (free_composite_vlsm_machine IM)
      by apply free_composite_vlsm_spec.
    +message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
Free:= free_composite_vlsm IM: VLSM message
s: state (equivocators_no_equivocations_vlsm IM)
trX: list transition_item
initial_descriptors: equivocator_descriptors IM
HtrX: finite_valid_trace
  {|
    vlsm_type :=
      free_composite_vlsm
        (sub_IM IM (enum index));
    vlsm_machine :=
      free_composite_vlsm
        (sub_IM IM (enum index))
  |}
  (equivocators_state_project
     (λ i : sub_index (enum index), IM (`i))
     (λ i : sub_index (enum index),
        initial_descriptors (`i))
     (composite_state_sub_projection
        (equivocator_IM IM) 
        (enum index) s))
  (finite_trace_sub_projection IM 
     (enum index) trX)
Heq: VLSM_eq
  (free_composite_vlsm (sub_IM IM (enum index)))
  (preloaded_vlsm
     (free_composite_vlsm
        (sub_IM IM (enum index)))
     (λ _ : message, False))
Hproj: VLSM_embedding
  (composite_vlsm (sub_IM IM (enum index))
     (free_sub_free_constraint IM
        (free_constraint IM)))
  (composite_vlsm IM (free_constraint IM))
  (lift_sub_label IM (enum index))
  (free_sub_free_state IM)
HtrX': finite_valid_trace
  (composite_vlsm IM (free_constraint IM))
  (equivocators_state_project IM
     initial_descriptors s)
  (VLSM_embedding_finite_trace_project Hproj
     (finite_trace_sub_projection IM
        (enum index) trX))
trX =
VLSM_embedding_finite_trace_project Hproj
  (finite_trace_sub_projection IM (enum index) trX) clear.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
trX: list transition_item
Hproj: VLSM_embedding
  (composite_vlsm (sub_IM IM (enum index))
     (free_sub_free_constraint IM
        (free_constraint IM)))
  (composite_vlsm IM (free_constraint IM))
  (lift_sub_label IM (enum index))
  (free_sub_free_state IM)
trX =
VLSM_embedding_finite_trace_project Hproj
  (finite_trace_sub_projection IM (enum index) trX)
      induction trX; [done |].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
a: transition_item
trX: list transition_item
Hproj: VLSM_embedding
  (composite_vlsm (sub_IM IM (enum index))
     (free_sub_free_constraint IM
        (free_constraint IM)))
  (composite_vlsm IM (free_constraint IM))
  (lift_sub_label IM (enum index))
  (free_sub_free_state IM)
IHtrX: trX =
VLSM_embedding_finite_trace_project Hproj
  (finite_trace_sub_projection IM 
     (enum index) trX)
a :: trX =
VLSM_embedding_finite_trace_project Hproj
  (finite_trace_sub_projection IM (enum index)
     (a :: trX))
      simpl.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
a: transition_item
trX: list transition_item
Hproj: VLSM_embedding
  (composite_vlsm (sub_IM IM (enum index))
     (free_sub_free_constraint IM
        (free_constraint IM)))
  (composite_vlsm IM (free_constraint IM))
  (lift_sub_label IM (enum index))
  (free_sub_free_state IM)
IHtrX: trX =
VLSM_embedding_finite_trace_project Hproj
  (finite_trace_sub_projection IM 
     (enum index) trX)
a :: trX =
VLSM_embedding_finite_trace_project Hproj
  match
    pre_VLSM_projection_transition_item_project
      (composite_type IM)
      (composite_type (sub_IM IM (enum index)))
      (composite_label_sub_projection_option IM
         (enum index))
      (composite_state_sub_projection IM (enum index))
      a
  with
  | Some y =>
      y
      :: finite_trace_sub_projection IM (enum index)
           trX
  | None =>
      finite_trace_sub_projection IM (enum index) trX
  end
      unfold pre_VLSM_projection_transition_item_project,
        composite_label_sub_projection_option.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
a: transition_item
trX: list transition_item
Hproj: VLSM_embedding
  (composite_vlsm (sub_IM IM (enum index))
     (free_sub_free_constraint IM
        (free_constraint IM)))
  (composite_vlsm IM (free_constraint IM))
  (lift_sub_label IM (enum index))
  (free_sub_free_state IM)
IHtrX: trX =
VLSM_embedding_finite_trace_project Hproj
  (finite_trace_sub_projection IM 
     (enum index) trX)
a :: trX =
VLSM_embedding_finite_trace_project Hproj
  match
    match
      match decide (projT1 (l a) ∈ enum index) with
      | left i_in =>
          Some
            (composite_label_sub_projection IM
               (enum index) (l a) i_in)
      | right _ => None
      end
    with
    | Some lY =>
        Some
          {|
            l := lY;
            input := input a;
            destination :=
              composite_state_sub_projection IM
                (enum index) (destination a);
            output := output a
          |}
    | None => None
    end
  with
  | Some y =>
      y
      :: finite_trace_sub_projection IM (enum index)
           trX
  | None =>
      finite_trace_sub_projection IM (enum index) trX
  end
      simpl.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
a: transition_item
trX: list transition_item
Hproj: VLSM_embedding
  (composite_vlsm (sub_IM IM (enum index))
     (free_sub_free_constraint IM
        (free_constraint IM)))
  (composite_vlsm IM (free_constraint IM))
  (lift_sub_label IM (enum index))
  (free_sub_free_state IM)
IHtrX: trX =
VLSM_embedding_finite_trace_project Hproj
  (finite_trace_sub_projection IM 
     (enum index) trX)
a :: trX =
VLSM_embedding_finite_trace_project Hproj
  match
    match
      match decide (projT1 (l a) ∈ enum index) with
      | left i_in =>
          Some
            (composite_label_sub_projection IM
               (enum index) (l a) i_in)
      | right _ => None
      end
    with
    | Some lY =>
        Some
          {|
            l := lY;
            input := input a;
            destination :=
              composite_state_sub_projection IM
                (enum index) (destination a);
            output := output a
          |}
    | None => None
    end
  with
  | Some y =>
      y
      :: finite_trace_sub_projection IM (enum index)
           trX
  | None =>
      finite_trace_sub_projection IM (enum index) trX
  end
      case_decide as Hla; [| contradict Hla; apply elem_of_enum].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
a: transition_item
trX: list transition_item
Hproj: VLSM_embedding
  (composite_vlsm (sub_IM IM (enum index))
     (free_sub_free_constraint IM
        (free_constraint IM)))
  (composite_vlsm IM (free_constraint IM))
  (lift_sub_label IM (enum index))
  (free_sub_free_state IM)
IHtrX: trX =
VLSM_embedding_finite_trace_project Hproj
  (finite_trace_sub_projection IM 
     (enum index) trX)
Hla: projT1 (l a) ∈ enum index
a :: trX =
VLSM_embedding_finite_trace_project Hproj
  ({|
     l :=
       composite_label_sub_projection IM (enum index)
         (l a) Hla;
     input := input a;
     destination :=
       composite_state_sub_projection IM (enum index)
         (destination a);
     output := output a
   |}
   :: finite_trace_sub_projection IM (enum index) trX)
      cbn; f_equal; [| done].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
a: transition_item
trX: list transition_item
Hproj: VLSM_embedding
  (composite_vlsm (sub_IM IM (enum index))
     (free_sub_free_constraint IM
        (free_constraint IM)))
  (composite_vlsm IM (free_constraint IM))
  (lift_sub_label IM (enum index))
  (free_sub_free_state IM)
IHtrX: trX =
VLSM_embedding_finite_trace_project Hproj
  (finite_trace_sub_projection IM 
     (enum index) trX)
Hla: projT1 (l a) ∈ enum index
a =
pre_VLSM_embedding_transition_item_project
  (composite_type (sub_IM IM (enum index)))
  (composite_type IM) (lift_sub_label IM (enum index))
  (free_sub_free_state IM)
  {|
    l :=
      composite_label_sub_projection IM (enum index)
        (l a) Hla;
    input := input a;
    destination :=
      composite_state_sub_projection IM (enum index)
        (destination a);
    output := output a
  |}
      by destruct a, l as [i li].
Qed.

Lemma equivocators_valid_trace_from_project
  (final_descriptors : equivocator_descriptors IM)
  (is final_state : state (equivocators_no_equivocations_vlsm IM))
  (tr : list (composite_transition_item (equivocator_IM IM)))
  (Hproper : not_equivocating_equivocator_descriptors IM final_descriptors final_state)
  (Htr : finite_valid_trace_from_to (equivocators_no_equivocations_vlsm IM) is final_state tr)
  : exists
    isX final_stateX
    (trX : list (composite_transition_item IM))
    (initial_descriptors : equivocator_descriptors IM),
    isX = equivocators_state_project IM initial_descriptors is /\
    proper_equivocator_descriptors IM initial_descriptors is /\
    equivocators_trace_project IM final_descriptors tr = Some (trX, initial_descriptors) /\
    equivocators_state_project IM final_descriptors final_state = final_stateX /\
    finite_valid_trace_from_to Free isX final_stateX trX.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
final_descriptors: equivocator_descriptors IM
is, final_state: state
  (equivocators_no_equivocations_vlsm IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Hproper: not_equivocating_equivocator_descriptors IM
  final_descriptors final_state
Htr: finite_valid_trace_from_to
  (equivocators_no_equivocations_vlsm IM) is
  final_state tr
∃ (isX final_stateX : state (free_composite_vlsm IM)) 
  (trX : list (composite_transition_item IM)) (initial_descriptors : 
                                               equivocator_descriptors
                                                 IM),
  isX =
  equivocators_state_project IM initial_descriptors is
  ∧ proper_equivocator_descriptors IM
      initial_descriptors is
    ∧ equivocators_trace_project IM final_descriptors
        tr = Some (trX, initial_descriptors)
      ∧ equivocators_state_project IM
          final_descriptors final_state = final_stateX
        ∧ finite_valid_trace_from_to Free isX
            final_stateX trX
Proof.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
final_descriptors: equivocator_descriptors IM
is, final_state: state
  (equivocators_no_equivocations_vlsm IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Hproper: not_equivocating_equivocator_descriptors IM
  final_descriptors final_state
Htr: finite_valid_trace_from_to
  (equivocators_no_equivocations_vlsm IM) is
  final_state tr
∃ (isX final_stateX : state (free_composite_vlsm IM)) 
  (trX : list (composite_transition_item IM)) (initial_descriptors : 
                                               equivocator_descriptors
                                                 IM),
  isX =
  equivocators_state_project IM initial_descriptors is
  ∧ proper_equivocator_descriptors IM
      initial_descriptors is
    ∧ equivocators_trace_project IM final_descriptors
        tr = Some (trX, initial_descriptors)
      ∧ equivocators_state_project IM
          final_descriptors final_state = final_stateX
        ∧ finite_valid_trace_from_to Free isX
            final_stateX trX
  apply valid_trace_get_last in Htr as Hfinal_state.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
final_descriptors: equivocator_descriptors IM
is, final_state: state
  (equivocators_no_equivocations_vlsm IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Hproper: not_equivocating_equivocator_descriptors IM
  final_descriptors final_state
Htr: finite_valid_trace_from_to
  (equivocators_no_equivocations_vlsm IM) is
  final_state tr
Hfinal_state: finite_trace_last is tr = final_state
∃ (isX final_stateX : state (free_composite_vlsm IM)) 
  (trX : list (composite_transition_item IM)) (initial_descriptors : 
                                               equivocator_descriptors
                                                 IM),
  isX =
  equivocators_state_project IM initial_descriptors is
  ∧ proper_equivocator_descriptors IM
      initial_descriptors is
    ∧ equivocators_trace_project IM final_descriptors
        tr = Some (trX, initial_descriptors)
      ∧ equivocators_state_project IM
          final_descriptors final_state = final_stateX
        ∧ finite_valid_trace_from_to Free isX
            final_stateX trX apply valid_trace_forget_last in Htr.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
final_descriptors: equivocator_descriptors IM
is, final_state: state
  (equivocators_no_equivocations_vlsm IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Hproper: not_equivocating_equivocator_descriptors IM
  final_descriptors final_state
Htr: finite_valid_trace_from
  (equivocators_no_equivocations_vlsm IM) is tr
Hfinal_state: finite_trace_last is tr = final_state
∃ (isX final_stateX : state (free_composite_vlsm IM)) 
  (trX : list (composite_transition_item IM)) (initial_descriptors : 
                                               equivocator_descriptors
                                                 IM),
  isX =
  equivocators_state_project IM initial_descriptors is
  ∧ proper_equivocator_descriptors IM
      initial_descriptors is
    ∧ equivocators_trace_project IM final_descriptors
        tr = Some (trX, initial_descriptors)
      ∧ equivocators_state_project IM
          final_descriptors final_state = final_stateX
        ∧ finite_valid_trace_from_to Free isX
            final_stateX trX
  subst final_state.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
final_descriptors: equivocator_descriptors IM
is: state (equivocators_no_equivocations_vlsm IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Hproper: not_equivocating_equivocator_descriptors IM
  final_descriptors
  (finite_trace_last is tr)
Htr: finite_valid_trace_from
  (equivocators_no_equivocations_vlsm IM) is tr
∃ (isX final_stateX : state (free_composite_vlsm IM)) 
  (trX : list (composite_transition_item IM)) (initial_descriptors : 
                                               equivocator_descriptors
                                                 IM),
  isX =
  equivocators_state_project IM initial_descriptors is
  ∧ proper_equivocator_descriptors IM
      initial_descriptors is
    ∧ equivocators_trace_project IM final_descriptors
        tr = Some (trX, initial_descriptors)
      ∧ equivocators_state_project IM
          final_descriptors (finite_trace_last is tr) =
        final_stateX
        ∧ finite_valid_trace_from_to Free isX
            final_stateX trX
  specialize (VLSM_partial_projection_finite_valid_trace_from
    (equivocators_no_equivocations_vlsm_X_vlsm_partial_projection final_descriptors) is tr) as Hsim.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
final_descriptors: equivocator_descriptors IM
is: state (equivocators_no_equivocations_vlsm IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Hproper: not_equivocating_equivocator_descriptors IM
  final_descriptors
  (finite_trace_last is tr)
Htr: finite_valid_trace_from
  (equivocators_no_equivocations_vlsm IM) is tr
Hsim: ∀ (sY : state Free) (trY : 
                     list transition_item),
  equivocators_partial_trace_project IM
    final_descriptors (
    is, tr) = Some (sY, trY)
  → finite_valid_trace_from
      (equivocators_no_equivocations_vlsm IM)
      is tr
    → finite_valid_trace_from Free sY trY
∃ (isX final_stateX : state (free_composite_vlsm IM)) 
  (trX : list (composite_transition_item IM)) (initial_descriptors : 
                                               equivocator_descriptors
                                                 IM),
  isX =
  equivocators_state_project IM initial_descriptors is
  ∧ proper_equivocator_descriptors IM
      initial_descriptors is
    ∧ equivocators_trace_project IM final_descriptors
        tr = Some (trX, initial_descriptors)
      ∧ equivocators_state_project IM
          final_descriptors (finite_trace_last is tr) =
        final_stateX
        ∧ finite_valid_trace_from_to Free isX
            final_stateX trX
  unfold equivocators_partial_trace_project in Hsim.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
final_descriptors: equivocator_descriptors IM
is: state (equivocators_no_equivocations_vlsm IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Hproper: not_equivocating_equivocator_descriptors IM
  final_descriptors
  (finite_trace_last is tr)
Htr: finite_valid_trace_from
  (equivocators_no_equivocations_vlsm IM) is tr
Hsim: ∀ (sY : state Free) (trY : 
                     list transition_item),
  (if
    decide
      (not_equivocating_equivocator_descriptors
         IM final_descriptors
         (finite_trace_last is tr))
   then
    match
      equivocators_trace_project IM
        final_descriptors tr
    with
    | Some (trX, initial_descriptors) =>
        Some
          (equivocators_state_project IM
             initial_descriptors is, trX)
    | None => None
    end
   else None) = Some (sY, trY)
  → finite_valid_trace_from
      (equivocators_no_equivocations_vlsm IM)
      is tr
    → finite_valid_trace_from Free sY trY
∃ (isX final_stateX : state (free_composite_vlsm IM)) 
  (trX : list (composite_transition_item IM)) (initial_descriptors : 
                                               equivocator_descriptors
                                                 IM),
  isX =
  equivocators_state_project IM initial_descriptors is
  ∧ proper_equivocator_descriptors IM
      initial_descriptors is
    ∧ equivocators_trace_project IM final_descriptors
        tr = Some (trX, initial_descriptors)
      ∧ equivocators_state_project IM
          final_descriptors (finite_trace_last is tr) =
        final_stateX
        ∧ finite_valid_trace_from_to Free isX
            final_stateX trX
  rewrite decide_True in Hsim by done.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
final_descriptors: equivocator_descriptors IM
is: state (equivocators_no_equivocations_vlsm IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Hproper: not_equivocating_equivocator_descriptors IM
  final_descriptors
  (finite_trace_last is tr)
Htr: finite_valid_trace_from
  (equivocators_no_equivocations_vlsm IM) is tr
Hsim: ∀ (sY : state Free) (trY : 
                     list transition_item),
  match
    equivocators_trace_project IM
      final_descriptors tr
  with
  | Some (trX, initial_descriptors) =>
      Some
        (equivocators_state_project IM
           initial_descriptors is, trX)
  | None => None
  end = Some (sY, trY)
  → finite_valid_trace_from
      (equivocators_no_equivocations_vlsm IM)
      is tr
    → finite_valid_trace_from Free sY trY
∃ (isX final_stateX : state (free_composite_vlsm IM)) 
  (trX : list (composite_transition_item IM)) (initial_descriptors : 
                                               equivocator_descriptors
                                                 IM),
  isX =
  equivocators_state_project IM initial_descriptors is
  ∧ proper_equivocator_descriptors IM
      initial_descriptors is
    ∧ equivocators_trace_project IM final_descriptors
        tr = Some (trX, initial_descriptors)
      ∧ equivocators_state_project IM
          final_descriptors (finite_trace_last is tr) =
        final_stateX
        ∧ finite_valid_trace_from_to Free isX
            final_stateX trX
  assert (HPreFree_tr : finite_constrained_trace_from FreeE is tr).message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
final_descriptors: equivocator_descriptors IM
is: state (equivocators_no_equivocations_vlsm IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Hproper: not_equivocating_equivocator_descriptors IM
  final_descriptors
  (finite_trace_last is tr)
Htr: finite_valid_trace_from
  (equivocators_no_equivocations_vlsm IM) is tr
Hsim: ∀ (sY : state Free) (trY : 
                     list transition_item),
  match
    equivocators_trace_project IM
      final_descriptors tr
  with
  | Some (trX, initial_descriptors) =>
      Some
        (equivocators_state_project IM
           initial_descriptors is, trX)
  | None => None
  end = Some (sY, trY)
  → finite_valid_trace_from
      (equivocators_no_equivocations_vlsm IM)
      is tr
    → finite_valid_trace_from Free sY trY
finite_constrained_trace_from FreeE is tr
message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
final_descriptors: equivocator_descriptors IM
is: state (equivocators_no_equivocations_vlsm IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Hproper: not_equivocating_equivocator_descriptors IM
  final_descriptors
  (finite_trace_last is tr)
Htr: finite_valid_trace_from
  (equivocators_no_equivocations_vlsm IM) is tr
Hsim: ∀ (sY : state Free) (trY : 
                     list transition_item),
  match
    equivocators_trace_project IM
      final_descriptors tr
  with
  | Some (trX, initial_descriptors) =>
      Some
        (equivocators_state_project IM
           initial_descriptors is, trX)
  | None => None
  end = Some (sY, trY)
  → finite_valid_trace_from
      (equivocators_no_equivocations_vlsm IM)
      is tr
    → finite_valid_trace_from Free sY trY
HPreFree_tr: finite_constrained_trace_from FreeE is
  tr
∃ (isX final_stateX : state (free_composite_vlsm IM)) 
  (trX : list (composite_transition_item IM)) 
  (initial_descriptors : 
   equivocator_descriptors IM),
  isX =
  equivocators_state_project IM initial_descriptors is
  ∧ proper_equivocator_descriptors IM
      initial_descriptors is
    ∧ equivocators_trace_project IM final_descriptors
        tr = Some (trX, initial_descriptors)
      ∧ equivocators_state_project IM
          final_descriptors 
          (finite_trace_last is tr) = final_stateX
        ∧ finite_valid_trace_from_to Free isX
            final_stateX trX
  {message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
final_descriptors: equivocator_descriptors IM
is: state (equivocators_no_equivocations_vlsm IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Hproper: not_equivocating_equivocator_descriptors IM
  final_descriptors
  (finite_trace_last is tr)
Htr: finite_valid_trace_from
  (equivocators_no_equivocations_vlsm IM) is tr
Hsim: ∀ (sY : state Free) (trY : 
                     list transition_item),
  match
    equivocators_trace_project IM
      final_descriptors tr
  with
  | Some (trX, initial_descriptors) =>
      Some
        (equivocators_state_project IM
           initial_descriptors is, trX)
  | None => None
  end = Some (sY, trY)
  → finite_valid_trace_from
      (equivocators_no_equivocations_vlsm IM)
      is tr
    → finite_valid_trace_from Free sY trY
finite_constrained_trace_from FreeE is tr
    apply VLSM_incl_finite_valid_trace_from; [| done].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
final_descriptors: equivocator_descriptors IM
is: state (equivocators_no_equivocations_vlsm IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Hproper: not_equivocating_equivocator_descriptors IM
  final_descriptors
  (finite_trace_last is tr)
Htr: finite_valid_trace_from
  (equivocators_no_equivocations_vlsm IM) is tr
Hsim: ∀ (sY : state Free) (trY : 
                     list transition_item),
  match
    equivocators_trace_project IM
      final_descriptors tr
  with
  | Some (trX, initial_descriptors) =>
      Some
        (equivocators_state_project IM
           initial_descriptors is, trX)
  | None => None
  end = Some (sY, trY)
  → finite_valid_trace_from
      (equivocators_no_equivocations_vlsm IM)
      is tr
    → finite_valid_trace_from Free sY trY
VLSM_incl_part
  (constrained_vlsm_machine
     (free_composite_vlsm (equivocator_IM IM))
     (equivocators_no_equivocations_constraint IM))
  (preloaded_vlsm_machine FreeE (λ _ : message, True))
    by apply constrained_preloaded_incl.
  }message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
final_descriptors: equivocator_descriptors IM
is: state (equivocators_no_equivocations_vlsm IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Hproper: not_equivocating_equivocator_descriptors IM
  final_descriptors
  (finite_trace_last is tr)
Htr: finite_valid_trace_from
  (equivocators_no_equivocations_vlsm IM) is tr
Hsim: ∀ (sY : state Free) (trY : 
                     list transition_item),
  match
    equivocators_trace_project IM
      final_descriptors tr
  with
  | Some (trX, initial_descriptors) =>
      Some
        (equivocators_state_project IM
           initial_descriptors is, trX)
  | None => None
  end = Some (sY, trY)
  → finite_valid_trace_from
      (equivocators_no_equivocations_vlsm IM)
      is tr
    → finite_valid_trace_from Free sY trY
HPreFree_tr: finite_constrained_trace_from FreeE is
  tr
∃ (isX final_stateX : state (free_composite_vlsm IM)) 
  (trX : list (composite_transition_item IM)) (initial_descriptors : 
                                               equivocator_descriptors
                                                 IM),
  isX =
  equivocators_state_project IM initial_descriptors is
  ∧ proper_equivocator_descriptors IM
      initial_descriptors is
    ∧ equivocators_trace_project IM final_descriptors
        tr = Some (trX, initial_descriptors)
      ∧ equivocators_state_project IM
          final_descriptors (finite_trace_last is tr) =
        final_stateX
        ∧ finite_valid_trace_from_to Free isX
            final_stateX trX
  apply not_equivocating_equivocator_descriptors_proper in Hproper.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
final_descriptors: equivocator_descriptors IM
is: state (equivocators_no_equivocations_vlsm IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Hproper: proper_equivocator_descriptors IM
  final_descriptors
  (finite_trace_last is tr)
Htr: finite_valid_trace_from
  (equivocators_no_equivocations_vlsm IM) is tr
Hsim: ∀ (sY : state Free) (trY : 
                     list transition_item),
  match
    equivocators_trace_project IM
      final_descriptors tr
  with
  | Some (trX, initial_descriptors) =>
      Some
        (equivocators_state_project IM
           initial_descriptors is, trX)
  | None => None
  end = Some (sY, trY)
  → finite_valid_trace_from
      (equivocators_no_equivocations_vlsm IM)
      is tr
    → finite_valid_trace_from Free sY trY
HPreFree_tr: finite_constrained_trace_from FreeE is
  tr
∃ (isX final_stateX : state (free_composite_vlsm IM)) 
  (trX : list (composite_transition_item IM)) (initial_descriptors : 
                                               equivocator_descriptors
                                                 IM),
  isX =
  equivocators_state_project IM initial_descriptors is
  ∧ proper_equivocator_descriptors IM
      initial_descriptors is
    ∧ equivocators_trace_project IM final_descriptors
        tr = Some (trX, initial_descriptors)
      ∧ equivocators_state_project IM
          final_descriptors (finite_trace_last is tr) =
        final_stateX
        ∧ finite_valid_trace_from_to Free isX
            final_stateX trX
  destruct
    (preloaded_equivocators_valid_trace_from_project _
      _ _ _ Hproper HPreFree_tr)
    as [trX [initial_descriptors [Htr_project [Hinitial_desciptors Hfinal_project]]]].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
final_descriptors: equivocator_descriptors IM
is: state (equivocators_no_equivocations_vlsm IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Hproper: proper_equivocator_descriptors IM
  final_descriptors
  (finite_trace_last is tr)
Htr: finite_valid_trace_from
  (equivocators_no_equivocations_vlsm IM) is tr
Hsim: ∀ (sY : state Free) (trY : 
                     list transition_item),
  match
    equivocators_trace_project IM
      final_descriptors tr
  with
  | Some (trX, initial_descriptors) =>
      Some
        (equivocators_state_project IM
           initial_descriptors is, trX)
  | None => None
  end = Some (sY, trY)
  → finite_valid_trace_from
      (equivocators_no_equivocations_vlsm IM)
      is tr
    → finite_valid_trace_from Free sY trY
HPreFree_tr: finite_constrained_trace_from FreeE is
  tr
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors tr =
Some (trX, initial_descriptors)
Hinitial_desciptors: proper_equivocator_descriptors
  IM initial_descriptors is
Hfinal_project: equivocators_state_project IM
  final_descriptors
  (finite_trace_last is tr) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
∃ (isX final_stateX : state (free_composite_vlsm IM)) 
  (trX : list (composite_transition_item IM)) (initial_descriptors : 
                                               equivocator_descriptors
                                                 IM),
  isX =
  equivocators_state_project IM initial_descriptors is
  ∧ proper_equivocator_descriptors IM
      initial_descriptors is
    ∧ equivocators_trace_project IM final_descriptors
        tr = Some (trX, initial_descriptors)
      ∧ equivocators_state_project IM
          final_descriptors (finite_trace_last is tr) =
        final_stateX
        ∧ finite_valid_trace_from_to Free isX
            final_stateX trX
  eexists.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
final_descriptors: equivocator_descriptors IM
is: state (equivocators_no_equivocations_vlsm IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Hproper: proper_equivocator_descriptors IM
  final_descriptors
  (finite_trace_last is tr)
Htr: finite_valid_trace_from
  (equivocators_no_equivocations_vlsm IM) is tr
Hsim: ∀ (sY : state Free) (trY : 
                     list transition_item),
  match
    equivocators_trace_project IM
      final_descriptors tr
  with
  | Some (trX, initial_descriptors) =>
      Some
        (equivocators_state_project IM
           initial_descriptors is, trX)
  | None => None
  end = Some (sY, trY)
  → finite_valid_trace_from
      (equivocators_no_equivocations_vlsm IM)
      is tr
    → finite_valid_trace_from Free sY trY
HPreFree_tr: finite_constrained_trace_from FreeE is
  tr
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors tr =
Some (trX, initial_descriptors)
Hinitial_desciptors: proper_equivocator_descriptors
  IM initial_descriptors is
Hfinal_project: equivocators_state_project IM
  final_descriptors
  (finite_trace_last is tr) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
∃ (final_stateX : state (free_composite_vlsm IM)) 
  (trX0 : list (composite_transition_item IM)) (initial_descriptors0 : 
                                                equivocator_descriptors
                                                 IM),
  ?isX =
  equivocators_state_project IM initial_descriptors0
    is
  ∧ proper_equivocator_descriptors IM
      initial_descriptors0 is
    ∧ equivocators_trace_project IM final_descriptors
        tr = Some (trX0, initial_descriptors0)
      ∧ equivocators_state_project IM
          final_descriptors (finite_trace_last is tr) =
        final_stateX
        ∧ finite_valid_trace_from_to Free ?isX
            final_stateX trX0 eexists.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
final_descriptors: equivocator_descriptors IM
is: state (equivocators_no_equivocations_vlsm IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Hproper: proper_equivocator_descriptors IM
  final_descriptors
  (finite_trace_last is tr)
Htr: finite_valid_trace_from
  (equivocators_no_equivocations_vlsm IM) is tr
Hsim: ∀ (sY : state Free) (trY : 
                     list transition_item),
  match
    equivocators_trace_project IM
      final_descriptors tr
  with
  | Some (trX, initial_descriptors) =>
      Some
        (equivocators_state_project IM
           initial_descriptors is, trX)
  | None => None
  end = Some (sY, trY)
  → finite_valid_trace_from
      (equivocators_no_equivocations_vlsm IM)
      is tr
    → finite_valid_trace_from Free sY trY
HPreFree_tr: finite_constrained_trace_from FreeE is
  tr
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors tr =
Some (trX, initial_descriptors)
Hinitial_desciptors: proper_equivocator_descriptors
  IM initial_descriptors is
Hfinal_project: equivocators_state_project IM
  final_descriptors
  (finite_trace_last is tr) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
∃ (trX0 : list (composite_transition_item IM)) (initial_descriptors0 : 
                                                equivocator_descriptors
                                                 IM),
  ?isX =
  equivocators_state_project IM initial_descriptors0
    is
  ∧ proper_equivocator_descriptors IM
      initial_descriptors0 is
    ∧ equivocators_trace_project IM final_descriptors
        tr = Some (trX0, initial_descriptors0)
      ∧ equivocators_state_project IM
          final_descriptors (finite_trace_last is tr) =
        ?final_stateX
        ∧ finite_valid_trace_from_to Free ?isX
            ?final_stateX trX0 eexists.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
final_descriptors: equivocator_descriptors IM
is: state (equivocators_no_equivocations_vlsm IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Hproper: proper_equivocator_descriptors IM
  final_descriptors
  (finite_trace_last is tr)
Htr: finite_valid_trace_from
  (equivocators_no_equivocations_vlsm IM) is tr
Hsim: ∀ (sY : state Free) (trY : 
                     list transition_item),
  match
    equivocators_trace_project IM
      final_descriptors tr
  with
  | Some (trX, initial_descriptors) =>
      Some
        (equivocators_state_project IM
           initial_descriptors is, trX)
  | None => None
  end = Some (sY, trY)
  → finite_valid_trace_from
      (equivocators_no_equivocations_vlsm IM)
      is tr
    → finite_valid_trace_from Free sY trY
HPreFree_tr: finite_constrained_trace_from FreeE is
  tr
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors tr =
Some (trX, initial_descriptors)
Hinitial_desciptors: proper_equivocator_descriptors
  IM initial_descriptors is
Hfinal_project: equivocators_state_project IM
  final_descriptors
  (finite_trace_last is tr) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
∃ initial_descriptors0 : equivocator_descriptors IM,
  ?isX =
  equivocators_state_project IM initial_descriptors0
    is
  ∧ proper_equivocator_descriptors IM
      initial_descriptors0 is
    ∧ equivocators_trace_project IM final_descriptors
        tr = Some (?trX, initial_descriptors0)
      ∧ equivocators_state_project IM
          final_descriptors (finite_trace_last is tr) =
        ?final_stateX
        ∧ finite_valid_trace_from_to Free ?isX
            ?final_stateX ?trX eexists.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
final_descriptors: equivocator_descriptors IM
is: state (equivocators_no_equivocations_vlsm IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Hproper: proper_equivocator_descriptors IM
  final_descriptors
  (finite_trace_last is tr)
Htr: finite_valid_trace_from
  (equivocators_no_equivocations_vlsm IM) is tr
Hsim: ∀ (sY : state Free) (trY : 
                     list transition_item),
  match
    equivocators_trace_project IM
      final_descriptors tr
  with
  | Some (trX, initial_descriptors) =>
      Some
        (equivocators_state_project IM
           initial_descriptors is, trX)
  | None => None
  end = Some (sY, trY)
  → finite_valid_trace_from
      (equivocators_no_equivocations_vlsm IM)
      is tr
    → finite_valid_trace_from Free sY trY
HPreFree_tr: finite_constrained_trace_from FreeE is
  tr
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors tr =
Some (trX, initial_descriptors)
Hinitial_desciptors: proper_equivocator_descriptors
  IM initial_descriptors is
Hfinal_project: equivocators_state_project IM
  final_descriptors
  (finite_trace_last is tr) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
?isX =
equivocators_state_project IM ?initial_descriptors is
∧ proper_equivocator_descriptors IM
    ?initial_descriptors is
  ∧ equivocators_trace_project IM final_descriptors tr =
    Some (?trX, ?initial_descriptors)
    ∧ equivocators_state_project IM final_descriptors
        (finite_trace_last is tr) = ?final_stateX
      ∧ finite_valid_trace_from_to Free ?isX
          ?final_stateX ?trX split; [done |].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
final_descriptors: equivocator_descriptors IM
is: state (equivocators_no_equivocations_vlsm IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Hproper: proper_equivocator_descriptors IM
  final_descriptors
  (finite_trace_last is tr)
Htr: finite_valid_trace_from
  (equivocators_no_equivocations_vlsm IM) is tr
Hsim: ∀ (sY : state Free) (trY : 
                     list transition_item),
  match
    equivocators_trace_project IM
      final_descriptors tr
  with
  | Some (trX, initial_descriptors) =>
      Some
        (equivocators_state_project IM
           initial_descriptors is, trX)
  | None => None
  end = Some (sY, trY)
  → finite_valid_trace_from
      (equivocators_no_equivocations_vlsm IM)
      is tr
    → finite_valid_trace_from Free sY trY
HPreFree_tr: finite_constrained_trace_from FreeE is
  tr
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors tr =
Some (trX, initial_descriptors)
Hinitial_desciptors: proper_equivocator_descriptors
  IM initial_descriptors is
Hfinal_project: equivocators_state_project IM
  final_descriptors
  (finite_trace_last is tr) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
proper_equivocator_descriptors IM ?initial_descriptors
  is
∧ equivocators_trace_project IM final_descriptors tr =
  Some (?trX, ?initial_descriptors)
  ∧ equivocators_state_project IM final_descriptors
      (finite_trace_last is tr) = ?final_stateX
    ∧ finite_valid_trace_from_to Free
        (equivocators_state_project IM
           ?initial_descriptors is) ?final_stateX 
        ?trX split; [by apply Hinitial_desciptors |].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
final_descriptors: equivocator_descriptors IM
is: state (equivocators_no_equivocations_vlsm IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Hproper: proper_equivocator_descriptors IM
  final_descriptors
  (finite_trace_last is tr)
Htr: finite_valid_trace_from
  (equivocators_no_equivocations_vlsm IM) is tr
Hsim: ∀ (sY : state Free) (trY : 
                     list transition_item),
  match
    equivocators_trace_project IM
      final_descriptors tr
  with
  | Some (trX, initial_descriptors) =>
      Some
        (equivocators_state_project IM
           initial_descriptors is, trX)
  | None => None
  end = Some (sY, trY)
  → finite_valid_trace_from
      (equivocators_no_equivocations_vlsm IM)
      is tr
    → finite_valid_trace_from Free sY trY
HPreFree_tr: finite_constrained_trace_from FreeE is
  tr
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors tr =
Some (trX, initial_descriptors)
Hinitial_desciptors: proper_equivocator_descriptors
  IM initial_descriptors is
Hfinal_project: equivocators_state_project IM
  final_descriptors
  (finite_trace_last is tr) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
equivocators_trace_project IM final_descriptors tr =
Some (?trX, initial_descriptors)
∧ equivocators_state_project IM final_descriptors
    (finite_trace_last is tr) = ?final_stateX
  ∧ finite_valid_trace_from_to Free
      (equivocators_state_project IM
         initial_descriptors is) ?final_stateX ?trX
  split; [by apply Htr_project |].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
final_descriptors: equivocator_descriptors IM
is: state (equivocators_no_equivocations_vlsm IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Hproper: proper_equivocator_descriptors IM
  final_descriptors
  (finite_trace_last is tr)
Htr: finite_valid_trace_from
  (equivocators_no_equivocations_vlsm IM) is tr
Hsim: ∀ (sY : state Free) (trY : 
                     list transition_item),
  match
    equivocators_trace_project IM
      final_descriptors tr
  with
  | Some (trX, initial_descriptors) =>
      Some
        (equivocators_state_project IM
           initial_descriptors is, trX)
  | None => None
  end = Some (sY, trY)
  → finite_valid_trace_from
      (equivocators_no_equivocations_vlsm IM)
      is tr
    → finite_valid_trace_from Free sY trY
HPreFree_tr: finite_constrained_trace_from FreeE is
  tr
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors tr =
Some (trX, initial_descriptors)
Hinitial_desciptors: proper_equivocator_descriptors
  IM initial_descriptors is
Hfinal_project: equivocators_state_project IM
  final_descriptors
  (finite_trace_last is tr) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
equivocators_state_project IM final_descriptors
  (finite_trace_last is tr) = ?final_stateX
∧ finite_valid_trace_from_to Free
    (equivocators_state_project IM initial_descriptors
       is) ?final_stateX trX split; [by apply Hfinal_project |].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
final_descriptors: equivocator_descriptors IM
is: state (equivocators_no_equivocations_vlsm IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Hproper: proper_equivocator_descriptors IM
  final_descriptors
  (finite_trace_last is tr)
Htr: finite_valid_trace_from
  (equivocators_no_equivocations_vlsm IM) is tr
Hsim: ∀ (sY : state Free) (trY : 
                     list transition_item),
  match
    equivocators_trace_project IM
      final_descriptors tr
  with
  | Some (trX, initial_descriptors) =>
      Some
        (equivocators_state_project IM
           initial_descriptors is, trX)
  | None => None
  end = Some (sY, trY)
  → finite_valid_trace_from
      (equivocators_no_equivocations_vlsm IM)
      is tr
    → finite_valid_trace_from Free sY trY
HPreFree_tr: finite_constrained_trace_from FreeE is
  tr
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors tr =
Some (trX, initial_descriptors)
Hinitial_desciptors: proper_equivocator_descriptors
  IM initial_descriptors is
Hfinal_project: equivocators_state_project IM
  final_descriptors
  (finite_trace_last is tr) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
finite_valid_trace_from_to Free
  (equivocators_state_project IM initial_descriptors
     is)
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX) trX
  apply valid_trace_add_default_last.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
final_descriptors: equivocator_descriptors IM
is: state (equivocators_no_equivocations_vlsm IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Hproper: proper_equivocator_descriptors IM
  final_descriptors
  (finite_trace_last is tr)
Htr: finite_valid_trace_from
  (equivocators_no_equivocations_vlsm IM) is tr
Hsim: ∀ (sY : state Free) (trY : 
                     list transition_item),
  match
    equivocators_trace_project IM
      final_descriptors tr
  with
  | Some (trX, initial_descriptors) =>
      Some
        (equivocators_state_project IM
           initial_descriptors is, trX)
  | None => None
  end = Some (sY, trY)
  → finite_valid_trace_from
      (equivocators_no_equivocations_vlsm IM)
      is tr
    → finite_valid_trace_from Free sY trY
HPreFree_tr: finite_constrained_trace_from FreeE is
  tr
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors tr =
Some (trX, initial_descriptors)
Hinitial_desciptors: proper_equivocator_descriptors
  IM initial_descriptors is
Hfinal_project: equivocators_state_project IM
  final_descriptors
  (finite_trace_last is tr) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
finite_valid_trace_from (free_composite_vlsm IM)
  (equivocators_state_project IM initial_descriptors
     is) trX
  apply Hsim; [| done].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
final_descriptors: equivocator_descriptors IM
is: state (equivocators_no_equivocations_vlsm IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Hproper: proper_equivocator_descriptors IM
  final_descriptors
  (finite_trace_last is tr)
Htr: finite_valid_trace_from
  (equivocators_no_equivocations_vlsm IM) is tr
Hsim: ∀ (sY : state Free) (trY : 
                     list transition_item),
  match
    equivocators_trace_project IM
      final_descriptors tr
  with
  | Some (trX, initial_descriptors) =>
      Some
        (equivocators_state_project IM
           initial_descriptors is, trX)
  | None => None
  end = Some (sY, trY)
  → finite_valid_trace_from
      (equivocators_no_equivocations_vlsm IM)
      is tr
    → finite_valid_trace_from Free sY trY
HPreFree_tr: finite_constrained_trace_from FreeE is
  tr
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors tr =
Some (trX, initial_descriptors)
Hinitial_desciptors: proper_equivocator_descriptors
  IM initial_descriptors is
Hfinal_project: equivocators_state_project IM
  final_descriptors
  (finite_trace_last is tr) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
match
  equivocators_trace_project IM final_descriptors tr
with
| Some (trX, initial_descriptors) =>
    Some
      (equivocators_state_project IM
         initial_descriptors is, trX)
| None => None
end =
Some
  (equivocators_state_project IM initial_descriptors
     is, trX)
  by rewrite Htr_project.
Qed.

Lemma PreFreeE_Free_vlsm_projection_type
  : VLSM_projection_type PreFreeE _
      (equivocators_total_label_project IM) (equivocators_total_state_project IM).message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
VLSM_projection_type PreFreeE (composite_type IM)
  (equivocators_total_label_project IM)
  (equivocators_total_state_project IM)
Proof.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
VLSM_projection_type PreFreeE (composite_type IM)
  (equivocators_total_label_project IM)
  (equivocators_total_state_project IM)
  apply basic_VLSM_projection_type.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
weak_projection_transition_consistency_None PreFreeE
  (composite_type IM)
  (equivocators_total_label_project IM)
  (equivocators_total_state_project IM)
  intros l Hl s om s' om' [[_ [_ Hv]] Ht].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
l: label PreFreeE
Hl: equivocators_total_label_project IM l = None
s: state PreFreeE
om: option message
s': state PreFreeE
om': option message
Hv: valid l (s, om)
Ht: transition l (s, om) = (s', om')
equivocators_total_state_project IM s' =
equivocators_total_state_project IM s
  destruct l as [i [sn | ji li | ji li]]; cbn in Hv, Ht.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
i: index
sn: state (IM i)
Hl: equivocators_total_label_project IM
  (existT i (Spawn sn)) = None
s: state PreFreeE
om: option message
s': state PreFreeE
om': option message
Hv: initial_state_prop sn ∧ om = None
Ht: (state_update (equivocator_IM IM) s i
   (equivocator_state_extend (s i) sn), None) =
(s', om')
equivocators_total_state_project IM s' =
equivocators_total_state_project IM s
message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
i: index
ji: nat
li: label (IM i)
Hl: equivocators_total_label_project IM
  (existT i (ContinueWith ji li)) = None
s: state PreFreeE
om: option message
s': state PreFreeE
om': option message
Hv: match equivocator_state_project (s i) ji with
| Some si => valid li (si, om)
| None => False
end
Ht: (let (si', om') :=
   match equivocator_state_project (s i) ji with
   | Some si =>
       let (si', om') :=
         transition li (si, om) in
       (equivocator_state_update (s i) ji si',
        om')
   | None => (s i, om)
   end in
 (state_update (equivocator_IM IM) s i si', om')) =
(s', om')
equivocators_total_state_project IM s' =
equivocators_total_state_project IM s
message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
i: index
ji: nat
li: label (IM i)
Hl: equivocators_total_label_project IM
  (existT i (ForkWith ji li)) = None
s: state PreFreeE
om: option message
s': state PreFreeE
om': option message
Hv: match equivocator_state_project (s i) ji with
| Some si => valid li (si, om)
| None => False
end
Ht: (let (si', om') :=
   match equivocator_state_project (s i) ji with
   | Some si =>
       let (si', om') :=
         transition li (si, om) in
       (equivocator_state_extend (s i) si', om')
   | None => (s i, om)
   end in
 (state_update (equivocator_IM IM) s i si', om')) =
(s', om')
equivocators_total_state_project IM s' =
equivocators_total_state_project IM s
  -message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
i: index
sn: state (IM i)
Hl: equivocators_total_label_project IM
  (existT i (Spawn sn)) = None
s: state PreFreeE
om: option message
s': state PreFreeE
om': option message
Hv: initial_state_prop sn ∧ om = None
Ht: (state_update (equivocator_IM IM) s i
   (equivocator_state_extend (s i) sn), None) =
(s', om')
equivocators_total_state_project IM s' =
equivocators_total_state_project IM s inversion_clear Ht.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
i: index
sn: state (IM i)
Hl: equivocators_total_label_project IM
  (existT i (Spawn sn)) = None
s: state PreFreeE
om: option message
s': state PreFreeE
om': option message
Hv: initial_state_prop sn ∧ om = None
equivocators_total_state_project IM
  (state_update (equivocator_IM IM) s i
     (equivocator_state_extend (s i) sn)) =
equivocators_total_state_project IM s unfold equivocators_total_state_project.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
i: index
sn: state (IM i)
Hl: equivocators_total_label_project IM
  (existT i (Spawn sn)) = None
s: state PreFreeE
om: option message
s': state PreFreeE
om': option message
Hv: initial_state_prop sn ∧ om = None
equivocators_state_project IM (zero_descriptor IM)
  (state_update (equivocator_IM IM) s i
     (equivocator_state_extend (s i) sn)) =
equivocators_state_project IM (zero_descriptor IM) s
    by state_update_simpl.
  -message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
i: index
ji: nat
li: label (IM i)
Hl: equivocators_total_label_project IM
  (existT i (ContinueWith ji li)) = None
s: state PreFreeE
om: option message
s': state PreFreeE
om': option message
Hv: match equivocator_state_project (s i) ji with
| Some si => valid li (si, om)
| None => False
end
Ht: (let (si', om') :=
   match equivocator_state_project (s i) ji with
   | Some si =>
       let (si', om') :=
         transition li (si, om) in
       (equivocator_state_update (s i) ji si',
        om')
   | None => (s i, om)
   end in
 (state_update (equivocator_IM IM) s i si', om')) =
(s', om')
equivocators_total_state_project IM s' =
equivocators_total_state_project IM s simpl in Hl.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
i: index
ji: nat
li: label (IM i)
Hl: option_map (existT i)
  match ji with
  | 0 => Some li
  | S _ => None
  end = None
s: state PreFreeE
om: option message
s': state PreFreeE
om': option message
Hv: match equivocator_state_project (s i) ji with
| Some si => valid li (si, om)
| None => False
end
Ht: (let (si', om') :=
   match equivocator_state_project (s i) ji with
   | Some si =>
       let (si', om') :=
         transition li (si, om) in
       (equivocator_state_update (s i) ji si',
        om')
   | None => (s i, om)
   end in
 (state_update (equivocator_IM IM) s i si', om')) =
(s', om')
equivocators_total_state_project IM s' =
equivocators_total_state_project IM s destruct ji as [| ji]; [by inversion Hl |].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
i: index
ji: nat
li: label (IM i)
Hl: option_map (existT i) None = None
s: state PreFreeE
om: option message
s': state PreFreeE
om': option message
Hv: match equivocator_state_project (s i) (S ji) with
| Some si => valid li (si, om)
| None => False
end
Ht: (let (si', om') :=
   match
     equivocator_state_project (s i) (S ji)
   with
   | Some si =>
       let (si', om') :=
         transition li (si, om) in
       (equivocator_state_update (s i) (S ji) si',
        om')
   | None => (s i, om)
   end in
 (state_update (equivocator_IM IM) s i si', om')) =
(s', om')
equivocators_total_state_project IM s' =
equivocators_total_state_project IM s clear Hl.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
i: index
ji: nat
li: label (IM i)
s: state PreFreeE
om: option message
s': state PreFreeE
om': option message
Hv: match equivocator_state_project (s i) (S ji) with
| Some si => valid li (si, om)
| None => False
end
Ht: (let (si', om') :=
   match
     equivocator_state_project (s i) (S ji)
   with
   | Some si =>
       let (si', om') :=
         transition li (si, om) in
       (equivocator_state_update (s i) (S ji) si',
        om')
   | None => (s i, om)
   end in
 (state_update (equivocator_IM IM) s i si', om')) =
(s', om')
equivocators_total_state_project IM s' =
equivocators_total_state_project IM s
    destruct (equivocator_state_project _ _) as [si |]; [| done].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
i: index
ji: nat
li: label (IM i)
s: state PreFreeE
om: option message
s': state PreFreeE
om': option message
si: state (IM i)
Hv: valid li (si, om)
Ht: (let (si', om') :=
   let (si', om') := transition li (si, om) in
   (equivocator_state_update (s i) (S ji) si',
    om') in
 (state_update (equivocator_IM IM) s i si', om')) =
(s', om')
equivocators_total_state_project IM s' =
equivocators_total_state_project IM s
    destruct (transition _ _ _) as (si', _om').message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
i: index
ji: nat
li: label (IM i)
s: state PreFreeE
om: option message
s': state PreFreeE
om': option message
si: state (IM i)
Hv: valid li (si, om)
si': state (IM i)
_om': option message
Ht: (state_update (equivocator_IM IM) s i
   (equivocator_state_update (s i) (S ji) si'),
 _om') = (s', om')
equivocators_total_state_project IM s' =
equivocators_total_state_project IM s
    inversion_clear Ht.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
i: index
ji: nat
li: label (IM i)
s: state PreFreeE
om: option message
s': state PreFreeE
om': option message
si: state (IM i)
Hv: valid li (si, om)
si': state (IM i)
_om': option message
equivocators_total_state_project IM
  (state_update (equivocator_IM IM) s i
     (equivocator_state_update (s i) (S ji) si')) =
equivocators_total_state_project IM s  unfold equivocators_total_state_project.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
i: index
ji: nat
li: label (IM i)
s: state PreFreeE
om: option message
s': state PreFreeE
om': option message
si: state (IM i)
Hv: valid li (si, om)
si': state (IM i)
_om': option message
equivocators_state_project IM (zero_descriptor IM)
  (state_update (equivocator_IM IM) s i
     (equivocator_state_update (s i) (S ji) si')) =
equivocators_state_project IM (zero_descriptor IM) s
    by state_update_simpl.
  -message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
i: index
ji: nat
li: label (IM i)
Hl: equivocators_total_label_project IM
  (existT i (ForkWith ji li)) = None
s: state PreFreeE
om: option message
s': state PreFreeE
om': option message
Hv: match equivocator_state_project (s i) ji with
| Some si => valid li (si, om)
| None => False
end
Ht: (let (si', om') :=
   match equivocator_state_project (s i) ji with
   | Some si =>
       let (si', om') :=
         transition li (si, om) in
       (equivocator_state_extend (s i) si', om')
   | None => (s i, om)
   end in
 (state_update (equivocator_IM IM) s i si', om')) =
(s', om')
equivocators_total_state_project IM s' =
equivocators_total_state_project IM s destruct (equivocator_state_project _ _) as [si |]; [| done].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
i: index
ji: nat
li: label (IM i)
Hl: equivocators_total_label_project IM
  (existT i (ForkWith ji li)) = None
s: state PreFreeE
om: option message
s': state PreFreeE
om': option message
si: state (IM i)
Hv: valid li (si, om)
Ht: (let (si', om') :=
   let (si', om') := transition li (si, om) in
   (equivocator_state_extend (s i) si', om') in
 (state_update (equivocator_IM IM) s i si', om')) =
(s', om')
equivocators_total_state_project IM s' =
equivocators_total_state_project IM s
    destruct (transition _ _ _) as (si', _om').message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
i: index
ji: nat
li: label (IM i)
Hl: equivocators_total_label_project IM
  (existT i (ForkWith ji li)) = None
s: state PreFreeE
om: option message
s': state PreFreeE
om': option message
si: state (IM i)
Hv: valid li (si, om)
si': state (IM i)
_om': option message
Ht: (state_update (equivocator_IM IM) s i
   (equivocator_state_extend (s i) si'), _om') =
(s', om')
equivocators_total_state_project IM s' =
equivocators_total_state_project IM s
    inversion_clear Ht.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
i: index
ji: nat
li: label (IM i)
Hl: equivocators_total_label_project IM
  (existT i (ForkWith ji li)) = None
s: state PreFreeE
om: option message
s': state PreFreeE
om': option message
si: state (IM i)
Hv: valid li (si, om)
si': state (IM i)
_om': option message
equivocators_total_state_project IM
  (state_update (equivocator_IM IM) s i
     (equivocator_state_extend (s i) si')) =
equivocators_total_state_project IM s  unfold equivocators_total_state_project.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
i: index
ji: nat
li: label (IM i)
Hl: equivocators_total_label_project IM
  (existT i (ForkWith ji li)) = None
s: state PreFreeE
om: option message
s': state PreFreeE
om': option message
si: state (IM i)
Hv: valid li (si, om)
si': state (IM i)
_om': option message
equivocators_state_project IM (zero_descriptor IM)
  (state_update (equivocator_IM IM) s i
     (equivocator_state_extend (s i) si')) =
equivocators_state_project IM (zero_descriptor IM) s
    by state_update_simpl.
Qed.

Lemma equivocators_no_equivocations_vlsm_X_vlsm_projection
  : VLSM_projection (equivocators_no_equivocations_vlsm IM) Free
      (equivocators_total_label_project IM) (equivocators_total_state_project IM).message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
VLSM_projection
  (equivocators_no_equivocations_vlsm IM) Free
  (equivocators_total_label_project IM)
  (equivocators_total_state_project IM)
Proof.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
VLSM_projection
  (equivocators_no_equivocations_vlsm IM) Free
  (equivocators_total_label_project IM)
  (equivocators_total_state_project IM)
  constructor; [constructor |].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
∀ (sX : state (equivocators_no_equivocations_vlsm IM)) 
  (trX : list transition_item),
  finite_valid_trace_from
    (equivocators_no_equivocations_vlsm IM) sX trX
  → equivocators_total_state_project IM
      (finite_trace_last sX trX) =
    finite_trace_last
      (equivocators_total_state_project IM sX)
      (pre_VLSM_projection_finite_trace_project
         (equivocators_no_equivocations_vlsm IM) Free
         (equivocators_total_label_project IM)
         (equivocators_total_state_project IM) trX)
message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
∀ (sX : state (equivocators_no_equivocations_vlsm IM)) 
  (trX : list transition_item),
  finite_valid_trace
    (equivocators_no_equivocations_vlsm IM) sX trX
  → finite_valid_trace Free
      (equivocators_total_state_project IM sX)
      (pre_VLSM_projection_finite_trace_project
         (equivocators_no_equivocations_vlsm IM) Free
         (equivocators_total_label_project IM)
         (equivocators_total_state_project IM) trX)
  -message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
∀ (sX : state (equivocators_no_equivocations_vlsm IM)) 
  (trX : list transition_item),
  finite_valid_trace_from
    (equivocators_no_equivocations_vlsm IM) sX trX
  → equivocators_total_state_project IM
      (finite_trace_last sX trX) =
    finite_trace_last
      (equivocators_total_state_project IM sX)
      (pre_VLSM_projection_finite_trace_project
         (equivocators_no_equivocations_vlsm IM) Free
         (equivocators_total_label_project IM)
         (equivocators_total_state_project IM) trX) intros * Htr.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
sX: state (equivocators_no_equivocations_vlsm IM)
trX: list transition_item
Htr: finite_valid_trace_from
  (equivocators_no_equivocations_vlsm IM) sX trX
equivocators_total_state_project IM
  (finite_trace_last sX trX) =
finite_trace_last
  (equivocators_total_state_project IM sX)
  (pre_VLSM_projection_finite_trace_project
     (equivocators_no_equivocations_vlsm IM) Free
     (equivocators_total_label_project IM)
     (equivocators_total_state_project IM) trX) apply PreFreeE_Free_vlsm_projection_type.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
sX: state (equivocators_no_equivocations_vlsm IM)
trX: list transition_item
Htr: finite_valid_trace_from
  (equivocators_no_equivocations_vlsm IM) sX trX
finite_valid_trace_from PreFreeE sX trX
    apply VLSM_incl_finite_valid_trace_from; [| done].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
sX: state (equivocators_no_equivocations_vlsm IM)
trX: list transition_item
Htr: finite_valid_trace_from
  (equivocators_no_equivocations_vlsm IM) sX trX
VLSM_incl_part
  (constrained_vlsm_machine
     (free_composite_vlsm (equivocator_IM IM))
     (equivocators_no_equivocations_constraint IM))
  (preloaded_vlsm_machine FreeE (λ _ : message, True))
    by apply constrained_preloaded_incl.
  -message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
∀ (sX : state (equivocators_no_equivocations_vlsm IM)) 
  (trX : list transition_item),
  finite_valid_trace
    (equivocators_no_equivocations_vlsm IM) sX trX
  → finite_valid_trace Free
      (equivocators_total_state_project IM sX)
      (pre_VLSM_projection_finite_trace_project
         (equivocators_no_equivocations_vlsm IM) Free
         (equivocators_total_label_project IM)
         (equivocators_total_state_project IM) trX) intros * Htr.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
sX: state (equivocators_no_equivocations_vlsm IM)
trX: list transition_item
Htr: finite_valid_trace
  (equivocators_no_equivocations_vlsm IM) sX trX
finite_valid_trace Free
  (equivocators_total_state_project IM sX)
  (pre_VLSM_projection_finite_trace_project
     (equivocators_no_equivocations_vlsm IM) Free
     (equivocators_total_label_project IM)
     (equivocators_total_state_project IM) trX)
    assert (Hpre_tr : finite_constrained_trace FreeE sX trX).message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
sX: state (equivocators_no_equivocations_vlsm IM)
trX: list transition_item
Htr: finite_valid_trace
  (equivocators_no_equivocations_vlsm IM) sX trX
finite_constrained_trace FreeE sX trX
message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
sX: state (equivocators_no_equivocations_vlsm IM)
trX: list transition_item
Htr: finite_valid_trace
  (equivocators_no_equivocations_vlsm IM) sX trX
Hpre_tr: finite_constrained_trace FreeE sX trX
finite_valid_trace Free
  (equivocators_total_state_project IM sX)
  (pre_VLSM_projection_finite_trace_project
     (equivocators_no_equivocations_vlsm IM) Free
     (equivocators_total_label_project IM)
     (equivocators_total_state_project IM) trX)
    {message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
sX: state (equivocators_no_equivocations_vlsm IM)
trX: list transition_item
Htr: finite_valid_trace
  (equivocators_no_equivocations_vlsm IM) sX trX
finite_constrained_trace FreeE sX trX
      apply VLSM_incl_finite_valid_trace; [| done].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
sX: state (equivocators_no_equivocations_vlsm IM)
trX: list transition_item
Htr: finite_valid_trace
  (equivocators_no_equivocations_vlsm IM) sX trX
VLSM_incl_part
  (constrained_vlsm_machine
     (free_composite_vlsm (equivocator_IM IM))
     (equivocators_no_equivocations_constraint IM))
  (preloaded_vlsm_machine FreeE (λ _ : message, True))
      by apply constrained_preloaded_incl.
    }message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
sX: state (equivocators_no_equivocations_vlsm IM)
trX: list transition_item
Htr: finite_valid_trace
  (equivocators_no_equivocations_vlsm IM) sX trX
Hpre_tr: finite_constrained_trace FreeE sX trX
finite_valid_trace Free
  (equivocators_total_state_project IM sX)
  (pre_VLSM_projection_finite_trace_project
     (equivocators_no_equivocations_vlsm IM) Free
     (equivocators_total_label_project IM)
     (equivocators_total_state_project IM) trX)
    specialize
     (VLSM_partial_projection_finite_valid_trace
      (equivocators_no_equivocations_vlsm_X_vlsm_partial_projection (zero_descriptor IM))
       sX trX (equivocators_total_state_project IM sX) (equivocators_total_trace_project IM trX))
     as Hsim.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
sX: state (equivocators_no_equivocations_vlsm IM)
trX: list transition_item
Htr: finite_valid_trace
  (equivocators_no_equivocations_vlsm IM) sX trX
Hpre_tr: finite_constrained_trace FreeE sX trX
Hsim: equivocators_partial_trace_project IM
  (zero_descriptor IM) (
  sX, trX) =
Some
  (equivocators_total_state_project IM sX,
   equivocators_total_trace_project IM trX)
→ finite_valid_trace
    (equivocators_no_equivocations_vlsm IM) sX
    trX
  → finite_valid_trace Free
      (equivocators_total_state_project IM sX)
      (equivocators_total_trace_project IM trX)
finite_valid_trace Free
  (equivocators_total_state_project IM sX)
  (pre_VLSM_projection_finite_trace_project
     (equivocators_no_equivocations_vlsm IM) Free
     (equivocators_total_label_project IM)
     (equivocators_total_state_project IM) trX)
    spec Hsim.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
sX: state (equivocators_no_equivocations_vlsm IM)
trX: list transition_item
Htr: finite_valid_trace
  (equivocators_no_equivocations_vlsm IM) sX trX
Hpre_tr: finite_constrained_trace FreeE sX trX
Hsim: equivocators_partial_trace_project IM
  (zero_descriptor IM) (
  sX, trX) =
Some
  (equivocators_total_state_project IM sX,
   equivocators_total_trace_project IM trX)
→ finite_valid_trace
    (equivocators_no_equivocations_vlsm IM) sX
    trX
  → finite_valid_trace Free
      (equivocators_total_state_project IM sX)
      (equivocators_total_trace_project IM trX)
equivocators_partial_trace_project IM
  (zero_descriptor IM) (sX, trX) =
Some
  (equivocators_total_state_project IM sX,
   equivocators_total_trace_project IM trX)
message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
sX: state (equivocators_no_equivocations_vlsm IM)
trX: list transition_item
Htr: finite_valid_trace
  (equivocators_no_equivocations_vlsm IM) sX trX
Hpre_tr: finite_constrained_trace FreeE sX trX
Hsim: finite_valid_trace
  (equivocators_no_equivocations_vlsm IM) sX
  trX
→ finite_valid_trace Free
    (equivocators_total_state_project IM sX)
    (equivocators_total_trace_project IM trX)
finite_valid_trace Free
  (equivocators_total_state_project IM sX)
  (pre_VLSM_projection_finite_trace_project
     (equivocators_no_equivocations_vlsm IM) Free
     (equivocators_total_label_project IM)
     (equivocators_total_state_project IM) trX)
    {message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
sX: state (equivocators_no_equivocations_vlsm IM)
trX: list transition_item
Htr: finite_valid_trace
  (equivocators_no_equivocations_vlsm IM) sX trX
Hpre_tr: finite_constrained_trace FreeE sX trX
Hsim: equivocators_partial_trace_project IM
  (zero_descriptor IM) (
  sX, trX) =
Some
  (equivocators_total_state_project IM sX,
   equivocators_total_trace_project IM trX)
→ finite_valid_trace
    (equivocators_no_equivocations_vlsm IM) sX
    trX
  → finite_valid_trace Free
      (equivocators_total_state_project IM sX)
      (equivocators_total_trace_project IM trX)
equivocators_partial_trace_project IM
  (zero_descriptor IM) (sX, trX) =
Some
  (equivocators_total_state_project IM sX,
   equivocators_total_trace_project IM trX) simpl.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
sX: state (equivocators_no_equivocations_vlsm IM)
trX: list transition_item
Htr: finite_valid_trace
  (equivocators_no_equivocations_vlsm IM) sX trX
Hpre_tr: finite_constrained_trace FreeE sX trX
Hsim: equivocators_partial_trace_project IM
  (zero_descriptor IM) (
  sX, trX) =
Some
  (equivocators_total_state_project IM sX,
   equivocators_total_trace_project IM trX)
→ finite_valid_trace
    (equivocators_no_equivocations_vlsm IM) sX
    trX
  → finite_valid_trace Free
      (equivocators_total_state_project IM sX)
      (equivocators_total_trace_project IM trX)
(if
  decide
    (not_equivocating_equivocator_descriptors IM
       (zero_descriptor IM) (finite_trace_last sX trX))
 then
  match
    equivocators_trace_project IM (zero_descriptor IM)
      trX
  with
  | Some (trX, initial_descriptors) =>
      Some
        (equivocators_state_project IM
           initial_descriptors sX, trX)
  | None => None
  end
 else None) =
Some
  (equivocators_total_state_project IM sX,
   equivocators_total_trace_project IM trX) rewrite decide_True by apply zero_descriptor_not_equivocating.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
sX: state (equivocators_no_equivocations_vlsm IM)
trX: list transition_item
Htr: finite_valid_trace
  (equivocators_no_equivocations_vlsm IM) sX trX
Hpre_tr: finite_constrained_trace FreeE sX trX
Hsim: equivocators_partial_trace_project IM
  (zero_descriptor IM) (
  sX, trX) =
Some
  (equivocators_total_state_project IM sX,
   equivocators_total_trace_project IM trX)
→ finite_valid_trace
    (equivocators_no_equivocations_vlsm IM) sX
    trX
  → finite_valid_trace Free
      (equivocators_total_state_project IM sX)
      (equivocators_total_trace_project IM trX)
match
  equivocators_trace_project IM (zero_descriptor IM)
    trX
with
| Some (trX, initial_descriptors) =>
    Some
      (equivocators_state_project IM
         initial_descriptors sX, trX)
| None => None
end =
Some
  (equivocators_total_state_project IM sX,
   equivocators_total_trace_project IM trX)
      by rewrite (equivocators_total_trace_project_characterization IM (proj1 Hpre_tr)).
    }message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
sX: state (equivocators_no_equivocations_vlsm IM)
trX: list transition_item
Htr: finite_valid_trace
  (equivocators_no_equivocations_vlsm IM) sX trX
Hpre_tr: finite_constrained_trace FreeE sX trX
Hsim: finite_valid_trace
  (equivocators_no_equivocations_vlsm IM) sX
  trX
→ finite_valid_trace Free
    (equivocators_total_state_project IM sX)
    (equivocators_total_trace_project IM trX)
finite_valid_trace Free
  (equivocators_total_state_project IM sX)
  (pre_VLSM_projection_finite_trace_project
     (equivocators_no_equivocations_vlsm IM) Free
     (equivocators_total_label_project IM)
     (equivocators_total_state_project IM) trX)
    apply Hsim in Htr.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
sX: state (equivocators_no_equivocations_vlsm IM)
trX: list transition_item
Htr: finite_valid_trace Free
  (equivocators_total_state_project IM sX)
  (equivocators_total_trace_project IM trX)
Hpre_tr: finite_constrained_trace FreeE sX trX
Hsim: finite_valid_trace
  (equivocators_no_equivocations_vlsm IM) sX
  trX
→ finite_valid_trace Free
    (equivocators_total_state_project IM sX)
    (equivocators_total_trace_project IM trX)
finite_valid_trace Free
  (equivocators_total_state_project IM sX)
  (pre_VLSM_projection_finite_trace_project
     (equivocators_no_equivocations_vlsm IM) Free
     (equivocators_total_label_project IM)
     (equivocators_total_state_project IM) trX)
    remember (pre_VLSM_projection_finite_trace_project _ _ _ _ _) as tr.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
sX: state (equivocators_no_equivocations_vlsm IM)
trX: list transition_item
Htr: finite_valid_trace Free
  (equivocators_total_state_project IM sX)
  (equivocators_total_trace_project IM trX)
Hpre_tr: finite_constrained_trace FreeE sX trX
Hsim: finite_valid_trace
  (equivocators_no_equivocations_vlsm IM) sX
  trX
→ finite_valid_trace Free
    (equivocators_total_state_project IM sX)
    (equivocators_total_trace_project IM trX)
tr: list transition_item
Heqtr: tr =
pre_VLSM_projection_finite_trace_project
  (equivocators_no_equivocations_vlsm IM) Free
  (equivocators_total_label_project IM)
  (equivocators_total_state_project IM) trX
finite_valid_trace Free
  (equivocators_total_state_project IM sX) tr
    replace tr with (equivocators_total_trace_project IM trX); [done |].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
sX: state (equivocators_no_equivocations_vlsm IM)
trX: list transition_item
Htr: finite_valid_trace Free
  (equivocators_total_state_project IM sX)
  (equivocators_total_trace_project IM trX)
Hpre_tr: finite_constrained_trace FreeE sX trX
Hsim: finite_valid_trace
  (equivocators_no_equivocations_vlsm IM) sX
  trX
→ finite_valid_trace Free
    (equivocators_total_state_project IM sX)
    (equivocators_total_trace_project IM trX)
tr: list transition_item
Heqtr: tr =
pre_VLSM_projection_finite_trace_project
  (equivocators_no_equivocations_vlsm IM) Free
  (equivocators_total_label_project IM)
  (equivocators_total_state_project IM) trX
equivocators_total_trace_project IM trX = tr
    subst.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
sX: state (equivocators_no_equivocations_vlsm IM)
trX: list transition_item
Htr: finite_valid_trace Free
  (equivocators_total_state_project IM sX)
  (equivocators_total_trace_project IM trX)
Hpre_tr: finite_constrained_trace FreeE sX trX
Hsim: finite_valid_trace
  (equivocators_no_equivocations_vlsm IM) sX
  trX
→ finite_valid_trace Free
    (equivocators_total_state_project IM sX)
    (equivocators_total_trace_project IM trX)
equivocators_total_trace_project IM trX =
pre_VLSM_projection_finite_trace_project
  (equivocators_no_equivocations_vlsm IM) Free
  (equivocators_total_label_project IM)
  (equivocators_total_state_project IM) trX symmetry.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
sX: state (equivocators_no_equivocations_vlsm IM)
trX: list transition_item
Htr: finite_valid_trace Free
  (equivocators_total_state_project IM sX)
  (equivocators_total_trace_project IM trX)
Hpre_tr: finite_constrained_trace FreeE sX trX
Hsim: finite_valid_trace
  (equivocators_no_equivocations_vlsm IM) sX
  trX
→ finite_valid_trace Free
    (equivocators_total_state_project IM sX)
    (equivocators_total_trace_project IM trX)
pre_VLSM_projection_finite_trace_project
  (equivocators_no_equivocations_vlsm IM) Free
  (equivocators_total_label_project IM)
  (equivocators_total_state_project IM) trX =
equivocators_total_trace_project IM trX
    by eapply (equivocators_total_VLSM_projection_finite_trace_project IM), Hpre_tr.
Qed.

Lemma preloaded_equivocators_no_equivocations_vlsm_X_vlsm_projection
  : VLSM_projection PreFreeE PreFree
      (equivocators_total_label_project IM) (equivocators_total_state_project IM).message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
VLSM_projection PreFreeE PreFree
  (equivocators_total_label_project IM)
  (equivocators_total_state_project IM)
Proof.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
VLSM_projection PreFreeE PreFree
  (equivocators_total_label_project IM)
  (equivocators_total_state_project IM)
  constructor; [constructor; intros |].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
sX: state PreFreeE
trX: list transition_item
H1: finite_valid_trace_from PreFreeE sX trX
equivocators_total_state_project IM
  (finite_trace_last sX trX) =
finite_trace_last
  (equivocators_total_state_project IM sX)
  (pre_VLSM_projection_finite_trace_project PreFreeE
     PreFree (equivocators_total_label_project IM)
     (equivocators_total_state_project IM) trX)
message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
∀ (sX : state PreFreeE) (trX : list transition_item),
  finite_valid_trace PreFreeE sX trX
  → finite_valid_trace PreFree
      (equivocators_total_state_project IM sX)
      (pre_VLSM_projection_finite_trace_project
         PreFreeE PreFree
         (equivocators_total_label_project IM)
         (equivocators_total_state_project IM) trX)
  -message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
sX: state PreFreeE
trX: list transition_item
H1: finite_valid_trace_from PreFreeE sX trX
equivocators_total_state_project IM
  (finite_trace_last sX trX) =
finite_trace_last
  (equivocators_total_state_project IM sX)
  (pre_VLSM_projection_finite_trace_project PreFreeE
     PreFree (equivocators_total_label_project IM)
     (equivocators_total_state_project IM) trX) by apply PreFreeE_Free_vlsm_projection_type.
  -message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
∀ (sX : state PreFreeE) (trX : list transition_item),
  finite_valid_trace PreFreeE sX trX
  → finite_valid_trace PreFree
      (equivocators_total_state_project IM sX)
      (pre_VLSM_projection_finite_trace_project
         PreFreeE PreFree
         (equivocators_total_label_project IM)
         (equivocators_total_state_project IM) trX) intros * Htr.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
sX: state PreFreeE
trX: list transition_item
Htr: finite_valid_trace PreFreeE sX trX
finite_valid_trace PreFree
  (equivocators_total_state_project IM sX)
  (pre_VLSM_projection_finite_trace_project PreFreeE
     PreFree (equivocators_total_label_project IM)
     (equivocators_total_state_project IM) trX)
    specialize (VLSM_partial_projection_finite_valid_trace
      (PreFreeE_PreFree_vlsm_partial_projection IM (zero_descriptor IM))
      sX trX (equivocators_total_state_project IM sX) (equivocators_total_trace_project IM trX))
      as Hsim.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
sX: state PreFreeE
trX: list transition_item
Htr: finite_valid_trace PreFreeE sX trX
Hsim: equivocators_partial_trace_project IM
  (zero_descriptor IM) (
  sX, trX) =
Some
  (equivocators_total_state_project IM sX,
   equivocators_total_trace_project IM trX)
→ finite_valid_trace
    (preloaded_with_all_messages_vlsm
       (free_composite_vlsm (equivocator_IM IM)))
    sX trX
  → finite_valid_trace
      (preloaded_with_all_messages_vlsm
         (free_composite_vlsm IM))
      (equivocators_total_state_project IM sX)
      (equivocators_total_trace_project IM trX)
finite_valid_trace PreFree
  (equivocators_total_state_project IM sX)
  (pre_VLSM_projection_finite_trace_project PreFreeE
     PreFree (equivocators_total_label_project IM)
     (equivocators_total_state_project IM) trX)
    spec Hsim.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
sX: state PreFreeE
trX: list transition_item
Htr: finite_valid_trace PreFreeE sX trX
Hsim: equivocators_partial_trace_project IM
  (zero_descriptor IM) (
  sX, trX) =
Some
  (equivocators_total_state_project IM sX,
   equivocators_total_trace_project IM trX)
→ finite_valid_trace
    (preloaded_with_all_messages_vlsm
       (free_composite_vlsm (equivocator_IM IM)))
    sX trX
  → finite_valid_trace
      (preloaded_with_all_messages_vlsm
         (free_composite_vlsm IM))
      (equivocators_total_state_project IM sX)
      (equivocators_total_trace_project IM trX)
equivocators_partial_trace_project IM
  (zero_descriptor IM) (sX, trX) =
Some
  (equivocators_total_state_project IM sX,
   equivocators_total_trace_project IM trX)
message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
sX: state PreFreeE
trX: list transition_item
Htr: finite_valid_trace PreFreeE sX trX
Hsim: finite_valid_trace
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
  sX trX
→ finite_valid_trace
    (preloaded_with_all_messages_vlsm
       (free_composite_vlsm IM))
    (equivocators_total_state_project IM sX)
    (equivocators_total_trace_project IM trX)
finite_valid_trace PreFree
  (equivocators_total_state_project IM sX)
  (pre_VLSM_projection_finite_trace_project PreFreeE
     PreFree (equivocators_total_label_project IM)
     (equivocators_total_state_project IM) trX)
    {message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
sX: state PreFreeE
trX: list transition_item
Htr: finite_valid_trace PreFreeE sX trX
Hsim: equivocators_partial_trace_project IM
  (zero_descriptor IM) (
  sX, trX) =
Some
  (equivocators_total_state_project IM sX,
   equivocators_total_trace_project IM trX)
→ finite_valid_trace
    (preloaded_with_all_messages_vlsm
       (free_composite_vlsm (equivocator_IM IM)))
    sX trX
  → finite_valid_trace
      (preloaded_with_all_messages_vlsm
         (free_composite_vlsm IM))
      (equivocators_total_state_project IM sX)
      (equivocators_total_trace_project IM trX)
equivocators_partial_trace_project IM
  (zero_descriptor IM) (sX, trX) =
Some
  (equivocators_total_state_project IM sX,
   equivocators_total_trace_project IM trX) simpl.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
sX: state PreFreeE
trX: list transition_item
Htr: finite_valid_trace PreFreeE sX trX
Hsim: equivocators_partial_trace_project IM
  (zero_descriptor IM) (
  sX, trX) =
Some
  (equivocators_total_state_project IM sX,
   equivocators_total_trace_project IM trX)
→ finite_valid_trace
    (preloaded_with_all_messages_vlsm
       (free_composite_vlsm (equivocator_IM IM)))
    sX trX
  → finite_valid_trace
      (preloaded_with_all_messages_vlsm
         (free_composite_vlsm IM))
      (equivocators_total_state_project IM sX)
      (equivocators_total_trace_project IM trX)
(if
  decide
    (not_equivocating_equivocator_descriptors IM
       (zero_descriptor IM) (finite_trace_last sX trX))
 then
  match
    equivocators_trace_project IM (zero_descriptor IM)
      trX
  with
  | Some (trX, initial_descriptors) =>
      Some
        (equivocators_state_project IM
           initial_descriptors sX, trX)
  | None => None
  end
 else None) =
Some
  (equivocators_total_state_project IM sX,
   equivocators_total_trace_project IM trX) rewrite decide_True by apply zero_descriptor_not_equivocating.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
sX: state PreFreeE
trX: list transition_item
Htr: finite_valid_trace PreFreeE sX trX
Hsim: equivocators_partial_trace_project IM
  (zero_descriptor IM) (
  sX, trX) =
Some
  (equivocators_total_state_project IM sX,
   equivocators_total_trace_project IM trX)
→ finite_valid_trace
    (preloaded_with_all_messages_vlsm
       (free_composite_vlsm (equivocator_IM IM)))
    sX trX
  → finite_valid_trace
      (preloaded_with_all_messages_vlsm
         (free_composite_vlsm IM))
      (equivocators_total_state_project IM sX)
      (equivocators_total_trace_project IM trX)
match
  equivocators_trace_project IM (zero_descriptor IM)
    trX
with
| Some (trX, initial_descriptors) =>
    Some
      (equivocators_state_project IM
         initial_descriptors sX, trX)
| None => None
end =
Some
  (equivocators_total_state_project IM sX,
   equivocators_total_trace_project IM trX)
      by rewrite (equivocators_total_trace_project_characterization IM (proj1 Htr)).
    }message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
sX: state PreFreeE
trX: list transition_item
Htr: finite_valid_trace PreFreeE sX trX
Hsim: finite_valid_trace
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
  sX trX
→ finite_valid_trace
    (preloaded_with_all_messages_vlsm
       (free_composite_vlsm IM))
    (equivocators_total_state_project IM sX)
    (equivocators_total_trace_project IM trX)
finite_valid_trace PreFree
  (equivocators_total_state_project IM sX)
  (pre_VLSM_projection_finite_trace_project PreFreeE
     PreFree (equivocators_total_label_project IM)
     (equivocators_total_state_project IM) trX)
    apply Hsim in Htr as Hpr.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
sX: state PreFreeE
trX: list transition_item
Htr: finite_valid_trace PreFreeE sX trX
Hsim: finite_valid_trace
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
  sX trX
→ finite_valid_trace
    (preloaded_with_all_messages_vlsm
       (free_composite_vlsm IM))
    (equivocators_total_state_project IM sX)
    (equivocators_total_trace_project IM trX)
Hpr: finite_valid_trace
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm IM))
  (equivocators_total_state_project IM sX)
  (equivocators_total_trace_project IM trX)
finite_valid_trace PreFree
  (equivocators_total_state_project IM sX)
  (pre_VLSM_projection_finite_trace_project PreFreeE
     PreFree (equivocators_total_label_project IM)
     (equivocators_total_state_project IM) trX)
    remember (pre_VLSM_projection_finite_trace_project _ _ _ _ _) as tr.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
sX: state PreFreeE
trX: list transition_item
Htr: finite_valid_trace PreFreeE sX trX
Hsim: finite_valid_trace
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
  sX trX
→ finite_valid_trace
    (preloaded_with_all_messages_vlsm
       (free_composite_vlsm IM))
    (equivocators_total_state_project IM sX)
    (equivocators_total_trace_project IM trX)
Hpr: finite_valid_trace
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm IM))
  (equivocators_total_state_project IM sX)
  (equivocators_total_trace_project IM trX)
tr: list transition_item
Heqtr: tr =
pre_VLSM_projection_finite_trace_project
  PreFreeE PreFree
  (equivocators_total_label_project IM)
  (equivocators_total_state_project IM) trX
finite_valid_trace PreFree
  (equivocators_total_state_project IM sX) tr
    replace tr with (equivocators_total_trace_project IM trX); [done |].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
sX: state PreFreeE
trX: list transition_item
Htr: finite_valid_trace PreFreeE sX trX
Hsim: finite_valid_trace
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
  sX trX
→ finite_valid_trace
    (preloaded_with_all_messages_vlsm
       (free_composite_vlsm IM))
    (equivocators_total_state_project IM sX)
    (equivocators_total_trace_project IM trX)
Hpr: finite_valid_trace
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm IM))
  (equivocators_total_state_project IM sX)
  (equivocators_total_trace_project IM trX)
tr: list transition_item
Heqtr: tr =
pre_VLSM_projection_finite_trace_project
  PreFreeE PreFree
  (equivocators_total_label_project IM)
  (equivocators_total_state_project IM) trX
equivocators_total_trace_project IM trX = tr
    subst.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
sX: state PreFreeE
trX: list transition_item
Htr: finite_valid_trace PreFreeE sX trX
Hsim: finite_valid_trace
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
  sX trX
→ finite_valid_trace
    (preloaded_with_all_messages_vlsm
       (free_composite_vlsm IM))
    (equivocators_total_state_project IM sX)
    (equivocators_total_trace_project IM trX)
Hpr: finite_valid_trace
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm IM))
  (equivocators_total_state_project IM sX)
  (equivocators_total_trace_project IM trX)
equivocators_total_trace_project IM trX =
pre_VLSM_projection_finite_trace_project PreFreeE
  PreFree (equivocators_total_label_project IM)
  (equivocators_total_state_project IM) trX symmetry.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
PreFreeE:= preloaded_with_all_messages_vlsm FreeE: VLSM message
Free:= free_composite_vlsm IM: VLSM message
PreFree:= preloaded_with_all_messages_vlsm Free: VLSM message
sX: state PreFreeE
trX: list transition_item
Htr: finite_valid_trace PreFreeE sX trX
Hsim: finite_valid_trace
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
  sX trX
→ finite_valid_trace
    (preloaded_with_all_messages_vlsm
       (free_composite_vlsm IM))
    (equivocators_total_state_project IM sX)
    (equivocators_total_trace_project IM trX)
Hpr: finite_valid_trace
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm IM))
  (equivocators_total_state_project IM sX)
  (equivocators_total_trace_project IM trX)
pre_VLSM_projection_finite_trace_project PreFreeE
  PreFree (equivocators_total_label_project IM)
  (equivocators_total_state_project IM) trX =
equivocators_total_trace_project IM trX
    by eapply equivocators_total_VLSM_projection_finite_trace_project, Htr.
Qed.

End sec_equivocators_composition_vlsm_projection.