From stdpp Require Import prelude finite.[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]
From Coq Require Import FinFun FunctionalExtensionality.
From VLSM.Lib Require Import Preamble ListExtras.[Loading ML file coq-itauto.plugin ... done]
From VLSM.Core Require Import VLSM VLSMProjections Composition SubProjectionTraces.
From VLSM.Core Require Import Equivocation Equivocation.NoEquivocation.
From VLSM.Core Require Import Equivocators.Equivocators Equivocators.MessageProperties.[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.EquivocatorsComposition.
From VLSM.Core Require Import Equivocators.EquivocatorsCompositionProjections.
From VLSM.Core Require Import Equivocators.FullReplayTraces.

Core: Equivocators Simulating Regular Components

In this module, we prove a general simulation result parameterized by constraints with good properties, then we instantiate the general result for the free composition of regular components.

Section sec_generalized_constraints.

Generic simulation

In this section we prove a general simulation result, namely that a composition of equivocators constrained by a constraintE can simulate a composition of the corresponding regular components constrained by a constraintX if they are related through the zero_descriptor_constraint_lifting_property and the replayable_message_prop.

Context
  {message : Type}
  `{finite.Finite index}
  (IM : index -> VLSM message)
  `{forall i : index, HasBeenSentCapability (IM i)}
  (seed : message -> Prop)
  (constraintX : composite_label IM -> composite_state IM * option message -> Prop)
  (CX := preloaded_vlsm (composite_vlsm IM constraintX) seed)
  (constraintE : composite_label (equivocator_IM IM) ->
                  composite_state (equivocator_IM IM) * option message -> Prop)
  (CE := preloaded_vlsm (composite_vlsm (equivocator_IM IM) constraintE) seed)
  (FreeE := free_composite_vlsm (equivocator_IM IM))
  .

Definition last_in_trace_except_from
  {T : VLSMType message} exception (tr : list (transition_item T)) iom : Prop :=
    match iom with
    | None => True
    | Some im =>
      finite_trace_last_output tr = Some im
      \/ exception im
    end.

A property requiring that constraintE holds for any transition using an original copy of an equivocator if the message is initial or has_been_sent for the given state.

Definition zero_descriptor_constraint_lifting_prop : Prop :=
  forall
    es (Hes : valid_state_prop CE es)
    om (Hom : sent_except_from (equivocator_IM IM) (@initial_message_prop _ _ CE) es om)
    eqv li,
    constraintE (existT eqv (ContinueWith 0 li)) (es, om).

The replayable_message_property plays an important role in designing a general, abstract, proof for trace simulation (Lemma generalized_equivocators_finite_valid_trace_init_to_rev), as it specifies that given a message m received in a regular composite state s for which the constraint constraintX is satisfied, then any trace of the equivocators-composition (constrained by constraintE>) producing m can be "replayed" on top of an equivocators-composition state corresponding to s, with no transitions being performed on the original copies.

Definition replayable_message_prop : Prop :=
  forall si s tr
    (HtrX : finite_valid_trace_init_to CX si s tr)
    eqv_state_s
    (Hstate_valid : valid_state_prop CE eqv_state_s)
    (Hstate_final_project : equivocators_total_state_project IM eqv_state_s = s)
    eqv_msg_is eqv_msg_s eqv_msg_tr
    (Hmsg_trace : finite_valid_trace_init_to CE eqv_msg_is eqv_msg_s eqv_msg_tr)
    iom
    (Hfinal_msg : last_in_trace_except_from (@initial_message_prop _ _ CE) eqv_msg_tr iom)
    l
    (HcX : constraintX l (s, iom)),
    exists eqv_msg_tr lst_msg_tr,
      finite_valid_trace_from_to CE eqv_state_s lst_msg_tr eqv_msg_tr /\
      equivocators_total_trace_project IM eqv_msg_tr = [] /\
      equivocators_total_state_project IM lst_msg_tr = s /\
      sent_except_from (equivocator_IM IM) (@initial_message_prop _ _ CE) lst_msg_tr iom.

The main result of this section, showing that every trace of the composition of regular components can be obtained as a zero_descriptor projection of a valid trace for the composition of equivocators.

The proof proceeds by replaying the transitions of the trace on the original copy of the machines, replaying traces corresponding to the messages between those transition through equivocation to guarantee the no-message-equivocation constraint.

Lemma generalized_equivocators_finite_valid_trace_init_to_rev
  (Hsubsumption : zero_descriptor_constraint_lifting_prop)
  (Hreplayable : replayable_message_prop)
  isX sX trX
  (HtrX : finite_valid_trace_init_to CX isX sX trX)
  : exists is, equivocators_total_state_project IM is = isX /\
    exists s, equivocators_total_state_project IM s = sX /\
    exists tr, equivocators_total_trace_project IM tr = trX /\
    finite_valid_trace_init_to CE is s tr /\
    finite_trace_last_output trX = finite_trace_last_output tr.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
seed: message → Prop
constraintX: composite_label IM
→ composite_state IM * option message
  → Prop
CX:= preloaded_vlsm (composite_vlsm IM constraintX)
  seed: VLSM message
constraintE: composite_label (equivocator_IM IM)
→ composite_state (equivocator_IM IM) *
  option message → Prop
CE:= preloaded_vlsm
  (composite_vlsm (equivocator_IM IM) constraintE)
  seed: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hsubsumption: zero_descriptor_constraint_lifting_prop
Hreplayable: replayable_message_prop
isX, sX: state CX
trX: list transition_item
HtrX: finite_valid_trace_init_to CX isX sX trX
∃ is : state (free_composite_vlsm (equivocator_IM IM)),
  equivocators_total_state_project IM is = isX
  ∧ (∃ s : state
             (free_composite_vlsm (equivocator_IM IM)),
       equivocators_total_state_project IM s = sX
       ∧ (∃ tr : list
                   (composite_transition_item
                      (equivocator_IM IM)),
            equivocators_total_trace_project IM tr =
            trX
            ∧ finite_valid_trace_init_to CE is s tr
              ∧ finite_trace_last_output trX =
                finite_trace_last_output tr))
Proof.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
seed: message → Prop
constraintX: composite_label IM
→ composite_state IM * option message
  → Prop
CX:= preloaded_vlsm (composite_vlsm IM constraintX)
  seed: VLSM message
constraintE: composite_label (equivocator_IM IM)
→ composite_state (equivocator_IM IM) *
  option message → Prop
CE:= preloaded_vlsm
  (composite_vlsm (equivocator_IM IM) constraintE)
  seed: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hsubsumption: zero_descriptor_constraint_lifting_prop
Hreplayable: replayable_message_prop
isX, sX: state CX
trX: list transition_item
HtrX: finite_valid_trace_init_to CX isX sX trX
∃ is : state (free_composite_vlsm (equivocator_IM IM)),
  equivocators_total_state_project IM is = isX
  ∧ (∃ s : state
             (free_composite_vlsm (equivocator_IM IM)),
       equivocators_total_state_project IM s = sX
       ∧ (∃ tr : list
                   (composite_transition_item
                      (equivocator_IM IM)),
            equivocators_total_trace_project IM tr =
            trX
            ∧ finite_valid_trace_init_to CE is s tr
              ∧ finite_trace_last_output trX =
                finite_trace_last_output tr))
  assert (HinclE : VLSM_incl CE (preloaded_with_all_messages_vlsm FreeE))
    by apply constrained_preloaded_vlsm_incl_preloaded_with_all_messages.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
seed: message → Prop
constraintX: composite_label IM
→ composite_state IM * option message
  → Prop
CX:= preloaded_vlsm (composite_vlsm IM constraintX)
  seed: VLSM message
constraintE: composite_label (equivocator_IM IM)
→ composite_state (equivocator_IM IM) *
  option message → Prop
CE:= preloaded_vlsm
  (composite_vlsm (equivocator_IM IM) constraintE)
  seed: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hsubsumption: zero_descriptor_constraint_lifting_prop
Hreplayable: replayable_message_prop
isX, sX: state CX
trX: list transition_item
HtrX: finite_valid_trace_init_to CX isX sX trX
HinclE: VLSM_incl CE
  (preloaded_with_all_messages_vlsm FreeE)
∃ is : state (free_composite_vlsm (equivocator_IM IM)),
  equivocators_total_state_project IM is = isX
  ∧ (∃ s : state
             (free_composite_vlsm (equivocator_IM IM)),
       equivocators_total_state_project IM s = sX
       ∧ (∃ tr : list
                   (composite_transition_item
                      (equivocator_IM IM)),
            equivocators_total_trace_project IM tr =
            trX
            ∧ finite_valid_trace_init_to CE is s tr
              ∧ finite_trace_last_output trX =
                finite_trace_last_output tr))
  induction HtrX using finite_valid_trace_init_to_rev_strong_ind.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
seed: message → Prop
constraintX: composite_label IM
→ composite_state IM * option message
  → Prop
CX:= preloaded_vlsm (composite_vlsm IM constraintX)
  seed: VLSM message
constraintE: composite_label (equivocator_IM IM)
→ composite_state (equivocator_IM IM) *
  option message → Prop
CE:= preloaded_vlsm
  (composite_vlsm (equivocator_IM IM) constraintE)
  seed: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hsubsumption: zero_descriptor_constraint_lifting_prop
Hreplayable: replayable_message_prop
is: state CX
His: initial_state_prop is
HinclE: VLSM_incl CE
  (preloaded_with_all_messages_vlsm FreeE)
∃ is0 : state
          (free_composite_vlsm (equivocator_IM IM)),
  equivocators_total_state_project IM is0 = is
  ∧ (∃ s : state
             (free_composite_vlsm (equivocator_IM IM)),
       equivocators_total_state_project IM s = is
       ∧ (∃ tr : list
                   (composite_transition_item
                      (equivocator_IM IM)),
            equivocators_total_trace_project IM tr =
            []
            ∧ finite_valid_trace_init_to CE is0 s tr
              ∧ finite_trace_last_output [] =
                finite_trace_last_output tr))
message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
seed: message → Prop
constraintX: composite_label IM
→ composite_state IM * option message
  → Prop
CX:= preloaded_vlsm (composite_vlsm IM constraintX)
  seed: VLSM message
constraintE: composite_label (equivocator_IM IM)
→ composite_state (equivocator_IM IM) *
  option message → Prop
CE:= preloaded_vlsm
  (composite_vlsm (equivocator_IM IM) constraintE)
  seed: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hsubsumption: zero_descriptor_constraint_lifting_prop
Hreplayable: replayable_message_prop
is, s: state CX
tr: list transition_item
HtrX1: finite_valid_trace_init_to CX is s tr
iom: option message
iom_si, iom_s: state CX
iom_tr: list transition_item
Heqiom: empty_initial_message_or_final_output CX
  iom_tr iom
HtrX2: finite_valid_trace_init_to CX iom_si iom_s
  iom_tr
sf: state CX
oom: option message
l: label CX
Ht: input_valid_transition CX l (s, iom) (sf, oom)
HinclE: VLSM_incl CE
  (preloaded_with_all_messages_vlsm FreeE)
IHHtrX1: ∃ is0 : state
          (free_composite_vlsm
             (equivocator_IM IM)),
  equivocators_total_state_project IM is0 =
  is
  ∧ (∃ s0 : state
              (free_composite_vlsm
                 (equivocator_IM IM)),
       equivocators_total_state_project IM
         s0 = s
       ∧ (∃ tr0 : list
                    (composite_transition_item
                      (equivocator_IM IM)),
            equivocators_total_trace_project
              IM tr0 = tr
            ∧ finite_valid_trace_init_to CE
                is0 s0 tr0
              ∧ finite_trace_last_output tr =
                finite_trace_last_output tr0))
IHHtrX2: ∃ is : state
         (free_composite_vlsm
            (equivocator_IM IM)),
  equivocators_total_state_project IM is =
  iom_si
  ∧ (∃ s : state
             (free_composite_vlsm
                (equivocator_IM IM)),
       equivocators_total_state_project IM s =
       iom_s
       ∧ (∃ tr : list
                   (composite_transition_item
                      (equivocator_IM IM)),
            equivocators_total_trace_project
              IM tr = iom_tr
            ∧ finite_valid_trace_init_to CE
                is s tr
              ∧ finite_trace_last_output
                  iom_tr =
                finite_trace_last_output tr))
∃ is0 : state
          (free_composite_vlsm (equivocator_IM IM)),
  equivocators_total_state_project IM is0 = is
  ∧ (∃ s : state
             (free_composite_vlsm (equivocator_IM IM)),
       equivocators_total_state_project IM s = sf
       ∧ (∃ tr0 : list
                    (composite_transition_item
                      (equivocator_IM IM)),
            equivocators_total_trace_project IM tr0 =
            tr ++
            [{|
               l := l;
               input := iom;
               destination := sf;
               output := oom
             |}]
            ∧ finite_valid_trace_init_to CE is0 s tr0
              ∧ finite_trace_last_output
                  (tr ++
                   [{|
                      l := l;
                      input := iom;
                      destination := sf;
                      output := oom
                    |}]) =
                finite_trace_last_output tr0))
  -message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
seed: message → Prop
constraintX: composite_label IM
→ composite_state IM * option message
  → Prop
CX:= preloaded_vlsm (composite_vlsm IM constraintX)
  seed: VLSM message
constraintE: composite_label (equivocator_IM IM)
→ composite_state (equivocator_IM IM) *
  option message → Prop
CE:= preloaded_vlsm
  (composite_vlsm (equivocator_IM IM) constraintE)
  seed: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hsubsumption: zero_descriptor_constraint_lifting_prop
Hreplayable: replayable_message_prop
is: state CX
His: initial_state_prop is
HinclE: VLSM_incl CE
  (preloaded_with_all_messages_vlsm FreeE)
∃ is0 : state
          (free_composite_vlsm (equivocator_IM IM)),
  equivocators_total_state_project IM is0 = is
  ∧ (∃ s : state
             (free_composite_vlsm (equivocator_IM IM)),
       equivocators_total_state_project IM s = is
       ∧ (∃ tr : list
                   (composite_transition_item
                      (equivocator_IM IM)),
            equivocators_total_trace_project IM tr =
            []
            ∧ finite_valid_trace_init_to CE is0 s tr
              ∧ finite_trace_last_output [] =
                finite_trace_last_output tr)) specialize (lift_initial_to_equivocators_state IM _ His) as Hs.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
seed: message → Prop
constraintX: composite_label IM
→ composite_state IM * option message
  → Prop
CX:= preloaded_vlsm (composite_vlsm IM constraintX)
  seed: VLSM message
constraintE: composite_label (equivocator_IM IM)
→ composite_state (equivocator_IM IM) *
  option message → Prop
CE:= preloaded_vlsm
  (composite_vlsm (equivocator_IM IM) constraintE)
  seed: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hsubsumption: zero_descriptor_constraint_lifting_prop
Hreplayable: replayable_message_prop
is: state CX
His: initial_state_prop is
HinclE: VLSM_incl CE
  (preloaded_with_all_messages_vlsm FreeE)
Hs: initial_state_prop
  (lift_to_equivocators_state IM is)
∃ is0 : state
          (free_composite_vlsm (equivocator_IM IM)),
  equivocators_total_state_project IM is0 = is
  ∧ (∃ s : state
             (free_composite_vlsm (equivocator_IM IM)),
       equivocators_total_state_project IM s = is
       ∧ (∃ tr : list
                   (composite_transition_item
                      (equivocator_IM IM)),
            equivocators_total_trace_project IM tr =
            []
            ∧ finite_valid_trace_init_to CE is0 s tr
              ∧ finite_trace_last_output [] =
                finite_trace_last_output tr))
    remember (lift_to_equivocators_state IM is) as s.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
seed: message → Prop
constraintX: composite_label IM
→ composite_state IM * option message
  → Prop
CX:= preloaded_vlsm (composite_vlsm IM constraintX)
  seed: VLSM message
constraintE: composite_label (equivocator_IM IM)
→ composite_state (equivocator_IM IM) *
  option message → Prop
CE:= preloaded_vlsm
  (composite_vlsm (equivocator_IM IM) constraintE)
  seed: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hsubsumption: zero_descriptor_constraint_lifting_prop
Hreplayable: replayable_message_prop
is: state CX
His: initial_state_prop is
HinclE: VLSM_incl CE
  (preloaded_with_all_messages_vlsm FreeE)
s: ∀ eqv : index, state (equivocator_IM IM eqv)
Heqs: s = lift_to_equivocators_state IM is
Hs: initial_state_prop s
∃ is0 : state
          (free_composite_vlsm (equivocator_IM IM)),
  equivocators_total_state_project IM is0 = is
  ∧ (∃ s : state
             (free_composite_vlsm (equivocator_IM IM)),
       equivocators_total_state_project IM s = is
       ∧ (∃ tr : list
                   (composite_transition_item
                      (equivocator_IM IM)),
            equivocators_total_trace_project IM tr =
            []
            ∧ finite_valid_trace_init_to CE is0 s tr
              ∧ finite_trace_last_output [] =
                finite_trace_last_output tr))
    cut (equivocators_state_project IM (zero_descriptor IM) s = is).message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
seed: message → Prop
constraintX: composite_label IM
→ composite_state IM * option message
  → Prop
CX:= preloaded_vlsm (composite_vlsm IM constraintX)
  seed: VLSM message
constraintE: composite_label (equivocator_IM IM)
→ composite_state (equivocator_IM IM) *
  option message → Prop
CE:= preloaded_vlsm
  (composite_vlsm (equivocator_IM IM) constraintE)
  seed: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hsubsumption: zero_descriptor_constraint_lifting_prop
Hreplayable: replayable_message_prop
is: state CX
His: initial_state_prop is
HinclE: VLSM_incl CE
  (preloaded_with_all_messages_vlsm FreeE)
s: ∀ eqv : index, state (equivocator_IM IM eqv)
Heqs: s = lift_to_equivocators_state IM is
Hs: initial_state_prop s
equivocators_state_project IM (zero_descriptor IM) s =
is
→ ∃ is0 : state
            (free_composite_vlsm (equivocator_IM IM)),
    equivocators_total_state_project IM is0 = is
    ∧ (∃ s : state
               (free_composite_vlsm
                  (equivocator_IM IM)),
         equivocators_total_state_project IM s = is
         ∧ (∃ tr : list
                     (composite_transition_item
                        (equivocator_IM IM)),
              equivocators_total_trace_project IM tr =
              []
              ∧ finite_valid_trace_init_to CE is0 s tr
                ∧ finite_trace_last_output [] =
                  finite_trace_last_output tr))
message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
seed: message → Prop
constraintX: composite_label IM
→ composite_state IM * option message
  → Prop
CX:= preloaded_vlsm (composite_vlsm IM constraintX)
  seed: VLSM message
constraintE: composite_label (equivocator_IM IM)
→ composite_state (equivocator_IM IM) *
  option message → Prop
CE:= preloaded_vlsm
  (composite_vlsm (equivocator_IM IM) constraintE)
  seed: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hsubsumption: zero_descriptor_constraint_lifting_prop
Hreplayable: replayable_message_prop
is: state CX
His: initial_state_prop is
HinclE: VLSM_incl CE
  (preloaded_with_all_messages_vlsm FreeE)
s: ∀ eqv : index, state (equivocator_IM IM eqv)
Heqs: s = lift_to_equivocators_state IM is
Hs: initial_state_prop s
equivocators_state_project IM (zero_descriptor IM) s =
is
    {message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
seed: message → Prop
constraintX: composite_label IM
→ composite_state IM * option message
  → Prop
CX:= preloaded_vlsm (composite_vlsm IM constraintX)
  seed: VLSM message
constraintE: composite_label (equivocator_IM IM)
→ composite_state (equivocator_IM IM) *
  option message → Prop
CE:= preloaded_vlsm
  (composite_vlsm (equivocator_IM IM) constraintE)
  seed: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hsubsumption: zero_descriptor_constraint_lifting_prop
Hreplayable: replayable_message_prop
is: state CX
His: initial_state_prop is
HinclE: VLSM_incl CE
  (preloaded_with_all_messages_vlsm FreeE)
s: ∀ eqv : index, state (equivocator_IM IM eqv)
Heqs: s = lift_to_equivocators_state IM is
Hs: initial_state_prop s
equivocators_state_project IM (zero_descriptor IM) s =
is
→ ∃ is0 : state
            (free_composite_vlsm (equivocator_IM IM)),
    equivocators_total_state_project IM is0 = is
    ∧ (∃ s : state
               (free_composite_vlsm
                  (equivocator_IM IM)),
         equivocators_total_state_project IM s = is
         ∧ (∃ tr : list
                     (composite_transition_item
                        (equivocator_IM IM)),
              equivocators_total_trace_project IM tr =
              []
              ∧ finite_valid_trace_init_to CE is0 s tr
                ∧ finite_trace_last_output [] =
                  finite_trace_last_output tr)) intro Hproject.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
seed: message → Prop
constraintX: composite_label IM
→ composite_state IM * option message
  → Prop
CX:= preloaded_vlsm (composite_vlsm IM constraintX)
  seed: VLSM message
constraintE: composite_label (equivocator_IM IM)
→ composite_state (equivocator_IM IM) *
  option message → Prop
CE:= preloaded_vlsm
  (composite_vlsm (equivocator_IM IM) constraintE)
  seed: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hsubsumption: zero_descriptor_constraint_lifting_prop
Hreplayable: replayable_message_prop
is: state CX
His: initial_state_prop is
HinclE: VLSM_incl CE
  (preloaded_with_all_messages_vlsm FreeE)
s: ∀ eqv : index, state (equivocator_IM IM eqv)
Heqs: s = lift_to_equivocators_state IM is
Hs: initial_state_prop s
Hproject: equivocators_state_project IM
  (zero_descriptor IM) s = is
∃ is0 : state
          (free_composite_vlsm (equivocator_IM IM)),
  equivocators_total_state_project IM is0 = is
  ∧ (∃ s : state
             (free_composite_vlsm (equivocator_IM IM)),
       equivocators_total_state_project IM s = is
       ∧ (∃ tr : list
                   (composite_transition_item
                      (equivocator_IM IM)),
            equivocators_total_trace_project IM tr =
            []
            ∧ finite_valid_trace_init_to CE is0 s tr
              ∧ finite_trace_last_output [] =
                finite_trace_last_output tr))
      exists s.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
seed: message → Prop
constraintX: composite_label IM
→ composite_state IM * option message
  → Prop
CX:= preloaded_vlsm (composite_vlsm IM constraintX)
  seed: VLSM message
constraintE: composite_label (equivocator_IM IM)
→ composite_state (equivocator_IM IM) *
  option message → Prop
CE:= preloaded_vlsm
  (composite_vlsm (equivocator_IM IM) constraintE)
  seed: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hsubsumption: zero_descriptor_constraint_lifting_prop
Hreplayable: replayable_message_prop
is: state CX
His: initial_state_prop is
HinclE: VLSM_incl CE
  (preloaded_with_all_messages_vlsm FreeE)
s: ∀ eqv : index, state (equivocator_IM IM eqv)
Heqs: s = lift_to_equivocators_state IM is
Hs: initial_state_prop s
Hproject: equivocators_state_project IM
  (zero_descriptor IM) s = is
equivocators_total_state_project IM s = is
∧ (∃ s0 : state
            (free_composite_vlsm (equivocator_IM IM)),
     equivocators_total_state_project IM s0 = is
     ∧ (∃ tr : list
                 (composite_transition_item
                    (equivocator_IM IM)),
          equivocators_total_trace_project IM tr = []
          ∧ finite_valid_trace_init_to CE s s0 tr
            ∧ finite_trace_last_output [] =
              finite_trace_last_output tr)) split; [done |].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
seed: message → Prop
constraintX: composite_label IM
→ composite_state IM * option message
  → Prop
CX:= preloaded_vlsm (composite_vlsm IM constraintX)
  seed: VLSM message
constraintE: composite_label (equivocator_IM IM)
→ composite_state (equivocator_IM IM) *
  option message → Prop
CE:= preloaded_vlsm
  (composite_vlsm (equivocator_IM IM) constraintE)
  seed: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hsubsumption: zero_descriptor_constraint_lifting_prop
Hreplayable: replayable_message_prop
is: state CX
His: initial_state_prop is
HinclE: VLSM_incl CE
  (preloaded_with_all_messages_vlsm FreeE)
s: ∀ eqv : index, state (equivocator_IM IM eqv)
Heqs: s = lift_to_equivocators_state IM is
Hs: initial_state_prop s
Hproject: equivocators_state_project IM
  (zero_descriptor IM) s = is
∃ s0 : state (free_composite_vlsm (equivocator_IM IM)),
  equivocators_total_state_project IM s0 = is
  ∧ (∃ tr : list
              (composite_transition_item
                 (equivocator_IM IM)),
       equivocators_total_trace_project IM tr = []
       ∧ finite_valid_trace_init_to CE s s0 tr
         ∧ finite_trace_last_output [] =
           finite_trace_last_output tr)
      exists s.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
seed: message → Prop
constraintX: composite_label IM
→ composite_state IM * option message
  → Prop
CX:= preloaded_vlsm (composite_vlsm IM constraintX)
  seed: VLSM message
constraintE: composite_label (equivocator_IM IM)
→ composite_state (equivocator_IM IM) *
  option message → Prop
CE:= preloaded_vlsm
  (composite_vlsm (equivocator_IM IM) constraintE)
  seed: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hsubsumption: zero_descriptor_constraint_lifting_prop
Hreplayable: replayable_message_prop
is: state CX
His: initial_state_prop is
HinclE: VLSM_incl CE
  (preloaded_with_all_messages_vlsm FreeE)
s: ∀ eqv : index, state (equivocator_IM IM eqv)
Heqs: s = lift_to_equivocators_state IM is
Hs: initial_state_prop s
Hproject: equivocators_state_project IM
  (zero_descriptor IM) s = is
equivocators_total_state_project IM s = is
∧ (∃ tr : list
            (composite_transition_item
               (equivocator_IM IM)),
     equivocators_total_trace_project IM tr = []
     ∧ finite_valid_trace_init_to CE s s tr
       ∧ finite_trace_last_output [] =
         finite_trace_last_output tr) split; [done |].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
seed: message → Prop
constraintX: composite_label IM
→ composite_state IM * option message
  → Prop
CX:= preloaded_vlsm (composite_vlsm IM constraintX)
  seed: VLSM message
constraintE: composite_label (equivocator_IM IM)
→ composite_state (equivocator_IM IM) *
  option message → Prop
CE:= preloaded_vlsm
  (composite_vlsm (equivocator_IM IM) constraintE)
  seed: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hsubsumption: zero_descriptor_constraint_lifting_prop
Hreplayable: replayable_message_prop
is: state CX
His: initial_state_prop is
HinclE: VLSM_incl CE
  (preloaded_with_all_messages_vlsm FreeE)
s: ∀ eqv : index, state (equivocator_IM IM eqv)
Heqs: s = lift_to_equivocators_state IM is
Hs: initial_state_prop s
Hproject: equivocators_state_project IM
  (zero_descriptor IM) s = is
∃ tr : list
         (composite_transition_item
            (equivocator_IM IM)),
  equivocators_total_trace_project IM tr = []
  ∧ finite_valid_trace_init_to CE s s tr
    ∧ finite_trace_last_output [] =
      finite_trace_last_output tr
      exists [].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
seed: message → Prop
constraintX: composite_label IM
→ composite_state IM * option message
  → Prop
CX:= preloaded_vlsm (composite_vlsm IM constraintX)
  seed: VLSM message
constraintE: composite_label (equivocator_IM IM)
→ composite_state (equivocator_IM IM) *
  option message → Prop
CE:= preloaded_vlsm
  (composite_vlsm (equivocator_IM IM) constraintE)
  seed: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hsubsumption: zero_descriptor_constraint_lifting_prop
Hreplayable: replayable_message_prop
is: state CX
His: initial_state_prop is
HinclE: VLSM_incl CE
  (preloaded_with_all_messages_vlsm FreeE)
s: ∀ eqv : index, state (equivocator_IM IM eqv)
Heqs: s = lift_to_equivocators_state IM is
Hs: initial_state_prop s
Hproject: equivocators_state_project IM
  (zero_descriptor IM) s = is
equivocators_total_trace_project IM [] = []
∧ finite_valid_trace_init_to CE s s []
  ∧ finite_trace_last_output [] =
    finite_trace_last_output [] split; [done |].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
seed: message → Prop
constraintX: composite_label IM
→ composite_state IM * option message
  → Prop
CX:= preloaded_vlsm (composite_vlsm IM constraintX)
  seed: VLSM message
constraintE: composite_label (equivocator_IM IM)
→ composite_state (equivocator_IM IM) *
  option message → Prop
CE:= preloaded_vlsm
  (composite_vlsm (equivocator_IM IM) constraintE)
  seed: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hsubsumption: zero_descriptor_constraint_lifting_prop
Hreplayable: replayable_message_prop
is: state CX
His: initial_state_prop is
HinclE: VLSM_incl CE
  (preloaded_with_all_messages_vlsm FreeE)
s: ∀ eqv : index, state (equivocator_IM IM eqv)
Heqs: s = lift_to_equivocators_state IM is
Hs: initial_state_prop s
Hproject: equivocators_state_project IM
  (zero_descriptor IM) s = is
finite_valid_trace_init_to CE s s []
∧ finite_trace_last_output [] =
  finite_trace_last_output [] do 2 (split; [| done]).message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
seed: message → Prop
constraintX: composite_label IM
→ composite_state IM * option message
  → Prop
CX:= preloaded_vlsm (composite_vlsm IM constraintX)
  seed: VLSM message
constraintE: composite_label (equivocator_IM IM)
→ composite_state (equivocator_IM IM) *
  option message → Prop
CE:= preloaded_vlsm
  (composite_vlsm (equivocator_IM IM) constraintE)
  seed: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hsubsumption: zero_descriptor_constraint_lifting_prop
Hreplayable: replayable_message_prop
is: state CX
His: initial_state_prop is
HinclE: VLSM_incl CE
  (preloaded_with_all_messages_vlsm FreeE)
s: ∀ eqv : index, state (equivocator_IM IM eqv)
Heqs: s = lift_to_equivocators_state IM is
Hs: initial_state_prop s
Hproject: equivocators_state_project IM
  (zero_descriptor IM) s = is
finite_valid_trace_from_to CE s s []
      constructor.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
seed: message → Prop
constraintX: composite_label IM
→ composite_state IM * option message
  → Prop
CX:= preloaded_vlsm (composite_vlsm IM constraintX)
  seed: VLSM message
constraintE: composite_label (equivocator_IM IM)
→ composite_state (equivocator_IM IM) *
  option message → Prop
CE:= preloaded_vlsm
  (composite_vlsm (equivocator_IM IM) constraintE)
  seed: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hsubsumption: zero_descriptor_constraint_lifting_prop
Hreplayable: replayable_message_prop
is: state CX
His: initial_state_prop is
HinclE: VLSM_incl CE
  (preloaded_with_all_messages_vlsm FreeE)
s: ∀ eqv : index, state (equivocator_IM IM eqv)
Heqs: s = lift_to_equivocators_state IM is
Hs: initial_state_prop s
Hproject: equivocators_state_project IM
  (zero_descriptor IM) s = is
valid_state_prop CE s
      by 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)
seed: message → Prop
constraintX: composite_label IM
→ composite_state IM * option message
  → Prop
CX:= preloaded_vlsm (composite_vlsm IM constraintX)
  seed: VLSM message
constraintE: composite_label (equivocator_IM IM)
→ composite_state (equivocator_IM IM) *
  option message → Prop
CE:= preloaded_vlsm
  (composite_vlsm (equivocator_IM IM) constraintE)
  seed: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hsubsumption: zero_descriptor_constraint_lifting_prop
Hreplayable: replayable_message_prop
is: state CX
His: initial_state_prop is
HinclE: VLSM_incl CE
  (preloaded_with_all_messages_vlsm FreeE)
s: ∀ eqv : index, state (equivocator_IM IM eqv)
Heqs: s = lift_to_equivocators_state IM is
Hs: initial_state_prop s
equivocators_state_project IM (zero_descriptor IM) s =
is
    by apply functional_extensionality_dep_good; subst.
  -message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
seed: message → Prop
constraintX: composite_label IM
→ composite_state IM * option message
  → Prop
CX:= preloaded_vlsm (composite_vlsm IM constraintX)
  seed: VLSM message
constraintE: composite_label (equivocator_IM IM)
→ composite_state (equivocator_IM IM) *
  option message → Prop
CE:= preloaded_vlsm
  (composite_vlsm (equivocator_IM IM) constraintE)
  seed: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hsubsumption: zero_descriptor_constraint_lifting_prop
Hreplayable: replayable_message_prop
is, s: state CX
tr: list transition_item
HtrX1: finite_valid_trace_init_to CX is s tr
iom: option message
iom_si, iom_s: state CX
iom_tr: list transition_item
Heqiom: empty_initial_message_or_final_output CX
  iom_tr iom
HtrX2: finite_valid_trace_init_to CX iom_si iom_s
  iom_tr
sf: state CX
oom: option message
l: label CX
Ht: input_valid_transition CX l (s, iom) (sf, oom)
HinclE: VLSM_incl CE
  (preloaded_with_all_messages_vlsm FreeE)
IHHtrX1: ∃ is0 : state
          (free_composite_vlsm
             (equivocator_IM IM)),
  equivocators_total_state_project IM is0 =
  is
  ∧ (∃ s0 : state
              (free_composite_vlsm
                 (equivocator_IM IM)),
       equivocators_total_state_project IM
         s0 = s
       ∧ (∃ tr0 : list
                    (composite_transition_item
                      (equivocator_IM IM)),
            equivocators_total_trace_project
              IM tr0 = tr
            ∧ finite_valid_trace_init_to CE
                is0 s0 tr0
              ∧ finite_trace_last_output tr =
                finite_trace_last_output tr0))
IHHtrX2: ∃ is : state
         (free_composite_vlsm
            (equivocator_IM IM)),
  equivocators_total_state_project IM is =
  iom_si
  ∧ (∃ s : state
             (free_composite_vlsm
                (equivocator_IM IM)),
       equivocators_total_state_project IM s =
       iom_s
       ∧ (∃ tr : list
                   (composite_transition_item
                      (equivocator_IM IM)),
            equivocators_total_trace_project
              IM tr = iom_tr
            ∧ finite_valid_trace_init_to CE
                is s tr
              ∧ finite_trace_last_output
                  iom_tr =
                finite_trace_last_output tr))
∃ is0 : state
          (free_composite_vlsm (equivocator_IM IM)),
  equivocators_total_state_project IM is0 = is
  ∧ (∃ s : state
             (free_composite_vlsm (equivocator_IM IM)),
       equivocators_total_state_project IM s = sf
       ∧ (∃ tr0 : list
                    (composite_transition_item
                       (equivocator_IM IM)),
            equivocators_total_trace_project IM tr0 =
            tr ++
            [{|
               l := l;
               input := iom;
               destination := sf;
               output := oom
             |}]
            ∧ finite_valid_trace_init_to CE is0 s tr0
              ∧ finite_trace_last_output
                  (tr ++
                   [{|
                      l := l;
                      input := iom;
                      destination := sf;
                      output := oom
                    |}]) =
                finite_trace_last_output tr0)) destruct IHHtrX1 as [eqv_state_is [Hstate_start_project [eqv_state_s
      [Hstate_final_project [eqv_state_tr [Hstate_project [Hstate_trace _]]]]]]].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
seed: message → Prop
constraintX: composite_label IM
→ composite_state IM * option message
  → Prop
CX:= preloaded_vlsm (composite_vlsm IM constraintX)
  seed: VLSM message
constraintE: composite_label (equivocator_IM IM)
→ composite_state (equivocator_IM IM) *
  option message → Prop
CE:= preloaded_vlsm
  (composite_vlsm (equivocator_IM IM) constraintE)
  seed: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hsubsumption: zero_descriptor_constraint_lifting_prop
Hreplayable: replayable_message_prop
is, s: state CX
tr: list transition_item
HtrX1: finite_valid_trace_init_to CX is s tr
iom: option message
iom_si, iom_s: state CX
iom_tr: list transition_item
Heqiom: empty_initial_message_or_final_output CX
  iom_tr iom
HtrX2: finite_valid_trace_init_to CX iom_si iom_s
  iom_tr
sf: state CX
oom: option message
l: label CX
Ht: input_valid_transition CX l (s, iom) (sf, oom)
HinclE: VLSM_incl CE
  (preloaded_with_all_messages_vlsm FreeE)
eqv_state_is: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_start_project: equivocators_total_state_project
  IM eqv_state_is = is
eqv_state_s: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_state_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hstate_project: equivocators_total_trace_project IM
  eqv_state_tr = tr
Hstate_trace: finite_valid_trace_init_to CE
  eqv_state_is eqv_state_s eqv_state_tr
IHHtrX2: ∃ is : state
         (free_composite_vlsm
            (equivocator_IM IM)),
  equivocators_total_state_project IM is =
  iom_si
  ∧ (∃ s : state
             (free_composite_vlsm
                (equivocator_IM IM)),
       equivocators_total_state_project IM s =
       iom_s
       ∧ (∃ tr : list
                   (composite_transition_item
                      (equivocator_IM IM)),
            equivocators_total_trace_project
              IM tr = iom_tr
            ∧ finite_valid_trace_init_to CE
                is s tr
              ∧ finite_trace_last_output
                  iom_tr =
                finite_trace_last_output tr))
∃ is0 : state
          (free_composite_vlsm (equivocator_IM IM)),
  equivocators_total_state_project IM is0 = is
  ∧ (∃ s : state
             (free_composite_vlsm (equivocator_IM IM)),
       equivocators_total_state_project IM s = sf
       ∧ (∃ tr0 : list
                    (composite_transition_item
                       (equivocator_IM IM)),
            equivocators_total_trace_project IM tr0 =
            tr ++
            [{|
               l := l;
               input := iom;
               destination := sf;
               output := oom
             |}]
            ∧ finite_valid_trace_init_to CE is0 s tr0
              ∧ finite_trace_last_output
                  (tr ++
                   [{|
                      l := l;
                      input := iom;
                      destination := sf;
                      output := oom
                    |}]) =
                finite_trace_last_output tr0))
    destruct IHHtrX2 as [eqv_msg_is [Hmsg_start_project [eqv_msg_s [_
      [eqv_msg_tr [Hmsg_project [Hmsg_trace Hfinal_msg]]]]]]].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
seed: message → Prop
constraintX: composite_label IM
→ composite_state IM * option message
  → Prop
CX:= preloaded_vlsm (composite_vlsm IM constraintX)
  seed: VLSM message
constraintE: composite_label (equivocator_IM IM)
→ composite_state (equivocator_IM IM) *
  option message → Prop
CE:= preloaded_vlsm
  (composite_vlsm (equivocator_IM IM) constraintE)
  seed: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hsubsumption: zero_descriptor_constraint_lifting_prop
Hreplayable: replayable_message_prop
is, s: state CX
tr: list transition_item
HtrX1: finite_valid_trace_init_to CX is s tr
iom: option message
iom_si, iom_s: state CX
iom_tr: list transition_item
Heqiom: empty_initial_message_or_final_output CX
  iom_tr iom
HtrX2: finite_valid_trace_init_to CX iom_si iom_s
  iom_tr
sf: state CX
oom: option message
l: label CX
Ht: input_valid_transition CX l (s, iom) (sf, oom)
HinclE: VLSM_incl CE
  (preloaded_with_all_messages_vlsm FreeE)
eqv_state_is: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_start_project: equivocators_total_state_project
  IM eqv_state_is = is
eqv_state_s: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_state_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hstate_project: equivocators_total_trace_project IM
  eqv_state_tr = tr
Hstate_trace: finite_valid_trace_init_to CE
  eqv_state_is eqv_state_s eqv_state_tr
eqv_msg_is: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hmsg_start_project: equivocators_total_state_project
  IM eqv_msg_is = iom_si
eqv_msg_s: state
  (free_composite_vlsm (equivocator_IM IM))
eqv_msg_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hmsg_project: equivocators_total_trace_project IM
  eqv_msg_tr = iom_tr
Hmsg_trace: finite_valid_trace_init_to CE eqv_msg_is
  eqv_msg_s eqv_msg_tr
Hfinal_msg: finite_trace_last_output iom_tr =
finite_trace_last_output eqv_msg_tr
∃ is0 : state
          (free_composite_vlsm (equivocator_IM IM)),
  equivocators_total_state_project IM is0 = is
  ∧ (∃ s : state
             (free_composite_vlsm (equivocator_IM IM)),
       equivocators_total_state_project IM s = sf
       ∧ (∃ tr0 : list
                    (composite_transition_item
                       (equivocator_IM IM)),
            equivocators_total_trace_project IM tr0 =
            tr ++
            [{|
               l := l;
               input := iom;
               destination := sf;
               output := oom
             |}]
            ∧ finite_valid_trace_init_to CE is0 s tr0
              ∧ finite_trace_last_output
                  (tr ++
                   [{|
                      l := l;
                      input := iom;
                      destination := sf;
                      output := oom
                    |}]) =
                finite_trace_last_output tr0))
    exists eqv_state_is.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
seed: message → Prop
constraintX: composite_label IM
→ composite_state IM * option message
  → Prop
CX:= preloaded_vlsm (composite_vlsm IM constraintX)
  seed: VLSM message
constraintE: composite_label (equivocator_IM IM)
→ composite_state (equivocator_IM IM) *
  option message → Prop
CE:= preloaded_vlsm
  (composite_vlsm (equivocator_IM IM) constraintE)
  seed: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hsubsumption: zero_descriptor_constraint_lifting_prop
Hreplayable: replayable_message_prop
is, s: state CX
tr: list transition_item
HtrX1: finite_valid_trace_init_to CX is s tr
iom: option message
iom_si, iom_s: state CX
iom_tr: list transition_item
Heqiom: empty_initial_message_or_final_output CX
  iom_tr iom
HtrX2: finite_valid_trace_init_to CX iom_si iom_s
  iom_tr
sf: state CX
oom: option message
l: label CX
Ht: input_valid_transition CX l (s, iom) (sf, oom)
HinclE: VLSM_incl CE
  (preloaded_with_all_messages_vlsm FreeE)
eqv_state_is: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_start_project: equivocators_total_state_project
  IM eqv_state_is = is
eqv_state_s: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_state_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hstate_project: equivocators_total_trace_project IM
  eqv_state_tr = tr
Hstate_trace: finite_valid_trace_init_to CE
  eqv_state_is eqv_state_s eqv_state_tr
eqv_msg_is: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hmsg_start_project: equivocators_total_state_project
  IM eqv_msg_is = iom_si
eqv_msg_s: state
  (free_composite_vlsm (equivocator_IM IM))
eqv_msg_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hmsg_project: equivocators_total_trace_project IM
  eqv_msg_tr = iom_tr
Hmsg_trace: finite_valid_trace_init_to CE eqv_msg_is
  eqv_msg_s eqv_msg_tr
Hfinal_msg: finite_trace_last_output iom_tr =
finite_trace_last_output eqv_msg_tr
equivocators_total_state_project IM eqv_state_is = is
∧ (∃ s : state
           (free_composite_vlsm (equivocator_IM IM)),
     equivocators_total_state_project IM s = sf
     ∧ (∃ tr0 : list
                  (composite_transition_item
                     (equivocator_IM IM)),
          equivocators_total_trace_project IM tr0 =
          tr ++
          [{|
             l := l;
             input := iom;
             destination := sf;
             output := oom
           |}]
          ∧ finite_valid_trace_init_to CE eqv_state_is
              s tr0
            ∧ finite_trace_last_output
                (tr ++
                 [{|
                    l := l;
                    input := iom;
                    destination := sf;
                    output := oom
                  |}]) = finite_trace_last_output tr0)) split; [done |].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
seed: message → Prop
constraintX: composite_label IM
→ composite_state IM * option message
  → Prop
CX:= preloaded_vlsm (composite_vlsm IM constraintX)
  seed: VLSM message
constraintE: composite_label (equivocator_IM IM)
→ composite_state (equivocator_IM IM) *
  option message → Prop
CE:= preloaded_vlsm
  (composite_vlsm (equivocator_IM IM) constraintE)
  seed: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hsubsumption: zero_descriptor_constraint_lifting_prop
Hreplayable: replayable_message_prop
is, s: state CX
tr: list transition_item
HtrX1: finite_valid_trace_init_to CX is s tr
iom: option message
iom_si, iom_s: state CX
iom_tr: list transition_item
Heqiom: empty_initial_message_or_final_output CX
  iom_tr iom
HtrX2: finite_valid_trace_init_to CX iom_si iom_s
  iom_tr
sf: state CX
oom: option message
l: label CX
Ht: input_valid_transition CX l (s, iom) (sf, oom)
HinclE: VLSM_incl CE
  (preloaded_with_all_messages_vlsm FreeE)
eqv_state_is: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_start_project: equivocators_total_state_project
  IM eqv_state_is = is
eqv_state_s: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_state_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hstate_project: equivocators_total_trace_project IM
  eqv_state_tr = tr
Hstate_trace: finite_valid_trace_init_to CE
  eqv_state_is eqv_state_s eqv_state_tr
eqv_msg_is: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hmsg_start_project: equivocators_total_state_project
  IM eqv_msg_is = iom_si
eqv_msg_s: state
  (free_composite_vlsm (equivocator_IM IM))
eqv_msg_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hmsg_project: equivocators_total_trace_project IM
  eqv_msg_tr = iom_tr
Hmsg_trace: finite_valid_trace_init_to CE eqv_msg_is
  eqv_msg_s eqv_msg_tr
Hfinal_msg: finite_trace_last_output iom_tr =
finite_trace_last_output eqv_msg_tr
∃ s : state (free_composite_vlsm (equivocator_IM IM)),
  equivocators_total_state_project IM s = sf
  ∧ (∃ tr0 : list
               (composite_transition_item
                  (equivocator_IM IM)),
       equivocators_total_trace_project IM tr0 =
       tr ++
       [{|
          l := l;
          input := iom;
          destination := sf;
          output := oom
        |}]
       ∧ finite_valid_trace_init_to CE eqv_state_is s
           tr0
         ∧ finite_trace_last_output
             (tr ++
              [{|
                 l := l;
                 input := iom;
                 destination := sf;
                 output := oom
               |}]) = finite_trace_last_output tr0)
    apply valid_trace_last_pstate in Hstate_trace as Hstate_valid.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
seed: message → Prop
constraintX: composite_label IM
→ composite_state IM * option message
  → Prop
CX:= preloaded_vlsm (composite_vlsm IM constraintX)
  seed: VLSM message
constraintE: composite_label (equivocator_IM IM)
→ composite_state (equivocator_IM IM) *
  option message → Prop
CE:= preloaded_vlsm
  (composite_vlsm (equivocator_IM IM) constraintE)
  seed: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hsubsumption: zero_descriptor_constraint_lifting_prop
Hreplayable: replayable_message_prop
is, s: state CX
tr: list transition_item
HtrX1: finite_valid_trace_init_to CX is s tr
iom: option message
iom_si, iom_s: state CX
iom_tr: list transition_item
Heqiom: empty_initial_message_or_final_output CX
  iom_tr iom
HtrX2: finite_valid_trace_init_to CX iom_si iom_s
  iom_tr
sf: state CX
oom: option message
l: label CX
Ht: input_valid_transition CX l (s, iom) (sf, oom)
HinclE: VLSM_incl CE
  (preloaded_with_all_messages_vlsm FreeE)
eqv_state_is: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_start_project: equivocators_total_state_project
  IM eqv_state_is = is
eqv_state_s: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_state_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hstate_project: equivocators_total_trace_project IM
  eqv_state_tr = tr
Hstate_trace: finite_valid_trace_init_to CE
  eqv_state_is eqv_state_s eqv_state_tr
eqv_msg_is: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hmsg_start_project: equivocators_total_state_project
  IM eqv_msg_is = iom_si
eqv_msg_s: state
  (free_composite_vlsm (equivocator_IM IM))
eqv_msg_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hmsg_project: equivocators_total_trace_project IM
  eqv_msg_tr = iom_tr
Hmsg_trace: finite_valid_trace_init_to CE eqv_msg_is
  eqv_msg_s eqv_msg_tr
Hfinal_msg: finite_trace_last_output iom_tr =
finite_trace_last_output eqv_msg_tr
Hstate_valid: valid_state_prop CE eqv_state_s
∃ s : state (free_composite_vlsm (equivocator_IM IM)),
  equivocators_total_state_project IM s = sf
  ∧ (∃ tr0 : list
               (composite_transition_item
                  (equivocator_IM IM)),
       equivocators_total_trace_project IM tr0 =
       tr ++
       [{|
          l := l;
          input := iom;
          destination := sf;
          output := oom
        |}]
       ∧ finite_valid_trace_init_to CE eqv_state_is s
           tr0
         ∧ finite_trace_last_output
             (tr ++
              [{|
                 l := l;
                 input := iom;
                 destination := sf;
                 output := oom
               |}]) = finite_trace_last_output tr0)
    destruct Ht as [[Hs [Hiom [Hv Hc]]] Ht].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
seed: message → Prop
constraintX: composite_label IM
→ composite_state IM * option message
  → Prop
CX:= preloaded_vlsm (composite_vlsm IM constraintX)
  seed: VLSM message
constraintE: composite_label (equivocator_IM IM)
→ composite_state (equivocator_IM IM) *
  option message → Prop
CE:= preloaded_vlsm
  (composite_vlsm (equivocator_IM IM) constraintE)
  seed: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hsubsumption: zero_descriptor_constraint_lifting_prop
Hreplayable: replayable_message_prop
is, s: state CX
tr: list transition_item
HtrX1: finite_valid_trace_init_to CX is s tr
iom: option message
iom_si, iom_s: state CX
iom_tr: list transition_item
Heqiom: empty_initial_message_or_final_output CX
  iom_tr iom
HtrX2: finite_valid_trace_init_to CX iom_si iom_s
  iom_tr
sf: state CX
oom: option message
l: label CX
Hs: valid_state_prop CX s
Hiom: option_valid_message_prop CX iom
Hv: valid l (s, iom)
Hc: constraintX l (s, iom)
Ht: transition l (s, iom) = (sf, oom)
HinclE: VLSM_incl CE
  (preloaded_with_all_messages_vlsm FreeE)
eqv_state_is: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_start_project: equivocators_total_state_project
  IM eqv_state_is = is
eqv_state_s: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_state_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hstate_project: equivocators_total_trace_project IM
  eqv_state_tr = tr
Hstate_trace: finite_valid_trace_init_to CE
  eqv_state_is eqv_state_s eqv_state_tr
eqv_msg_is: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hmsg_start_project: equivocators_total_state_project
  IM eqv_msg_is = iom_si
eqv_msg_s: state
  (free_composite_vlsm (equivocator_IM IM))
eqv_msg_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hmsg_project: equivocators_total_trace_project IM
  eqv_msg_tr = iom_tr
Hmsg_trace: finite_valid_trace_init_to CE eqv_msg_is
  eqv_msg_s eqv_msg_tr
Hfinal_msg: finite_trace_last_output iom_tr =
finite_trace_last_output eqv_msg_tr
Hstate_valid: valid_state_prop CE eqv_state_s
∃ s : state (free_composite_vlsm (equivocator_IM IM)),
  equivocators_total_state_project IM s = sf
  ∧ (∃ tr0 : list
               (composite_transition_item
                  (equivocator_IM IM)),
       equivocators_total_trace_project IM tr0 =
       tr ++
       [{|
          l := l;
          input := iom;
          destination := sf;
          output := oom
        |}]
       ∧ finite_valid_trace_init_to CE eqv_state_is s
           tr0
         ∧ finite_trace_last_output
             (tr ++
              [{|
                 l := l;
                 input := iom;
                 destination := sf;
                 output := oom
               |}]) = finite_trace_last_output tr0)
    specialize
      (Hreplayable _ _ _ HtrX1 _ Hstate_valid Hstate_final_project _ _ _ Hmsg_trace iom)
      as Hreplay.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
seed: message → Prop
constraintX: composite_label IM
→ composite_state IM * option message
  → Prop
CX:= preloaded_vlsm (composite_vlsm IM constraintX)
  seed: VLSM message
constraintE: composite_label (equivocator_IM IM)
→ composite_state (equivocator_IM IM) *
  option message → Prop
CE:= preloaded_vlsm
  (composite_vlsm (equivocator_IM IM) constraintE)
  seed: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hsubsumption: zero_descriptor_constraint_lifting_prop
Hreplayable: replayable_message_prop
is, s: state CX
tr: list transition_item
HtrX1: finite_valid_trace_init_to CX is s tr
iom: option message
iom_si, iom_s: state CX
iom_tr: list transition_item
Heqiom: empty_initial_message_or_final_output CX
  iom_tr iom
HtrX2: finite_valid_trace_init_to CX iom_si iom_s
  iom_tr
sf: state CX
oom: option message
l: label CX
Hs: valid_state_prop CX s
Hiom: option_valid_message_prop CX iom
Hv: valid l (s, iom)
Hc: constraintX l (s, iom)
Ht: transition l (s, iom) = (sf, oom)
HinclE: VLSM_incl CE
  (preloaded_with_all_messages_vlsm FreeE)
eqv_state_is: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_start_project: equivocators_total_state_project
  IM eqv_state_is = is
eqv_state_s: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_state_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hstate_project: equivocators_total_trace_project IM
  eqv_state_tr = tr
Hstate_trace: finite_valid_trace_init_to CE
  eqv_state_is eqv_state_s eqv_state_tr
eqv_msg_is: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hmsg_start_project: equivocators_total_state_project
  IM eqv_msg_is = iom_si
eqv_msg_s: state
  (free_composite_vlsm (equivocator_IM IM))
eqv_msg_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hmsg_project: equivocators_total_trace_project IM
  eqv_msg_tr = iom_tr
Hmsg_trace: finite_valid_trace_init_to CE eqv_msg_is
  eqv_msg_s eqv_msg_tr
Hfinal_msg: finite_trace_last_output iom_tr =
finite_trace_last_output eqv_msg_tr
Hstate_valid: valid_state_prop CE eqv_state_s
Hreplay: last_in_trace_except_from
  initial_message_prop eqv_msg_tr iom
→ ∀ l : composite_label IM,
    constraintX l (s, iom)
    → ∃ (eqv_msg_tr : list transition_item) 
        (lst_msg_tr : state CE),
        finite_valid_trace_from_to CE
          eqv_state_s lst_msg_tr eqv_msg_tr
        ∧ equivocators_total_trace_project
            IM eqv_msg_tr = []
          ∧ equivocators_total_state_project
              IM lst_msg_tr = s
            ∧ sent_except_from
                (equivocator_IM IM)
                initial_message_prop
                lst_msg_tr iom
∃ s : state (free_composite_vlsm (equivocator_IM IM)),
  equivocators_total_state_project IM s = sf
  ∧ (∃ tr0 : list
               (composite_transition_item
                  (equivocator_IM IM)),
       equivocators_total_trace_project IM tr0 =
       tr ++
       [{|
          l := l;
          input := iom;
          destination := sf;
          output := oom
        |}]
       ∧ finite_valid_trace_init_to CE eqv_state_is s
           tr0
         ∧ finite_trace_last_output
             (tr ++
              [{|
                 l := l;
                 input := iom;
                 destination := sf;
                 output := oom
               |}]) = finite_trace_last_output tr0)
    spec Hreplay.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
seed: message → Prop
constraintX: composite_label IM
→ composite_state IM * option message
  → Prop
CX:= preloaded_vlsm (composite_vlsm IM constraintX)
  seed: VLSM message
constraintE: composite_label (equivocator_IM IM)
→ composite_state (equivocator_IM IM) *
  option message → Prop
CE:= preloaded_vlsm
  (composite_vlsm (equivocator_IM IM) constraintE)
  seed: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hsubsumption: zero_descriptor_constraint_lifting_prop
Hreplayable: replayable_message_prop
is, s: state CX
tr: list transition_item
HtrX1: finite_valid_trace_init_to CX is s tr
iom: option message
iom_si, iom_s: state CX
iom_tr: list transition_item
Heqiom: empty_initial_message_or_final_output CX
  iom_tr iom
HtrX2: finite_valid_trace_init_to CX iom_si iom_s
  iom_tr
sf: state CX
oom: option message
l: label CX
Hs: valid_state_prop CX s
Hiom: option_valid_message_prop CX iom
Hv: valid l (s, iom)
Hc: constraintX l (s, iom)
Ht: transition l (s, iom) = (sf, oom)
HinclE: VLSM_incl CE
  (preloaded_with_all_messages_vlsm FreeE)
eqv_state_is: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_start_project: equivocators_total_state_project
  IM eqv_state_is = is
eqv_state_s: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_state_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hstate_project: equivocators_total_trace_project IM
  eqv_state_tr = tr
Hstate_trace: finite_valid_trace_init_to CE
  eqv_state_is eqv_state_s eqv_state_tr
eqv_msg_is: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hmsg_start_project: equivocators_total_state_project
  IM eqv_msg_is = iom_si
eqv_msg_s: state
  (free_composite_vlsm (equivocator_IM IM))
eqv_msg_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hmsg_project: equivocators_total_trace_project IM
  eqv_msg_tr = iom_tr
Hmsg_trace: finite_valid_trace_init_to CE eqv_msg_is
  eqv_msg_s eqv_msg_tr
Hfinal_msg: finite_trace_last_output iom_tr =
finite_trace_last_output eqv_msg_tr
Hstate_valid: valid_state_prop CE eqv_state_s
Hreplay: last_in_trace_except_from
  initial_message_prop eqv_msg_tr iom
→ ∀ l : composite_label IM,
    constraintX l (s, iom)
    → ∃ (eqv_msg_tr : list transition_item) 
        (lst_msg_tr : state CE),
        finite_valid_trace_from_to CE
          eqv_state_s lst_msg_tr eqv_msg_tr
        ∧ equivocators_total_trace_project
            IM eqv_msg_tr = []
          ∧ equivocators_total_state_project
              IM lst_msg_tr = s
            ∧ sent_except_from
                (equivocator_IM IM)
                initial_message_prop
                lst_msg_tr iom
last_in_trace_except_from initial_message_prop
  eqv_msg_tr iom
message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
seed: message → Prop
constraintX: composite_label IM
→ composite_state IM * option message
  → Prop
CX:= preloaded_vlsm (composite_vlsm IM constraintX)
  seed: VLSM message
constraintE: composite_label (equivocator_IM IM)
→ composite_state (equivocator_IM IM) *
  option message → Prop
CE:= preloaded_vlsm
  (composite_vlsm (equivocator_IM IM) constraintE)
  seed: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hsubsumption: zero_descriptor_constraint_lifting_prop
Hreplayable: replayable_message_prop
is, s: state CX
tr: list transition_item
HtrX1: finite_valid_trace_init_to CX is s tr
iom: option message
iom_si, iom_s: state CX
iom_tr: list transition_item
Heqiom: empty_initial_message_or_final_output CX
  iom_tr iom
HtrX2: finite_valid_trace_init_to CX iom_si iom_s
  iom_tr
sf: state CX
oom: option message
l: label CX
Hs: valid_state_prop CX s
Hiom: option_valid_message_prop CX iom
Hv: valid l (s, iom)
Hc: constraintX l (s, iom)
Ht: transition l (s, iom) = (sf, oom)
HinclE: VLSM_incl CE
  (preloaded_with_all_messages_vlsm FreeE)
eqv_state_is: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_start_project: equivocators_total_state_project
  IM eqv_state_is = is
eqv_state_s: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_state_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hstate_project: equivocators_total_trace_project IM
  eqv_state_tr = tr
Hstate_trace: finite_valid_trace_init_to CE
  eqv_state_is eqv_state_s eqv_state_tr
eqv_msg_is: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hmsg_start_project: equivocators_total_state_project
  IM eqv_msg_is = iom_si
eqv_msg_s: state
  (free_composite_vlsm (equivocator_IM IM))
eqv_msg_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hmsg_project: equivocators_total_trace_project IM
  eqv_msg_tr = iom_tr
Hmsg_trace: finite_valid_trace_init_to CE eqv_msg_is
  eqv_msg_s eqv_msg_tr
Hfinal_msg: finite_trace_last_output iom_tr =
finite_trace_last_output eqv_msg_tr
Hstate_valid: valid_state_prop CE eqv_state_s
Hreplay: ∀ l : composite_label IM,
  constraintX l (s, iom)
  → ∃ (eqv_msg_tr : list transition_item) 
      (lst_msg_tr : state CE),
      finite_valid_trace_from_to CE
        eqv_state_s lst_msg_tr eqv_msg_tr
      ∧ equivocators_total_trace_project IM
          eqv_msg_tr = []
        ∧ equivocators_total_state_project
            IM lst_msg_tr = s
          ∧ sent_except_from
              (equivocator_IM IM)
              initial_message_prop
              lst_msg_tr iom
∃ s : state (free_composite_vlsm (equivocator_IM IM)),
  equivocators_total_state_project IM s = sf
  ∧ (∃ tr0 : list
               (composite_transition_item
                  (equivocator_IM IM)),
       equivocators_total_trace_project IM tr0 =
       tr ++
       [{|
          l := l;
          input := iom;
          destination := sf;
          output := oom
        |}]
       ∧ finite_valid_trace_init_to CE eqv_state_is s
           tr0
         ∧ finite_trace_last_output
             (tr ++
              [{|
                 l := l;
                 input := iom;
                 destination := sf;
                 output := oom
               |}]) = finite_trace_last_output tr0)
    {message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
seed: message → Prop
constraintX: composite_label IM
→ composite_state IM * option message
  → Prop
CX:= preloaded_vlsm (composite_vlsm IM constraintX)
  seed: VLSM message
constraintE: composite_label (equivocator_IM IM)
→ composite_state (equivocator_IM IM) *
  option message → Prop
CE:= preloaded_vlsm
  (composite_vlsm (equivocator_IM IM) constraintE)
  seed: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hsubsumption: zero_descriptor_constraint_lifting_prop
Hreplayable: replayable_message_prop
is, s: state CX
tr: list transition_item
HtrX1: finite_valid_trace_init_to CX is s tr
iom: option message
iom_si, iom_s: state CX
iom_tr: list transition_item
Heqiom: empty_initial_message_or_final_output CX
  iom_tr iom
HtrX2: finite_valid_trace_init_to CX iom_si iom_s
  iom_tr
sf: state CX
oom: option message
l: label CX
Hs: valid_state_prop CX s
Hiom: option_valid_message_prop CX iom
Hv: valid l (s, iom)
Hc: constraintX l (s, iom)
Ht: transition l (s, iom) = (sf, oom)
HinclE: VLSM_incl CE
  (preloaded_with_all_messages_vlsm FreeE)
eqv_state_is: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_start_project: equivocators_total_state_project
  IM eqv_state_is = is
eqv_state_s: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_state_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hstate_project: equivocators_total_trace_project IM
  eqv_state_tr = tr
Hstate_trace: finite_valid_trace_init_to CE
  eqv_state_is eqv_state_s eqv_state_tr
eqv_msg_is: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hmsg_start_project: equivocators_total_state_project
  IM eqv_msg_is = iom_si
eqv_msg_s: state
  (free_composite_vlsm (equivocator_IM IM))
eqv_msg_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hmsg_project: equivocators_total_trace_project IM
  eqv_msg_tr = iom_tr
Hmsg_trace: finite_valid_trace_init_to CE eqv_msg_is
  eqv_msg_s eqv_msg_tr
Hfinal_msg: finite_trace_last_output iom_tr =
finite_trace_last_output eqv_msg_tr
Hstate_valid: valid_state_prop CE eqv_state_s
Hreplay: last_in_trace_except_from
  initial_message_prop eqv_msg_tr iom
→ ∀ l : composite_label IM,
    constraintX l (s, iom)
    → ∃ (eqv_msg_tr : list transition_item) 
        (lst_msg_tr : state CE),
        finite_valid_trace_from_to CE
          eqv_state_s lst_msg_tr eqv_msg_tr
        ∧ equivocators_total_trace_project
            IM eqv_msg_tr = []
          ∧ equivocators_total_state_project
              IM lst_msg_tr = s
            ∧ sent_except_from
                (equivocator_IM IM)
                initial_message_prop
                lst_msg_tr iom
last_in_trace_except_from initial_message_prop
  eqv_msg_tr iom clear -Heqiom Hfinal_msg.message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
seed: message → Prop
constraintX: composite_label IM
→ composite_state IM * option message
  → Prop
CX:= preloaded_vlsm (composite_vlsm IM constraintX)
  seed: VLSM message
constraintE: composite_label (equivocator_IM IM)
→ composite_state (equivocator_IM IM) *
  option message → Prop
CE:= preloaded_vlsm
  (composite_vlsm (equivocator_IM IM) constraintE)
  seed: VLSM message
iom: option message
iom_tr: list transition_item
Heqiom: empty_initial_message_or_final_output CX
  iom_tr iom
eqv_msg_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hfinal_msg: finite_trace_last_output iom_tr =
finite_trace_last_output eqv_msg_tr
last_in_trace_except_from initial_message_prop
  eqv_msg_tr iom
      destruct iom as [im |]; [| done].message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
seed: message → Prop
constraintX: composite_label IM
→ composite_state IM * option message
  → Prop
CX:= preloaded_vlsm (composite_vlsm IM constraintX)
  seed: VLSM message
constraintE: composite_label (equivocator_IM IM)
→ composite_state (equivocator_IM IM) *
  option message → Prop
CE:= preloaded_vlsm
  (composite_vlsm (equivocator_IM IM) constraintE)
  seed: VLSM message
im: message
iom_tr: list transition_item
Heqiom: empty_initial_message_or_final_output CX
  iom_tr (Some im)
eqv_msg_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hfinal_msg: finite_trace_last_output iom_tr =
finite_trace_last_output eqv_msg_tr
last_in_trace_except_from initial_message_prop
  eqv_msg_tr (Some im)
      unfold empty_initial_message_or_final_output in Heqiom.message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
seed: message → Prop
constraintX: composite_label IM
→ composite_state IM * option message
  → Prop
CX:= preloaded_vlsm (composite_vlsm IM constraintX)
  seed: VLSM message
constraintE: composite_label (equivocator_IM IM)
→ composite_state (equivocator_IM IM) *
  option message → Prop
CE:= preloaded_vlsm
  (composite_vlsm (equivocator_IM IM) constraintE)
  seed: VLSM message
im: message
iom_tr: list transition_item
Heqiom: match has_last_or_null iom_tr with
| inleft (existT x (x0 ↾ _)) =>
    output x0 = Some im
| inright _ =>
    option_initial_message_prop CX (Some im)
end
eqv_msg_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hfinal_msg: finite_trace_last_output iom_tr =
finite_trace_last_output eqv_msg_tr
last_in_trace_except_from initial_message_prop
  eqv_msg_tr (Some im)
      destruct_list_last iom_tr iom_tr' item Heqiom_tr.message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
seed: message → Prop
constraintX: composite_label IM
→ composite_state IM * option message
  → Prop
CX:= preloaded_vlsm (composite_vlsm IM constraintX)
  seed: VLSM message
constraintE: composite_label (equivocator_IM IM)
→ composite_state (equivocator_IM IM) *
  option message → Prop
CE:= preloaded_vlsm
  (composite_vlsm (equivocator_IM IM) constraintE)
  seed: VLSM message
im: message
iom_tr: list transition_item
Heqiom_tr: iom_tr = []
Heqiom: option_initial_message_prop CX (Some im)
eqv_msg_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hfinal_msg: finite_trace_last_output [] =
finite_trace_last_output eqv_msg_tr
last_in_trace_except_from initial_message_prop
  eqv_msg_tr (Some im)
message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
seed: message → Prop
constraintX: composite_label IM
→ composite_state IM * option message
  → Prop
CX:= preloaded_vlsm (composite_vlsm IM constraintX)
  seed: VLSM message
constraintE: composite_label (equivocator_IM IM)
→ composite_state (equivocator_IM IM) *
  option message → Prop
CE:= preloaded_vlsm
  (composite_vlsm (equivocator_IM IM) constraintE)
  seed: VLSM message
im: message
iom_tr, iom_tr': list transition_item
item: transition_item
Heqiom_tr: iom_tr = iom_tr' ++ [item]
Heqiom: output item = Some im
eqv_msg_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hfinal_msg: finite_trace_last_output
  (iom_tr' ++ [item]) =
finite_trace_last_output eqv_msg_tr
last_in_trace_except_from initial_message_prop
  eqv_msg_tr (Some im)
      -message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
seed: message → Prop
constraintX: composite_label IM
→ composite_state IM * option message
  → Prop
CX:= preloaded_vlsm (composite_vlsm IM constraintX)
  seed: VLSM message
constraintE: composite_label (equivocator_IM IM)
→ composite_state (equivocator_IM IM) *
  option message → Prop
CE:= preloaded_vlsm
  (composite_vlsm (equivocator_IM IM) constraintE)
  seed: VLSM message
im: message
iom_tr: list transition_item
Heqiom_tr: iom_tr = []
Heqiom: option_initial_message_prop CX (Some im)
eqv_msg_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hfinal_msg: finite_trace_last_output [] =
finite_trace_last_output eqv_msg_tr
last_in_trace_except_from initial_message_prop
  eqv_msg_tr (Some im) by right.
      -message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
seed: message → Prop
constraintX: composite_label IM
→ composite_state IM * option message
  → Prop
CX:= preloaded_vlsm (composite_vlsm IM constraintX)
  seed: VLSM message
constraintE: composite_label (equivocator_IM IM)
→ composite_state (equivocator_IM IM) *
  option message → Prop
CE:= preloaded_vlsm
  (composite_vlsm (equivocator_IM IM) constraintE)
  seed: VLSM message
im: message
iom_tr, iom_tr': list transition_item
item: transition_item
Heqiom_tr: iom_tr = iom_tr' ++ [item]
Heqiom: output item = Some im
eqv_msg_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hfinal_msg: finite_trace_last_output
  (iom_tr' ++ [item]) =
finite_trace_last_output eqv_msg_tr
last_in_trace_except_from initial_message_prop
  eqv_msg_tr (Some im) left.message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
seed: message → Prop
constraintX: composite_label IM
→ composite_state IM * option message
  → Prop
CX:= preloaded_vlsm (composite_vlsm IM constraintX)
  seed: VLSM message
constraintE: composite_label (equivocator_IM IM)
→ composite_state (equivocator_IM IM) *
  option message → Prop
CE:= preloaded_vlsm
  (composite_vlsm (equivocator_IM IM) constraintE)
  seed: VLSM message
im: message
iom_tr, iom_tr': list transition_item
item: transition_item
Heqiom_tr: iom_tr = iom_tr' ++ [item]
Heqiom: output item = Some im
eqv_msg_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hfinal_msg: finite_trace_last_output
  (iom_tr' ++ [item]) =
finite_trace_last_output eqv_msg_tr
finite_trace_last_output eqv_msg_tr = Some im simpl in *.message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
seed: message → Prop
constraintX: composite_label IM
→ composite_state IM * option message
  → Prop
CX:= preloaded_vlsm (composite_vlsm IM constraintX)
  seed: VLSM message
constraintE: composite_label (equivocator_IM IM)
→ composite_state (equivocator_IM IM) *
  option message → Prop
CE:= preloaded_vlsm
  (composite_vlsm (equivocator_IM IM) constraintE)
  seed: VLSM message
im: message
iom_tr, iom_tr': list transition_item
item: transition_item
Heqiom_tr: iom_tr = iom_tr' ++ [item]
Heqiom: output item = Some im
eqv_msg_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hfinal_msg: finite_trace_last_output
  (iom_tr' ++ [item]) =
finite_trace_last_output eqv_msg_tr
finite_trace_last_output eqv_msg_tr = Some im rewrite <- Hfinal_msg.message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
seed: message → Prop
constraintX: composite_label IM
→ composite_state IM * option message
  → Prop
CX:= preloaded_vlsm (composite_vlsm IM constraintX)
  seed: VLSM message
constraintE: composite_label (equivocator_IM IM)
→ composite_state (equivocator_IM IM) *
  option message → Prop
CE:= preloaded_vlsm
  (composite_vlsm (equivocator_IM IM) constraintE)
  seed: VLSM message
im: message
iom_tr, iom_tr': list transition_item
item: transition_item
Heqiom_tr: iom_tr = iom_tr' ++ [item]
Heqiom: output item = Some im
eqv_msg_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hfinal_msg: finite_trace_last_output
  (iom_tr' ++ [item]) =
finite_trace_last_output eqv_msg_tr
finite_trace_last_output (iom_tr' ++ [item]) = Some im
        by rewrite finite_trace_last_output_is_last.
    }message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
seed: message → Prop
constraintX: composite_label IM
→ composite_state IM * option message
  → Prop
CX:= preloaded_vlsm (composite_vlsm IM constraintX)
  seed: VLSM message
constraintE: composite_label (equivocator_IM IM)
→ composite_state (equivocator_IM IM) *
  option message → Prop
CE:= preloaded_vlsm
  (composite_vlsm (equivocator_IM IM) constraintE)
  seed: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hsubsumption: zero_descriptor_constraint_lifting_prop
Hreplayable: replayable_message_prop
is, s: state CX
tr: list transition_item
HtrX1: finite_valid_trace_init_to CX is s tr
iom: option message
iom_si, iom_s: state CX
iom_tr: list transition_item
Heqiom: empty_initial_message_or_final_output CX
  iom_tr iom
HtrX2: finite_valid_trace_init_to CX iom_si iom_s
  iom_tr
sf: state CX
oom: option message
l: label CX
Hs: valid_state_prop CX s
Hiom: option_valid_message_prop CX iom
Hv: valid l (s, iom)
Hc: constraintX l (s, iom)
Ht: transition l (s, iom) = (sf, oom)
HinclE: VLSM_incl CE
  (preloaded_with_all_messages_vlsm FreeE)
eqv_state_is: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_start_project: equivocators_total_state_project
  IM eqv_state_is = is
eqv_state_s: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_state_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hstate_project: equivocators_total_trace_project IM
  eqv_state_tr = tr
Hstate_trace: finite_valid_trace_init_to CE
  eqv_state_is eqv_state_s eqv_state_tr
eqv_msg_is: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hmsg_start_project: equivocators_total_state_project
  IM eqv_msg_is = iom_si
eqv_msg_s: state
  (free_composite_vlsm (equivocator_IM IM))
eqv_msg_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hmsg_project: equivocators_total_trace_project IM
  eqv_msg_tr = iom_tr
Hmsg_trace: finite_valid_trace_init_to CE eqv_msg_is
  eqv_msg_s eqv_msg_tr
Hfinal_msg: finite_trace_last_output iom_tr =
finite_trace_last_output eqv_msg_tr
Hstate_valid: valid_state_prop CE eqv_state_s
Hreplay: ∀ l : composite_label IM,
  constraintX l (s, iom)
  → ∃ (eqv_msg_tr : list transition_item) 
      (lst_msg_tr : state CE),
      finite_valid_trace_from_to CE
        eqv_state_s lst_msg_tr eqv_msg_tr
      ∧ equivocators_total_trace_project IM
          eqv_msg_tr = []
        ∧ equivocators_total_state_project
            IM lst_msg_tr = s
          ∧ sent_except_from
              (equivocator_IM IM)
              initial_message_prop
              lst_msg_tr iom
∃ s : state (free_composite_vlsm (equivocator_IM IM)),
  equivocators_total_state_project IM s = sf
  ∧ (∃ tr0 : list
               (composite_transition_item
                  (equivocator_IM IM)),
       equivocators_total_trace_project IM tr0 =
       tr ++
       [{|
          l := l;
          input := iom;
          destination := sf;
          output := oom
        |}]
       ∧ finite_valid_trace_init_to CE eqv_state_is s
           tr0
         ∧ finite_trace_last_output
             (tr ++
              [{|
                 l := l;
                 input := iom;
                 destination := sf;
                 output := oom
               |}]) = finite_trace_last_output tr0)
    specialize (Hreplay _ Hc)
      as [emsg_tr [es [Hmsg_trace_full_replay [Hemsg_tr_pr [Hes_pr Hbs_iom]]]]].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
seed: message → Prop
constraintX: composite_label IM
→ composite_state IM * option message
  → Prop
CX:= preloaded_vlsm (composite_vlsm IM constraintX)
  seed: VLSM message
constraintE: composite_label (equivocator_IM IM)
→ composite_state (equivocator_IM IM) *
  option message → Prop
CE:= preloaded_vlsm
  (composite_vlsm (equivocator_IM IM) constraintE)
  seed: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hsubsumption: zero_descriptor_constraint_lifting_prop
Hreplayable: replayable_message_prop
is, s: state CX
tr: list transition_item
HtrX1: finite_valid_trace_init_to CX is s tr
iom: option message
iom_si, iom_s: state CX
iom_tr: list transition_item
Heqiom: empty_initial_message_or_final_output CX
  iom_tr iom
HtrX2: finite_valid_trace_init_to CX iom_si iom_s
  iom_tr
sf: state CX
oom: option message
l: label CX
Hs: valid_state_prop CX s
Hiom: option_valid_message_prop CX iom
Hv: valid l (s, iom)
Hc: constraintX l (s, iom)
Ht: transition l (s, iom) = (sf, oom)
HinclE: VLSM_incl CE
  (preloaded_with_all_messages_vlsm FreeE)
eqv_state_is: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_start_project: equivocators_total_state_project
  IM eqv_state_is = is
eqv_state_s: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_state_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hstate_project: equivocators_total_trace_project IM
  eqv_state_tr = tr
Hstate_trace: finite_valid_trace_init_to CE
  eqv_state_is eqv_state_s eqv_state_tr
eqv_msg_is: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hmsg_start_project: equivocators_total_state_project
  IM eqv_msg_is = iom_si
eqv_msg_s: state
  (free_composite_vlsm (equivocator_IM IM))
eqv_msg_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hmsg_project: equivocators_total_trace_project IM
  eqv_msg_tr = iom_tr
Hmsg_trace: finite_valid_trace_init_to CE eqv_msg_is
  eqv_msg_s eqv_msg_tr
Hfinal_msg: finite_trace_last_output iom_tr =
finite_trace_last_output eqv_msg_tr
Hstate_valid: valid_state_prop CE eqv_state_s
emsg_tr: list transition_item
es: state CE
Hmsg_trace_full_replay: finite_valid_trace_from_to CE
  eqv_state_s es emsg_tr
Hemsg_tr_pr: equivocators_total_trace_project IM
  emsg_tr = []
Hes_pr: equivocators_total_state_project IM es = s
Hbs_iom: sent_except_from (equivocator_IM IM)
  initial_message_prop es iom
∃ s : state (free_composite_vlsm (equivocator_IM IM)),
  equivocators_total_state_project IM s = sf
  ∧ (∃ tr0 : list
               (composite_transition_item
                  (equivocator_IM IM)),
       equivocators_total_trace_project IM tr0 =
       tr ++
       [{|
          l := l;
          input := iom;
          destination := sf;
          output := oom
        |}]
       ∧ finite_valid_trace_init_to CE eqv_state_is s
           tr0
         ∧ finite_trace_last_output
             (tr ++
              [{|
                 l := l;
                 input := iom;
                 destination := sf;
                 output := oom
               |}]) = finite_trace_last_output tr0)
    intros .message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
seed: message → Prop
constraintX: composite_label IM
→ composite_state IM * option message
  → Prop
CX:= preloaded_vlsm (composite_vlsm IM constraintX)
  seed: VLSM message
constraintE: composite_label (equivocator_IM IM)
→ composite_state (equivocator_IM IM) *
  option message → Prop
CE:= preloaded_vlsm
  (composite_vlsm (equivocator_IM IM) constraintE)
  seed: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hsubsumption: zero_descriptor_constraint_lifting_prop
Hreplayable: replayable_message_prop
is, s: state CX
tr: list transition_item
HtrX1: finite_valid_trace_init_to CX is s tr
iom: option message
iom_si, iom_s: state CX
iom_tr: list transition_item
Heqiom: empty_initial_message_or_final_output CX
  iom_tr iom
HtrX2: finite_valid_trace_init_to CX iom_si iom_s
  iom_tr
sf: state CX
oom: option message
l: label CX
Hs: valid_state_prop CX s
Hiom: option_valid_message_prop CX iom
Hv: valid l (s, iom)
Hc: constraintX l (s, iom)
Ht: transition l (s, iom) = (sf, oom)
HinclE: VLSM_incl CE
  (preloaded_with_all_messages_vlsm FreeE)
eqv_state_is: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_start_project: equivocators_total_state_project
  IM eqv_state_is = is
eqv_state_s: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_state_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hstate_project: equivocators_total_trace_project IM
  eqv_state_tr = tr
Hstate_trace: finite_valid_trace_init_to CE
  eqv_state_is eqv_state_s eqv_state_tr
eqv_msg_is: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hmsg_start_project: equivocators_total_state_project
  IM eqv_msg_is = iom_si
eqv_msg_s: state
  (free_composite_vlsm (equivocator_IM IM))
eqv_msg_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hmsg_project: equivocators_total_trace_project IM
  eqv_msg_tr = iom_tr
Hmsg_trace: finite_valid_trace_init_to CE eqv_msg_is
  eqv_msg_s eqv_msg_tr
Hfinal_msg: finite_trace_last_output iom_tr =
finite_trace_last_output eqv_msg_tr
Hstate_valid: valid_state_prop CE eqv_state_s
emsg_tr: list transition_item
es: state CE
Hmsg_trace_full_replay: finite_valid_trace_from_to CE
  eqv_state_s es emsg_tr
Hemsg_tr_pr: equivocators_total_trace_project IM
  emsg_tr = []
Hes_pr: equivocators_total_state_project IM es = s
Hbs_iom: sent_except_from (equivocator_IM IM)
  initial_message_prop es iom
∃ s : state (free_composite_vlsm (equivocator_IM IM)),
  equivocators_total_state_project IM s = sf
  ∧ (∃ tr0 : list
               (composite_transition_item
                  (equivocator_IM IM)),
       equivocators_total_trace_project IM tr0 =
       tr ++
       [{|
          l := l;
          input := iom;
          destination := sf;
          output := oom
        |}]
       ∧ finite_valid_trace_init_to CE eqv_state_is s
           tr0
         ∧ finite_trace_last_output
             (tr ++
              [{|
                 l := l;
                 input := iom;
                 destination := sf;
                 output := oom
               |}]) = finite_trace_last_output tr0)
    specialize
      (finite_valid_trace_from_to_app CE  _ _ _ _ _ (proj1 Hstate_trace) Hmsg_trace_full_replay)
      as Happ.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
seed: message → Prop
constraintX: composite_label IM
→ composite_state IM * option message
  → Prop
CX:= preloaded_vlsm (composite_vlsm IM constraintX)
  seed: VLSM message
constraintE: composite_label (equivocator_IM IM)
→ composite_state (equivocator_IM IM) *
  option message → Prop
CE:= preloaded_vlsm
  (composite_vlsm (equivocator_IM IM) constraintE)
  seed: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hsubsumption: zero_descriptor_constraint_lifting_prop
Hreplayable: replayable_message_prop
is, s: state CX
tr: list transition_item
HtrX1: finite_valid_trace_init_to CX is s tr
iom: option message
iom_si, iom_s: state CX
iom_tr: list transition_item
Heqiom: empty_initial_message_or_final_output CX
  iom_tr iom
HtrX2: finite_valid_trace_init_to CX iom_si iom_s
  iom_tr
sf: state CX
oom: option message
l: label CX
Hs: valid_state_prop CX s
Hiom: option_valid_message_prop CX iom
Hv: valid l (s, iom)
Hc: constraintX l (s, iom)
Ht: transition l (s, iom) = (sf, oom)
HinclE: VLSM_incl CE
  (preloaded_with_all_messages_vlsm FreeE)
eqv_state_is: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_start_project: equivocators_total_state_project
  IM eqv_state_is = is
eqv_state_s: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_state_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hstate_project: equivocators_total_trace_project IM
  eqv_state_tr = tr
Hstate_trace: finite_valid_trace_init_to CE
  eqv_state_is eqv_state_s eqv_state_tr
eqv_msg_is: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hmsg_start_project: equivocators_total_state_project
  IM eqv_msg_is = iom_si
eqv_msg_s: state
  (free_composite_vlsm (equivocator_IM IM))
eqv_msg_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hmsg_project: equivocators_total_trace_project IM
  eqv_msg_tr = iom_tr
Hmsg_trace: finite_valid_trace_init_to CE eqv_msg_is
  eqv_msg_s eqv_msg_tr
Hfinal_msg: finite_trace_last_output iom_tr =
finite_trace_last_output eqv_msg_tr
Hstate_valid: valid_state_prop CE eqv_state_s
emsg_tr: list transition_item
es: state CE
Hmsg_trace_full_replay: finite_valid_trace_from_to CE
  eqv_state_s es emsg_tr
Hemsg_tr_pr: equivocators_total_trace_project IM
  emsg_tr = []
Hes_pr: equivocators_total_state_project IM es = s
Hbs_iom: sent_except_from (equivocator_IM IM)
  initial_message_prop es iom
Happ: finite_valid_trace_from_to CE eqv_state_is es
  (eqv_state_tr ++ emsg_tr)
∃ s : state (free_composite_vlsm (equivocator_IM IM)),
  equivocators_total_state_project IM s = sf
  ∧ (∃ tr0 : list
               (composite_transition_item
                  (equivocator_IM IM)),
       equivocators_total_trace_project IM tr0 =
       tr ++
       [{|
          l := l;
          input := iom;
          destination := sf;
          output := oom
        |}]
       ∧ finite_valid_trace_init_to CE eqv_state_is s
           tr0
         ∧ finite_trace_last_output
             (tr ++
              [{|
                 l := l;
                 input := iom;
                 destination := sf;
                 output := oom
               |}]) = finite_trace_last_output tr0)
    specialize
      (extend_right_finite_trace_from_to CE Happ) as Happ_extend.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
seed: message → Prop
constraintX: composite_label IM
→ composite_state IM * option message
  → Prop
CX:= preloaded_vlsm (composite_vlsm IM constraintX)
  seed: VLSM message
constraintE: composite_label (equivocator_IM IM)
→ composite_state (equivocator_IM IM) *
  option message → Prop
CE:= preloaded_vlsm
  (composite_vlsm (equivocator_IM IM) constraintE)
  seed: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hsubsumption: zero_descriptor_constraint_lifting_prop
Hreplayable: replayable_message_prop
is, s: state CX
tr: list transition_item
HtrX1: finite_valid_trace_init_to CX is s tr
iom: option message
iom_si, iom_s: state CX
iom_tr: list transition_item
Heqiom: empty_initial_message_or_final_output CX
  iom_tr iom
HtrX2: finite_valid_trace_init_to CX iom_si iom_s
  iom_tr
sf: state CX
oom: option message
l: label CX
Hs: valid_state_prop CX s
Hiom: option_valid_message_prop CX iom
Hv: valid l (s, iom)
Hc: constraintX l (s, iom)
Ht: transition l (s, iom) = (sf, oom)
HinclE: VLSM_incl CE
  (preloaded_with_all_messages_vlsm FreeE)
eqv_state_is: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_start_project: equivocators_total_state_project
  IM eqv_state_is = is
eqv_state_s: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_state_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hstate_project: equivocators_total_trace_project IM
  eqv_state_tr = tr
Hstate_trace: finite_valid_trace_init_to CE
  eqv_state_is eqv_state_s eqv_state_tr
eqv_msg_is: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hmsg_start_project: equivocators_total_state_project
  IM eqv_msg_is = iom_si
eqv_msg_s: state
  (free_composite_vlsm (equivocator_IM IM))
eqv_msg_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hmsg_project: equivocators_total_trace_project IM
  eqv_msg_tr = iom_tr
Hmsg_trace: finite_valid_trace_init_to CE eqv_msg_is
  eqv_msg_s eqv_msg_tr
Hfinal_msg: finite_trace_last_output iom_tr =
finite_trace_last_output eqv_msg_tr
Hstate_valid: valid_state_prop CE eqv_state_s
emsg_tr: list transition_item
es: state CE
Hmsg_trace_full_replay: finite_valid_trace_from_to CE
  eqv_state_s es emsg_tr
Hemsg_tr_pr: equivocators_total_trace_project IM
  emsg_tr = []
Hes_pr: equivocators_total_state_project IM es = s
Hbs_iom: sent_except_from (equivocator_IM IM)
  initial_message_prop es iom
Happ: finite_valid_trace_from_to CE eqv_state_is es
  (eqv_state_tr ++ emsg_tr)
Happ_extend: ∀ (l3 : label CE) (iom3 : 
                   option message) 
  (s3 : state CE) (oom3 : 
                   option message),
  input_valid_transition CE l3
    (es, iom3) (s3, oom3)
  → finite_valid_trace_from_to CE
      eqv_state_is s3
      ((eqv_state_tr ++ emsg_tr) ++
       [{|
          l := l3;
          input := iom3;
          destination := s3;
          output := oom3
        |}])
∃ s : state (free_composite_vlsm (equivocator_IM IM)),
  equivocators_total_state_project IM s = sf
  ∧ (∃ tr0 : list
               (composite_transition_item
                  (equivocator_IM IM)),
       equivocators_total_trace_project IM tr0 =
       tr ++
       [{|
          l := l;
          input := iom;
          destination := sf;
          output := oom
        |}]
       ∧ finite_valid_trace_init_to CE eqv_state_is s
           tr0
         ∧ finite_trace_last_output
             (tr ++
              [{|
                 l := l;
                 input := iom;
                 destination := sf;
                 output := oom
               |}]) = finite_trace_last_output tr0)
    destruct l as (eqv, li).message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
seed: message → Prop
constraintX: composite_label IM
→ composite_state IM * option message
  → Prop
CX:= preloaded_vlsm (composite_vlsm IM constraintX)
  seed: VLSM message
constraintE: composite_label (equivocator_IM IM)
→ composite_state (equivocator_IM IM) *
  option message → Prop
CE:= preloaded_vlsm
  (composite_vlsm (equivocator_IM IM) constraintE)
  seed: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hsubsumption: zero_descriptor_constraint_lifting_prop
Hreplayable: replayable_message_prop
is, s: state CX
tr: list transition_item
HtrX1: finite_valid_trace_init_to CX is s tr
iom: option message
iom_si, iom_s: state CX
iom_tr: list transition_item
Heqiom: empty_initial_message_or_final_output CX
  iom_tr iom
HtrX2: finite_valid_trace_init_to CX iom_si iom_s
  iom_tr
sf: state CX
oom: option message
eqv: index
li: label (IM eqv)
Hs: valid_state_prop CX s
Hiom: option_valid_message_prop CX iom
Hv: valid (existT eqv li) (s, iom)
Hc: constraintX (existT eqv li) (s, iom)
Ht: transition (existT eqv li) (s, iom) = (sf, oom)
HinclE: VLSM_incl CE
  (preloaded_with_all_messages_vlsm FreeE)
eqv_state_is: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_start_project: equivocators_total_state_project
  IM eqv_state_is = is
eqv_state_s: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_state_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hstate_project: equivocators_total_trace_project IM
  eqv_state_tr = tr
Hstate_trace: finite_valid_trace_init_to CE
  eqv_state_is eqv_state_s eqv_state_tr
eqv_msg_is: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hmsg_start_project: equivocators_total_state_project
  IM eqv_msg_is = iom_si
eqv_msg_s: state
  (free_composite_vlsm (equivocator_IM IM))
eqv_msg_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hmsg_project: equivocators_total_trace_project IM
  eqv_msg_tr = iom_tr
Hmsg_trace: finite_valid_trace_init_to CE eqv_msg_is
  eqv_msg_s eqv_msg_tr
Hfinal_msg: finite_trace_last_output iom_tr =
finite_trace_last_output eqv_msg_tr
Hstate_valid: valid_state_prop CE eqv_state_s
emsg_tr: list transition_item
es: state CE
Hmsg_trace_full_replay: finite_valid_trace_from_to CE
  eqv_state_s es emsg_tr
Hemsg_tr_pr: equivocators_total_trace_project IM
  emsg_tr = []
Hes_pr: equivocators_total_state_project IM es = s
Hbs_iom: sent_except_from (equivocator_IM IM)
  initial_message_prop es iom
Happ: finite_valid_trace_from_to CE eqv_state_is es
  (eqv_state_tr ++ emsg_tr)
Happ_extend: ∀ (l3 : label CE) (iom3 : 
                   option message) 
  (s3 : state CE) (oom3 : 
                   option message),
  input_valid_transition CE l3
    (es, iom3) (s3, oom3)
  → finite_valid_trace_from_to CE
      eqv_state_is s3
      ((eqv_state_tr ++ emsg_tr) ++
       [{|
          l := l3;
          input := iom3;
          destination := s3;
          output := oom3
        |}])
∃ s : state (free_composite_vlsm (equivocator_IM IM)),
  equivocators_total_state_project IM s = sf
  ∧ (∃ tr0 : list
               (composite_transition_item
                  (equivocator_IM IM)),
       equivocators_total_trace_project IM tr0 =
       tr ++
       [{|
          l := existT eqv li;
          input := iom;
          destination := sf;
          output := oom
        |}]
       ∧ finite_valid_trace_init_to CE eqv_state_is s
           tr0
         ∧ finite_trace_last_output
             (tr ++
              [{|
                 l := existT eqv li;
                 input := iom;
                 destination := sf;
                 output := oom
               |}]) = finite_trace_last_output tr0)
    pose
      (@existT _ (fun i : index => label (equivocator_IM IM i)) eqv (ContinueWith 0 li))
      as el.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
seed: message → Prop
constraintX: composite_label IM
→ composite_state IM * option message
  → Prop
CX:= preloaded_vlsm (composite_vlsm IM constraintX)
  seed: VLSM message
constraintE: composite_label (equivocator_IM IM)
→ composite_state (equivocator_IM IM) *
  option message → Prop
CE:= preloaded_vlsm
  (composite_vlsm (equivocator_IM IM) constraintE)
  seed: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hsubsumption: zero_descriptor_constraint_lifting_prop
Hreplayable: replayable_message_prop
is, s: state CX
tr: list transition_item
HtrX1: finite_valid_trace_init_to CX is s tr
iom: option message
iom_si, iom_s: state CX
iom_tr: list transition_item
Heqiom: empty_initial_message_or_final_output CX
  iom_tr iom
HtrX2: finite_valid_trace_init_to CX iom_si iom_s
  iom_tr
sf: state CX
oom: option message
eqv: index
li: label (IM eqv)
Hs: valid_state_prop CX s
Hiom: option_valid_message_prop CX iom
Hv: valid (existT eqv li) (s, iom)
Hc: constraintX (existT eqv li) (s, iom)
Ht: transition (existT eqv li) (s, iom) = (sf, oom)
HinclE: VLSM_incl CE
  (preloaded_with_all_messages_vlsm FreeE)
eqv_state_is: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_start_project: equivocators_total_state_project
  IM eqv_state_is = is
eqv_state_s: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_state_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hstate_project: equivocators_total_trace_project IM
  eqv_state_tr = tr
Hstate_trace: finite_valid_trace_init_to CE
  eqv_state_is eqv_state_s eqv_state_tr
eqv_msg_is: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hmsg_start_project: equivocators_total_state_project
  IM eqv_msg_is = iom_si
eqv_msg_s: state
  (free_composite_vlsm (equivocator_IM IM))
eqv_msg_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hmsg_project: equivocators_total_trace_project IM
  eqv_msg_tr = iom_tr
Hmsg_trace: finite_valid_trace_init_to CE eqv_msg_is
  eqv_msg_s eqv_msg_tr
Hfinal_msg: finite_trace_last_output iom_tr =
finite_trace_last_output eqv_msg_tr
Hstate_valid: valid_state_prop CE eqv_state_s
emsg_tr: list transition_item
es: state CE
Hmsg_trace_full_replay: finite_valid_trace_from_to CE
  eqv_state_s es emsg_tr
Hemsg_tr_pr: equivocators_total_trace_project IM
  emsg_tr = []
Hes_pr: equivocators_total_state_project IM es = s
Hbs_iom: sent_except_from (equivocator_IM IM)
  initial_message_prop es iom
Happ: finite_valid_trace_from_to CE eqv_state_is es
  (eqv_state_tr ++ emsg_tr)
Happ_extend: ∀ (l3 : label CE) (iom3 : 
                   option message) 
  (s3 : state CE) (oom3 : 
                   option message),
  input_valid_transition CE l3
    (es, iom3) (s3, oom3)
  → finite_valid_trace_from_to CE
      eqv_state_is s3
      ((eqv_state_tr ++ emsg_tr) ++
       [{|
          l := l3;
          input := iom3;
          destination := s3;
          output := oom3
        |}])
el:= existT eqv (ContinueWith 0 li): {i : index & label (equivocator_IM IM i)}
∃ s : state (free_composite_vlsm (equivocator_IM IM)),
  equivocators_total_state_project IM s = sf
  ∧ (∃ tr0 : list
               (composite_transition_item
                  (equivocator_IM IM)),
       equivocators_total_trace_project IM tr0 =
       tr ++
       [{|
          l := existT eqv li;
          input := iom;
          destination := sf;
          output := oom
        |}]
       ∧ finite_valid_trace_init_to CE eqv_state_is s
           tr0
         ∧ finite_trace_last_output
             (tr ++
              [{|
                 l := existT eqv li;
                 input := iom;
                 destination := sf;
                 output := oom
               |}]) = finite_trace_last_output tr0)
    destruct (transition CE el (es, iom))
      as (es', om') eqn: Hesom'.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
seed: message → Prop
constraintX: composite_label IM
→ composite_state IM * option message
  → Prop
CX:= preloaded_vlsm (composite_vlsm IM constraintX)
  seed: VLSM message
constraintE: composite_label (equivocator_IM IM)
→ composite_state (equivocator_IM IM) *
  option message → Prop
CE:= preloaded_vlsm
  (composite_vlsm (equivocator_IM IM) constraintE)
  seed: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hsubsumption: zero_descriptor_constraint_lifting_prop
Hreplayable: replayable_message_prop
is, s: state CX
tr: list transition_item
HtrX1: finite_valid_trace_init_to CX is s tr
iom: option message
iom_si, iom_s: state CX
iom_tr: list transition_item
Heqiom: empty_initial_message_or_final_output CX
  iom_tr iom
HtrX2: finite_valid_trace_init_to CX iom_si iom_s
  iom_tr
sf: state CX
oom: option message
eqv: index
li: label (IM eqv)
Hs: valid_state_prop CX s
Hiom: option_valid_message_prop CX iom
Hv: valid (existT eqv li) (s, iom)
Hc: constraintX (existT eqv li) (s, iom)
Ht: transition (existT eqv li) (s, iom) = (sf, oom)
HinclE: VLSM_incl CE
  (preloaded_with_all_messages_vlsm FreeE)
eqv_state_is: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_start_project: equivocators_total_state_project
  IM eqv_state_is = is
eqv_state_s: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_state_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hstate_project: equivocators_total_trace_project IM
  eqv_state_tr = tr
Hstate_trace: finite_valid_trace_init_to CE
  eqv_state_is eqv_state_s eqv_state_tr
eqv_msg_is: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hmsg_start_project: equivocators_total_state_project
  IM eqv_msg_is = iom_si
eqv_msg_s: state
  (free_composite_vlsm (equivocator_IM IM))
eqv_msg_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hmsg_project: equivocators_total_trace_project IM
  eqv_msg_tr = iom_tr
Hmsg_trace: finite_valid_trace_init_to CE eqv_msg_is
  eqv_msg_s eqv_msg_tr
Hfinal_msg: finite_trace_last_output iom_tr =
finite_trace_last_output eqv_msg_tr
Hstate_valid: valid_state_prop CE eqv_state_s
emsg_tr: list transition_item
es: state CE
Hmsg_trace_full_replay: finite_valid_trace_from_to CE
  eqv_state_s es emsg_tr
Hemsg_tr_pr: equivocators_total_trace_project IM
  emsg_tr = []
Hes_pr: equivocators_total_state_project IM es = s
Hbs_iom: sent_except_from (equivocator_IM IM)
  initial_message_prop es iom
Happ: finite_valid_trace_from_to CE eqv_state_is es
  (eqv_state_tr ++ emsg_tr)
Happ_extend: ∀ (l3 : label CE) (iom3 : 
                   option message) 
  (s3 : state CE) (oom3 : 
                   option message),
  input_valid_transition CE l3
    (es, iom3) (s3, oom3)
  → finite_valid_trace_from_to CE
      eqv_state_is s3
      ((eqv_state_tr ++ emsg_tr) ++
       [{|
          l := l3;
          input := iom3;
          destination := s3;
          output := oom3
        |}])
el:= existT eqv (ContinueWith 0 li): {i : index & label (equivocator_IM IM i)}
es': state CE
om': option message
Hesom': transition el (es, iom) = (es', om')
∃ s : state (free_composite_vlsm (equivocator_IM IM)),
  equivocators_total_state_project IM s = sf
  ∧ (∃ tr0 : list
               (composite_transition_item
                  (equivocator_IM IM)),
       equivocators_total_trace_project IM tr0 =
       tr ++
       [{|
          l := existT eqv li;
          input := iom;
          destination := sf;
          output := oom
        |}]
       ∧ finite_valid_trace_init_to CE eqv_state_is s
           tr0
         ∧ finite_trace_last_output
             (tr ++
              [{|
                 l := existT eqv li;
                 input := iom;
                 destination := sf;
                 output := oom
               |}]) = finite_trace_last_output tr0)
    specialize (Happ_extend  el iom es' om').message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
seed: message → Prop
constraintX: composite_label IM
→ composite_state IM * option message
  → Prop
CX:= preloaded_vlsm (composite_vlsm IM constraintX)
  seed: VLSM message
constraintE: composite_label (equivocator_IM IM)
→ composite_state (equivocator_IM IM) *
  option message → Prop
CE:= preloaded_vlsm
  (composite_vlsm (equivocator_IM IM) constraintE)
  seed: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hsubsumption: zero_descriptor_constraint_lifting_prop
Hreplayable: replayable_message_prop
is, s: state CX
tr: list transition_item
HtrX1: finite_valid_trace_init_to CX is s tr
iom: option message
iom_si, iom_s: state CX
iom_tr: list transition_item
Heqiom: empty_initial_message_or_final_output CX
  iom_tr iom
HtrX2: finite_valid_trace_init_to CX iom_si iom_s
  iom_tr
sf: state CX
oom: option message
eqv: index
li: label (IM eqv)
Hs: valid_state_prop CX s
Hiom: option_valid_message_prop CX iom
Hv: valid (existT eqv li) (s, iom)
Hc: constraintX (existT eqv li) (s, iom)
Ht: transition (existT eqv li) (s, iom) = (sf, oom)
HinclE: VLSM_incl CE
  (preloaded_with_all_messages_vlsm FreeE)
eqv_state_is: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_start_project: equivocators_total_state_project
  IM eqv_state_is = is
eqv_state_s: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_state_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hstate_project: equivocators_total_trace_project IM
  eqv_state_tr = tr
Hstate_trace: finite_valid_trace_init_to CE
  eqv_state_is eqv_state_s eqv_state_tr
eqv_msg_is: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hmsg_start_project: equivocators_total_state_project
  IM eqv_msg_is = iom_si
eqv_msg_s: state
  (free_composite_vlsm (equivocator_IM IM))
eqv_msg_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hmsg_project: equivocators_total_trace_project IM
  eqv_msg_tr = iom_tr
Hmsg_trace: finite_valid_trace_init_to CE eqv_msg_is
  eqv_msg_s eqv_msg_tr
Hfinal_msg: finite_trace_last_output iom_tr =
finite_trace_last_output eqv_msg_tr
Hstate_valid: valid_state_prop CE eqv_state_s
emsg_tr: list transition_item
es: state CE
Hmsg_trace_full_replay: finite_valid_trace_from_to CE
  eqv_state_s es emsg_tr
Hemsg_tr_pr: equivocators_total_trace_project IM
  emsg_tr = []
Hes_pr: equivocators_total_state_project IM es = s
Hbs_iom: sent_except_from (equivocator_IM IM)
  initial_message_prop es iom
Happ: finite_valid_trace_from_to CE eqv_state_is es
  (eqv_state_tr ++ emsg_tr)
el:= existT eqv (ContinueWith 0 li): {i : index & label (equivocator_IM IM i)}
es': state CE
om': option message
Happ_extend: input_valid_transition CE el (
  es, iom) (es', om')
→ finite_valid_trace_from_to CE
    eqv_state_is es'
    ((eqv_state_tr ++ emsg_tr) ++
     [{|
        l := el;
        input := iom;
        destination := es';
        output := om'
      |}])
Hesom': transition el (es, iom) = (es', om')
∃ s : state (free_composite_vlsm (equivocator_IM IM)),
  equivocators_total_state_project IM s = sf
  ∧ (∃ tr0 : list
               (composite_transition_item
                  (equivocator_IM IM)),
       equivocators_total_trace_project IM tr0 =
       tr ++
       [{|
          l := existT eqv li;
          input := iom;
          destination := sf;
          output := oom
        |}]
       ∧ finite_valid_trace_init_to CE eqv_state_is s
           tr0
         ∧ finite_trace_last_output
             (tr ++
              [{|
                 l := existT eqv li;
                 input := iom;
                 destination := sf;
                 output := oom
               |}]) = finite_trace_last_output tr0)
    apply valid_trace_get_last in Happ as Heqes.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
seed: message → Prop
constraintX: composite_label IM
→ composite_state IM * option message
  → Prop
CX:= preloaded_vlsm (composite_vlsm IM constraintX)
  seed: VLSM message
constraintE: composite_label (equivocator_IM IM)
→ composite_state (equivocator_IM IM) *
  option message → Prop
CE:= preloaded_vlsm
  (composite_vlsm (equivocator_IM IM) constraintE)
  seed: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hsubsumption: zero_descriptor_constraint_lifting_prop
Hreplayable: replayable_message_prop
is, s: state CX
tr: list transition_item
HtrX1: finite_valid_trace_init_to CX is s tr
iom: option message
iom_si, iom_s: state CX
iom_tr: list transition_item
Heqiom: empty_initial_message_or_final_output CX
  iom_tr iom
HtrX2: finite_valid_trace_init_to CX iom_si iom_s
  iom_tr
sf: state CX
oom: option message
eqv: index
li: label (IM eqv)
Hs: valid_state_prop CX s
Hiom: option_valid_message_prop CX iom
Hv: valid (existT eqv li) (s, iom)
Hc: constraintX (existT eqv li) (s, iom)
Ht: transition (existT eqv li) (s, iom) = (sf, oom)
HinclE: VLSM_incl CE
  (preloaded_with_all_messages_vlsm FreeE)
eqv_state_is: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_start_project: equivocators_total_state_project
  IM eqv_state_is = is
eqv_state_s: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_state_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hstate_project: equivocators_total_trace_project IM
  eqv_state_tr = tr
Hstate_trace: finite_valid_trace_init_to CE
  eqv_state_is eqv_state_s eqv_state_tr
eqv_msg_is: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hmsg_start_project: equivocators_total_state_project
  IM eqv_msg_is = iom_si
eqv_msg_s: state
  (free_composite_vlsm (equivocator_IM IM))
eqv_msg_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hmsg_project: equivocators_total_trace_project IM
  eqv_msg_tr = iom_tr
Hmsg_trace: finite_valid_trace_init_to CE eqv_msg_is
  eqv_msg_s eqv_msg_tr
Hfinal_msg: finite_trace_last_output iom_tr =
finite_trace_last_output eqv_msg_tr
Hstate_valid: valid_state_prop CE eqv_state_s
emsg_tr: list transition_item
es: state CE
Hmsg_trace_full_replay: finite_valid_trace_from_to CE
  eqv_state_s es emsg_tr
Hemsg_tr_pr: equivocators_total_trace_project IM
  emsg_tr = []
Hes_pr: equivocators_total_state_project IM es = s
Hbs_iom: sent_except_from (equivocator_IM IM)
  initial_message_prop es iom
Happ: finite_valid_trace_from_to CE eqv_state_is es
  (eqv_state_tr ++ emsg_tr)
el:= existT eqv (ContinueWith 0 li): {i : index & label (equivocator_IM IM i)}
es': state CE
om': option message
Happ_extend: input_valid_transition CE el (
  es, iom) (es', om')
→ finite_valid_trace_from_to CE
    eqv_state_is es'
    ((eqv_state_tr ++ emsg_tr) ++
     [{|
        l := el;
        input := iom;
        destination := es';
        output := om'
      |}])
Hesom': transition el (es, iom) = (es', om')
Heqes: finite_trace_last eqv_state_is
  (eqv_state_tr ++ emsg_tr) = es
∃ s : state (free_composite_vlsm (equivocator_IM IM)),
  equivocators_total_state_project IM s = sf
  ∧ (∃ tr0 : list
               (composite_transition_item
                  (equivocator_IM IM)),
       equivocators_total_trace_project IM tr0 =
       tr ++
       [{|
          l := existT eqv li;
          input := iom;
          destination := sf;
          output := oom
        |}]
       ∧ finite_valid_trace_init_to CE eqv_state_is s
           tr0
         ∧ finite_trace_last_output
             (tr ++
              [{|
                 l := existT eqv li;
                 input := iom;
                 destination := sf;
                 output := oom
               |}]) = finite_trace_last_output tr0)
    assert (Hes_pr_i : forall i, equivocators_total_state_project IM es i = s i)
      by (rewrite <- Hes_pr; done).message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
seed: message → Prop
constraintX: composite_label IM
→ composite_state IM * option message
  → Prop
CX:= preloaded_vlsm (composite_vlsm IM constraintX)
  seed: VLSM message
constraintE: composite_label (equivocator_IM IM)
→ composite_state (equivocator_IM IM) *
  option message → Prop
CE:= preloaded_vlsm
  (composite_vlsm (equivocator_IM IM) constraintE)
  seed: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hsubsumption: zero_descriptor_constraint_lifting_prop
Hreplayable: replayable_message_prop
is, s: state CX
tr: list transition_item
HtrX1: finite_valid_trace_init_to CX is s tr
iom: option message
iom_si, iom_s: state CX
iom_tr: list transition_item
Heqiom: empty_initial_message_or_final_output CX
  iom_tr iom
HtrX2: finite_valid_trace_init_to CX iom_si iom_s
  iom_tr
sf: state CX
oom: option message
eqv: index
li: label (IM eqv)
Hs: valid_state_prop CX s
Hiom: option_valid_message_prop CX iom
Hv: valid (existT eqv li) (s, iom)
Hc: constraintX (existT eqv li) (s, iom)
Ht: transition (existT eqv li) (s, iom) = (sf, oom)
HinclE: VLSM_incl CE
  (preloaded_with_all_messages_vlsm FreeE)
eqv_state_is: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_start_project: equivocators_total_state_project
  IM eqv_state_is = is
eqv_state_s: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_state_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hstate_project: equivocators_total_trace_project IM
  eqv_state_tr = tr
Hstate_trace: finite_valid_trace_init_to CE
  eqv_state_is eqv_state_s eqv_state_tr
eqv_msg_is: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hmsg_start_project: equivocators_total_state_project
  IM eqv_msg_is = iom_si
eqv_msg_s: state
  (free_composite_vlsm (equivocator_IM IM))
eqv_msg_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hmsg_project: equivocators_total_trace_project IM
  eqv_msg_tr = iom_tr
Hmsg_trace: finite_valid_trace_init_to CE eqv_msg_is
  eqv_msg_s eqv_msg_tr
Hfinal_msg: finite_trace_last_output iom_tr =
finite_trace_last_output eqv_msg_tr
Hstate_valid: valid_state_prop CE eqv_state_s
emsg_tr: list transition_item
es: state CE
Hmsg_trace_full_replay: finite_valid_trace_from_to CE
  eqv_state_s es emsg_tr
Hemsg_tr_pr: equivocators_total_trace_project IM
  emsg_tr = []
Hes_pr: equivocators_total_state_project IM es = s
Hbs_iom: sent_except_from (equivocator_IM IM)
  initial_message_prop es iom
Happ: finite_valid_trace_from_to CE eqv_state_is es
  (eqv_state_tr ++ emsg_tr)
el:= existT eqv (ContinueWith 0 li): {i : index & label (equivocator_IM IM i)}
es': state CE
om': option message
Happ_extend: input_valid_transition CE el (
  es, iom) (es', om')
→ finite_valid_trace_from_to CE
    eqv_state_is es'
    ((eqv_state_tr ++ emsg_tr) ++
     [{|
        l := el;
        input := iom;
        destination := es';
        output := om'
      |}])
Hesom': transition el (es, iom) = (es', om')
Heqes: finite_trace_last eqv_state_is
  (eqv_state_tr ++ emsg_tr) = es
Hes_pr_i: ∀ i : index,
  equivocators_total_state_project IM es i =
  s i
∃ s : state (free_composite_vlsm (equivocator_IM IM)),
  equivocators_total_state_project IM s = sf
  ∧ (∃ tr0 : list
               (composite_transition_item
                  (equivocator_IM IM)),
       equivocators_total_trace_project IM tr0 =
       tr ++
       [{|
          l := existT eqv li;
          input := iom;
          destination := sf;
          output := oom
        |}]
       ∧ finite_valid_trace_init_to CE eqv_state_is s
           tr0
         ∧ finite_trace_last_output
             (tr ++
              [{|
                 l := existT eqv li;
                 input := iom;
                 destination := sf;
                 output := oom
               |}]) = finite_trace_last_output tr0)
    exists es'.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
seed: message → Prop
constraintX: composite_label IM
→ composite_state IM * option message
  → Prop
CX:= preloaded_vlsm (composite_vlsm IM constraintX)
  seed: VLSM message
constraintE: composite_label (equivocator_IM IM)
→ composite_state (equivocator_IM IM) *
  option message → Prop
CE:= preloaded_vlsm
  (composite_vlsm (equivocator_IM IM) constraintE)
  seed: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hsubsumption: zero_descriptor_constraint_lifting_prop
Hreplayable: replayable_message_prop
is, s: state CX
tr: list transition_item
HtrX1: finite_valid_trace_init_to CX is s tr
iom: option message
iom_si, iom_s: state CX
iom_tr: list transition_item
Heqiom: empty_initial_message_or_final_output CX
  iom_tr iom
HtrX2: finite_valid_trace_init_to CX iom_si iom_s
  iom_tr
sf: state CX
oom: option message
eqv: index
li: label (IM eqv)
Hs: valid_state_prop CX s
Hiom: option_valid_message_prop CX iom
Hv: valid (existT eqv li) (s, iom)
Hc: constraintX (existT eqv li) (s, iom)
Ht: transition (existT eqv li) (s, iom) = (sf, oom)
HinclE: VLSM_incl CE
  (preloaded_with_all_messages_vlsm FreeE)
eqv_state_is: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_start_project: equivocators_total_state_project
  IM eqv_state_is = is
eqv_state_s: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_state_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hstate_project: equivocators_total_trace_project IM
  eqv_state_tr = tr
Hstate_trace: finite_valid_trace_init_to CE
  eqv_state_is eqv_state_s eqv_state_tr
eqv_msg_is: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hmsg_start_project: equivocators_total_state_project
  IM eqv_msg_is = iom_si
eqv_msg_s: state
  (free_composite_vlsm (equivocator_IM IM))
eqv_msg_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hmsg_project: equivocators_total_trace_project IM
  eqv_msg_tr = iom_tr
Hmsg_trace: finite_valid_trace_init_to CE eqv_msg_is
  eqv_msg_s eqv_msg_tr
Hfinal_msg: finite_trace_last_output iom_tr =
finite_trace_last_output eqv_msg_tr
Hstate_valid: valid_state_prop CE eqv_state_s
emsg_tr: list transition_item
es: state CE
Hmsg_trace_full_replay: finite_valid_trace_from_to CE
  eqv_state_s es emsg_tr
Hemsg_tr_pr: equivocators_total_trace_project IM
  emsg_tr = []
Hes_pr: equivocators_total_state_project IM es = s
Hbs_iom: sent_except_from (equivocator_IM IM)
  initial_message_prop es iom
Happ: finite_valid_trace_from_to CE eqv_state_is es
  (eqv_state_tr ++ emsg_tr)
el:= existT eqv (ContinueWith 0 li): {i : index & label (equivocator_IM IM i)}
es': state CE
om': option message
Happ_extend: input_valid_transition CE el (
  es, iom) (es', om')
→ finite_valid_trace_from_to CE
    eqv_state_is es'
    ((eqv_state_tr ++ emsg_tr) ++
     [{|
        l := el;
        input := iom;
        destination := es';
        output := om'
      |}])
Hesom': transition el (es, iom) = (es', om')
Heqes: finite_trace_last eqv_state_is
  (eqv_state_tr ++ emsg_tr) = es
Hes_pr_i: ∀ i : index,
  equivocators_total_state_project IM es i =
  s i
equivocators_total_state_project IM es' = sf
∧ (∃ tr0 : list
             (composite_transition_item
                (equivocator_IM IM)),
     equivocators_total_trace_project IM tr0 =
     tr ++
     [{|
        l := existT eqv li;
        input := iom;
        destination := sf;
        output := oom
      |}]
     ∧ finite_valid_trace_init_to CE eqv_state_is es'
         tr0
       ∧ finite_trace_last_output
           (tr ++
            [{|
               l := existT eqv li;
               input := iom;
               destination := sf;
               output := oom
             |}]) = finite_trace_last_output tr0)
    assert (Het := Hesom').message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
seed: message → Prop
constraintX: composite_label IM
→ composite_state IM * option message
  → Prop
CX:= preloaded_vlsm (composite_vlsm IM constraintX)
  seed: VLSM message
constraintE: composite_label (equivocator_IM IM)
→ composite_state (equivocator_IM IM) *
  option message → Prop
CE:= preloaded_vlsm
  (composite_vlsm (equivocator_IM IM) constraintE)
  seed: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hsubsumption: zero_descriptor_constraint_lifting_prop
Hreplayable: replayable_message_prop
is, s: state CX
tr: list transition_item
HtrX1: finite_valid_trace_init_to CX is s tr
iom: option message
iom_si, iom_s: state CX
iom_tr: list transition_item
Heqiom: empty_initial_message_or_final_output CX
  iom_tr iom
HtrX2: finite_valid_trace_init_to CX iom_si iom_s
  iom_tr
sf: state CX
oom: option message
eqv: index
li: label (IM eqv)
Hs: valid_state_prop CX s
Hiom: option_valid_message_prop CX iom
Hv: valid (existT eqv li) (s, iom)
Hc: constraintX (existT eqv li) (s, iom)
Ht: transition (existT eqv li) (s, iom) = (sf, oom)
HinclE: VLSM_incl CE
  (preloaded_with_all_messages_vlsm FreeE)
eqv_state_is: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_start_project: equivocators_total_state_project
  IM eqv_state_is = is
eqv_state_s: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_state_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hstate_project: equivocators_total_trace_project IM
  eqv_state_tr = tr
Hstate_trace: finite_valid_trace_init_to CE
  eqv_state_is eqv_state_s eqv_state_tr
eqv_msg_is: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hmsg_start_project: equivocators_total_state_project
  IM eqv_msg_is = iom_si
eqv_msg_s: state
  (free_composite_vlsm (equivocator_IM IM))
eqv_msg_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hmsg_project: equivocators_total_trace_project IM
  eqv_msg_tr = iom_tr
Hmsg_trace: finite_valid_trace_init_to CE eqv_msg_is
  eqv_msg_s eqv_msg_tr
Hfinal_msg: finite_trace_last_output iom_tr =
finite_trace_last_output eqv_msg_tr
Hstate_valid: valid_state_prop CE eqv_state_s
emsg_tr: list transition_item
es: state CE
Hmsg_trace_full_replay: finite_valid_trace_from_to CE
  eqv_state_s es emsg_tr
Hemsg_tr_pr: equivocators_total_trace_project IM
  emsg_tr = []
Hes_pr: equivocators_total_state_project IM es = s
Hbs_iom: sent_except_from (equivocator_IM IM)
  initial_message_prop es iom
Happ: finite_valid_trace_from_to CE eqv_state_is es
  (eqv_state_tr ++ emsg_tr)
el:= existT eqv (ContinueWith 0 li): {i : index & label (equivocator_IM IM i)}
es': state CE
om': option message
Happ_extend: input_valid_transition CE el (
  es, iom) (es', om')
→ finite_valid_trace_from_to CE
    eqv_state_is es'
    ((eqv_state_tr ++ emsg_tr) ++
     [{|
        l := el;
        input := iom;
        destination := es';
        output := om'
      |}])
Hesom': transition el (es, iom) = (es', om')
Heqes: finite_trace_last eqv_state_is
  (eqv_state_tr ++ emsg_tr) = es
Hes_pr_i: ∀ i : index,
  equivocators_total_state_project IM es i =
  s i
Het: transition el (es, iom) = (es', om')
equivocators_total_state_project IM es' = sf
∧ (∃ tr0 : list
             (composite_transition_item
                (equivocator_IM IM)),
     equivocators_total_trace_project IM tr0 =
     tr ++
     [{|
        l := existT eqv li;
        input := iom;
        destination := sf;
        output := oom
      |}]
     ∧ finite_valid_trace_init_to CE eqv_state_is es'
         tr0
       ∧ finite_trace_last_output
           (tr ++
            [{|
               l := existT eqv li;
               input := iom;
               destination := sf;
               output := oom
             |}]) = finite_trace_last_output tr0)
    specialize (Hes_pr_i eqv) as Hes_pr_eqv.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
seed: message → Prop
constraintX: composite_label IM
→ composite_state IM * option message
  → Prop
CX:= preloaded_vlsm (composite_vlsm IM constraintX)
  seed: VLSM message
constraintE: composite_label (equivocator_IM IM)
→ composite_state (equivocator_IM IM) *
  option message → Prop
CE:= preloaded_vlsm
  (composite_vlsm (equivocator_IM IM) constraintE)
  seed: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hsubsumption: zero_descriptor_constraint_lifting_prop
Hreplayable: replayable_message_prop
is, s: state CX
tr: list transition_item
HtrX1: finite_valid_trace_init_to CX is s tr
iom: option message
iom_si, iom_s: state CX
iom_tr: list transition_item
Heqiom: empty_initial_message_or_final_output CX
  iom_tr iom
HtrX2: finite_valid_trace_init_to CX iom_si iom_s
  iom_tr
sf: state CX
oom: option message
eqv: index
li: label (IM eqv)
Hs: valid_state_prop CX s
Hiom: option_valid_message_prop CX iom
Hv: valid (existT eqv li) (s, iom)
Hc: constraintX (existT eqv li) (s, iom)
Ht: transition (existT eqv li) (s, iom) = (sf, oom)
HinclE: VLSM_incl CE
  (preloaded_with_all_messages_vlsm FreeE)
eqv_state_is: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_start_project: equivocators_total_state_project
  IM eqv_state_is = is
eqv_state_s: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_state_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hstate_project: equivocators_total_trace_project IM
  eqv_state_tr = tr
Hstate_trace: finite_valid_trace_init_to CE
  eqv_state_is eqv_state_s eqv_state_tr
eqv_msg_is: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hmsg_start_project: equivocators_total_state_project
  IM eqv_msg_is = iom_si
eqv_msg_s: state
  (free_composite_vlsm (equivocator_IM IM))
eqv_msg_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hmsg_project: equivocators_total_trace_project IM
  eqv_msg_tr = iom_tr
Hmsg_trace: finite_valid_trace_init_to CE eqv_msg_is
  eqv_msg_s eqv_msg_tr
Hfinal_msg: finite_trace_last_output iom_tr =
finite_trace_last_output eqv_msg_tr
Hstate_valid: valid_state_prop CE eqv_state_s
emsg_tr: list transition_item
es: state CE
Hmsg_trace_full_replay: finite_valid_trace_from_to CE
  eqv_state_s es emsg_tr
Hemsg_tr_pr: equivocators_total_trace_project IM
  emsg_tr = []
Hes_pr: equivocators_total_state_project IM es = s
Hbs_iom: sent_except_from (equivocator_IM IM)
  initial_message_prop es iom
Happ: finite_valid_trace_from_to CE eqv_state_is es
  (eqv_state_tr ++ emsg_tr)
el:= existT eqv (ContinueWith 0 li): {i : index & label (equivocator_IM IM i)}
es': state CE
om': option message
Happ_extend: input_valid_transition CE el (
  es, iom) (es', om')
→ finite_valid_trace_from_to CE
    eqv_state_is es'
    ((eqv_state_tr ++ emsg_tr) ++
     [{|
        l := el;
        input := iom;
        destination := es';
        output := om'
      |}])
Hesom': transition el (es, iom) = (es', om')
Heqes: finite_trace_last eqv_state_is
  (eqv_state_tr ++ emsg_tr) = es
Hes_pr_i: ∀ i : index,
  equivocators_total_state_project IM es i =
  s i
Het: transition el (es, iom) = (es', om')
Hes_pr_eqv: equivocators_total_state_project IM es
  eqv = s eqv
equivocators_total_state_project IM es' = sf
∧ (∃ tr0 : list
             (composite_transition_item
                (equivocator_IM IM)),
     equivocators_total_trace_project IM tr0 =
     tr ++
     [{|
        l := existT eqv li;
        input := iom;
        destination := sf;
        output := oom
      |}]
     ∧ finite_valid_trace_init_to CE eqv_state_is es'
         tr0
       ∧ finite_trace_last_output
           (tr ++
            [{|
               l := existT eqv li;
               input := iom;
               destination := sf;
               output := oom
             |}]) = finite_trace_last_output tr0)
    cbn in Hesom', Hes_pr_eqv.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
seed: message → Prop
constraintX: composite_label IM
→ composite_state IM * option message
  → Prop
CX:= preloaded_vlsm (composite_vlsm IM constraintX)
  seed: VLSM message
constraintE: composite_label (equivocator_IM IM)
→ composite_state (equivocator_IM IM) *
  option message → Prop
CE:= preloaded_vlsm
  (composite_vlsm (equivocator_IM IM) constraintE)
  seed: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hsubsumption: zero_descriptor_constraint_lifting_prop
Hreplayable: replayable_message_prop
is, s: state CX
tr: list transition_item
HtrX1: finite_valid_trace_init_to CX is s tr
iom: option message
iom_si, iom_s: state CX
iom_tr: list transition_item
Heqiom: empty_initial_message_or_final_output CX
  iom_tr iom
HtrX2: finite_valid_trace_init_to CX iom_si iom_s
  iom_tr
sf: state CX
oom: option message
eqv: index
li: label (IM eqv)
Hs: valid_state_prop CX s
Hiom: option_valid_message_prop CX iom
Hv: valid (existT eqv li) (s, iom)
Hc: constraintX (existT eqv li) (s, iom)
Ht: transition (existT eqv li) (s, iom) = (sf, oom)
HinclE: VLSM_incl CE
  (preloaded_with_all_messages_vlsm FreeE)
eqv_state_is: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_start_project: equivocators_total_state_project
  IM eqv_state_is = is
eqv_state_s: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_state_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hstate_project: equivocators_total_trace_project IM
  eqv_state_tr = tr
Hstate_trace: finite_valid_trace_init_to CE
  eqv_state_is eqv_state_s eqv_state_tr
eqv_msg_is: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hmsg_start_project: equivocators_total_state_project
  IM eqv_msg_is = iom_si
eqv_msg_s: state
  (free_composite_vlsm (equivocator_IM IM))
eqv_msg_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hmsg_project: equivocators_total_trace_project IM
  eqv_msg_tr = iom_tr
Hmsg_trace: finite_valid_trace_init_to CE eqv_msg_is
  eqv_msg_s eqv_msg_tr
Hfinal_msg: finite_trace_last_output iom_tr =
finite_trace_last_output eqv_msg_tr
Hstate_valid: valid_state_prop CE eqv_state_s
emsg_tr: list transition_item
es: state CE
Hmsg_trace_full_replay: finite_valid_trace_from_to CE
  eqv_state_s es emsg_tr
Hemsg_tr_pr: equivocators_total_trace_project IM
  emsg_tr = []
Hes_pr: equivocators_total_state_project IM es = s
Hbs_iom: sent_except_from (equivocator_IM IM)
  initial_message_prop es iom
Happ: finite_valid_trace_from_to CE eqv_state_is es
  (eqv_state_tr ++ emsg_tr)
el:= existT eqv (ContinueWith 0 li): {i : index & label (equivocator_IM IM i)}
es': state CE
om': option message
Happ_extend: input_valid_transition CE el (
  es, iom) (es', om')
→ finite_valid_trace_from_to CE
    eqv_state_is es'
    ((eqv_state_tr ++ emsg_tr) ++
     [{|
        l := el;
        input := iom;
        destination := es';
        output := om'
      |}])
Hesom': (let (si', om') :=
   match
     equivocator_state_project (es eqv) 0
   with
   | Some si =>
       let (si', om') :=
         transition li (si, iom) in
       (equivocator_state_update 
          (es eqv) 0 si', om')
   | None => (es eqv, iom)
   end in
 (state_update (equivocator_IM IM) es eqv si',
  om')) = (es', om')
Heqes: finite_trace_last eqv_state_is
  (eqv_state_tr ++ emsg_tr) = es
Hes_pr_i: ∀ i : index,
  equivocators_total_state_project IM es i =
  s i
Het: transition el (es, iom) = (es', om')
Hes_pr_eqv: default (equivocator_state_zero (es eqv))
  (equivocator_state_project (es eqv) 0) =
s eqv
equivocators_total_state_project IM es' = sf
∧ (∃ tr0 : list
             (composite_transition_item
                (equivocator_IM IM)),
     equivocators_total_trace_project IM tr0 =
     tr ++
     [{|
        l := existT eqv li;
        input := iom;
        destination := sf;
        output := oom
      |}]
     ∧ finite_valid_trace_init_to CE eqv_state_is es'
         tr0
       ∧ finite_trace_last_output
           (tr ++
            [{|
               l := existT eqv li;
               input := iom;
               destination := sf;
               output := oom
             |}]) = finite_trace_last_output tr0)
    rewrite equivocator_state_project_zero in Hesom', Hes_pr_eqv.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
seed: message → Prop
constraintX: composite_label IM
→ composite_state IM * option message
  → Prop
CX:= preloaded_vlsm (composite_vlsm IM constraintX)
  seed: VLSM message
constraintE: composite_label (equivocator_IM IM)
→ composite_state (equivocator_IM IM) *
  option message → Prop
CE:= preloaded_vlsm
  (composite_vlsm (equivocator_IM IM) constraintE)
  seed: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hsubsumption: zero_descriptor_constraint_lifting_prop
Hreplayable: replayable_message_prop
is, s: state CX
tr: list transition_item
HtrX1: finite_valid_trace_init_to CX is s tr
iom: option message
iom_si, iom_s: state CX
iom_tr: list transition_item
Heqiom: empty_initial_message_or_final_output CX
  iom_tr iom
HtrX2: finite_valid_trace_init_to CX iom_si iom_s
  iom_tr
sf: state CX
oom: option message
eqv: index
li: label (IM eqv)
Hs: valid_state_prop CX s
Hiom: option_valid_message_prop CX iom
Hv: valid (existT eqv li) (s, iom)
Hc: constraintX (existT eqv li) (s, iom)
Ht: transition (existT eqv li) (s, iom) = (sf, oom)
HinclE: VLSM_incl CE
  (preloaded_with_all_messages_vlsm FreeE)
eqv_state_is: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_start_project: equivocators_total_state_project
  IM eqv_state_is = is
eqv_state_s: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_state_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hstate_project: equivocators_total_trace_project IM
  eqv_state_tr = tr
Hstate_trace: finite_valid_trace_init_to CE
  eqv_state_is eqv_state_s eqv_state_tr
eqv_msg_is: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hmsg_start_project: equivocators_total_state_project
  IM eqv_msg_is = iom_si
eqv_msg_s: state
  (free_composite_vlsm (equivocator_IM IM))
eqv_msg_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hmsg_project: equivocators_total_trace_project IM
  eqv_msg_tr = iom_tr
Hmsg_trace: finite_valid_trace_init_to CE eqv_msg_is
  eqv_msg_s eqv_msg_tr
Hfinal_msg: finite_trace_last_output iom_tr =
finite_trace_last_output eqv_msg_tr
Hstate_valid: valid_state_prop CE eqv_state_s
emsg_tr: list transition_item
es: state CE
Hmsg_trace_full_replay: finite_valid_trace_from_to CE
  eqv_state_s es emsg_tr
Hemsg_tr_pr: equivocators_total_trace_project IM
  emsg_tr = []
Hes_pr: equivocators_total_state_project IM es = s
Hbs_iom: sent_except_from (equivocator_IM IM)
  initial_message_prop es iom
Happ: finite_valid_trace_from_to CE eqv_state_is es
  (eqv_state_tr ++ emsg_tr)
el:= existT eqv (ContinueWith 0 li): {i : index & label (equivocator_IM IM i)}
es': state CE
om': option message
Happ_extend: input_valid_transition CE el (
  es, iom) (es', om')
→ finite_valid_trace_from_to CE
    eqv_state_is es'
    ((eqv_state_tr ++ emsg_tr) ++
     [{|
        l := el;
        input := iom;
        destination := es';
        output := om'
      |}])
Hesom': (let (si', om') :=
   let (si', om') :=
     transition li
       (equivocator_state_zero (es eqv), iom) in
   (equivocator_state_update (es eqv) 0 si',
    om') in
 (state_update (equivocator_IM IM) es eqv si',
  om')) = (es', om')
Heqes: finite_trace_last eqv_state_is
  (eqv_state_tr ++ emsg_tr) = es
Hes_pr_i: ∀ i : index,
  equivocators_total_state_project IM es i =
  s i
Het: transition el (es, iom) = (es', om')
Hes_pr_eqv: default (equivocator_state_zero (es eqv))
  (Some (equivocator_state_zero (es eqv))) =
s eqv
equivocators_total_state_project IM es' = sf
∧ (∃ tr0 : list
             (composite_transition_item
                (equivocator_IM IM)),
     equivocators_total_trace_project IM tr0 =
     tr ++
     [{|
        l := existT eqv li;
        input := iom;
        destination := sf;
        output := oom
      |}]
     ∧ finite_valid_trace_init_to CE eqv_state_is es'
         tr0
       ∧ finite_trace_last_output
           (tr ++
            [{|
               l := existT eqv li;
               input := iom;
               destination := sf;
               output := oom
             |}]) = finite_trace_last_output tr0)
    cbn in Hesom', Hes_pr_eqv.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
seed: message → Prop
constraintX: composite_label IM
→ composite_state IM * option message
  → Prop
CX:= preloaded_vlsm (composite_vlsm IM constraintX)
  seed: VLSM message
constraintE: composite_label (equivocator_IM IM)
→ composite_state (equivocator_IM IM) *
  option message → Prop
CE:= preloaded_vlsm
  (composite_vlsm (equivocator_IM IM) constraintE)
  seed: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hsubsumption: zero_descriptor_constraint_lifting_prop
Hreplayable: replayable_message_prop
is, s: state CX
tr: list transition_item
HtrX1: finite_valid_trace_init_to CX is s tr
iom: option message
iom_si, iom_s: state CX
iom_tr: list transition_item
Heqiom: empty_initial_message_or_final_output CX
  iom_tr iom
HtrX2: finite_valid_trace_init_to CX iom_si iom_s
  iom_tr
sf: state CX
oom: option message
eqv: index
li: label (IM eqv)
Hs: valid_state_prop CX s
Hiom: option_valid_message_prop CX iom
Hv: valid (existT eqv li) (s, iom)
Hc: constraintX (existT eqv li) (s, iom)
Ht: transition (existT eqv li) (s, iom) = (sf, oom)
HinclE: VLSM_incl CE
  (preloaded_with_all_messages_vlsm FreeE)
eqv_state_is: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_start_project: equivocators_total_state_project
  IM eqv_state_is = is
eqv_state_s: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_state_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hstate_project: equivocators_total_trace_project IM
  eqv_state_tr = tr
Hstate_trace: finite_valid_trace_init_to CE
  eqv_state_is eqv_state_s eqv_state_tr
eqv_msg_is: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hmsg_start_project: equivocators_total_state_project
  IM eqv_msg_is = iom_si
eqv_msg_s: state
  (free_composite_vlsm (equivocator_IM IM))
eqv_msg_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hmsg_project: equivocators_total_trace_project IM
  eqv_msg_tr = iom_tr
Hmsg_trace: finite_valid_trace_init_to CE eqv_msg_is
  eqv_msg_s eqv_msg_tr
Hfinal_msg: finite_trace_last_output iom_tr =
finite_trace_last_output eqv_msg_tr
Hstate_valid: valid_state_prop CE eqv_state_s
emsg_tr: list transition_item
es: state CE
Hmsg_trace_full_replay: finite_valid_trace_from_to CE
  eqv_state_s es emsg_tr
Hemsg_tr_pr: equivocators_total_trace_project IM
  emsg_tr = []
Hes_pr: equivocators_total_state_project IM es = s
Hbs_iom: sent_except_from (equivocator_IM IM)
  initial_message_prop es iom
Happ: finite_valid_trace_from_to CE eqv_state_is es
  (eqv_state_tr ++ emsg_tr)
el:= existT eqv (ContinueWith 0 li): {i : index & label (equivocator_IM IM i)}
es': state CE
om': option message
Happ_extend: input_valid_transition CE el (
  es, iom) (es', om')
→ finite_valid_trace_from_to CE
    eqv_state_is es'
    ((eqv_state_tr ++ emsg_tr) ++
     [{|
        l := el;
        input := iom;
        destination := es';
        output := om'
      |}])
Hesom': (let (si', om') :=
   let (si', om') :=
     transition li
       (equivocator_state_zero (es eqv), iom) in
   (equivocator_state_update (es eqv) 0 si',
    om') in
 (state_update (equivocator_IM IM) es eqv si',
  om')) = (es', om')
Heqes: finite_trace_last eqv_state_is
  (eqv_state_tr ++ emsg_tr) = es
Hes_pr_i: ∀ i : index,
  equivocators_total_state_project IM es i =
  s i
Het: transition el (es, iom) = (es', om')
Hes_pr_eqv: equivocator_state_zero (es eqv) = s eqv
equivocators_total_state_project IM es' = sf
∧ (∃ tr0 : list
             (composite_transition_item
                (equivocator_IM IM)),
     equivocators_total_trace_project IM tr0 =
     tr ++
     [{|
        l := existT eqv li;
        input := iom;
        destination := sf;
        output := oom
      |}]
     ∧ finite_valid_trace_init_to CE eqv_state_is es'
         tr0
       ∧ finite_trace_last_output
           (tr ++
            [{|
               l := existT eqv li;
               input := iom;
               destination := sf;
               output := oom
             |}]) = finite_trace_last_output tr0)
    rewrite Hes_pr_eqv in Hesom'.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
seed: message → Prop
constraintX: composite_label IM
→ composite_state IM * option message
  → Prop
CX:= preloaded_vlsm (composite_vlsm IM constraintX)
  seed: VLSM message
constraintE: composite_label (equivocator_IM IM)
→ composite_state (equivocator_IM IM) *
  option message → Prop
CE:= preloaded_vlsm
  (composite_vlsm (equivocator_IM IM) constraintE)
  seed: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hsubsumption: zero_descriptor_constraint_lifting_prop
Hreplayable: replayable_message_prop
is, s: state CX
tr: list transition_item
HtrX1: finite_valid_trace_init_to CX is s tr
iom: option message
iom_si, iom_s: state CX
iom_tr: list transition_item
Heqiom: empty_initial_message_or_final_output CX
  iom_tr iom
HtrX2: finite_valid_trace_init_to CX iom_si iom_s
  iom_tr
sf: state CX
oom: option message
eqv: index
li: label (IM eqv)
Hs: valid_state_prop CX s
Hiom: option_valid_message_prop CX iom
Hv: valid (existT eqv li) (s, iom)
Hc: constraintX (existT eqv li) (s, iom)
Ht: transition (existT eqv li) (s, iom) = (sf, oom)
HinclE: VLSM_incl CE
  (preloaded_with_all_messages_vlsm FreeE)
eqv_state_is: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_start_project: equivocators_total_state_project
  IM eqv_state_is = is
eqv_state_s: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_state_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hstate_project: equivocators_total_trace_project IM
  eqv_state_tr = tr
Hstate_trace: finite_valid_trace_init_to CE
  eqv_state_is eqv_state_s eqv_state_tr
eqv_msg_is: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hmsg_start_project: equivocators_total_state_project
  IM eqv_msg_is = iom_si
eqv_msg_s: state
  (free_composite_vlsm (equivocator_IM IM))
eqv_msg_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hmsg_project: equivocators_total_trace_project IM
  eqv_msg_tr = iom_tr
Hmsg_trace: finite_valid_trace_init_to CE eqv_msg_is
  eqv_msg_s eqv_msg_tr
Hfinal_msg: finite_trace_last_output iom_tr =
finite_trace_last_output eqv_msg_tr
Hstate_valid: valid_state_prop CE eqv_state_s
emsg_tr: list transition_item
es: state CE
Hmsg_trace_full_replay: finite_valid_trace_from_to CE
  eqv_state_s es emsg_tr
Hemsg_tr_pr: equivocators_total_trace_project IM
  emsg_tr = []
Hes_pr: equivocators_total_state_project IM es = s
Hbs_iom: sent_except_from (equivocator_IM IM)
  initial_message_prop es iom
Happ: finite_valid_trace_from_to CE eqv_state_is es
  (eqv_state_tr ++ emsg_tr)
el:= existT eqv (ContinueWith 0 li): {i : index & label (equivocator_IM IM i)}
es': state CE
om': option message
Happ_extend: input_valid_transition CE el (
  es, iom) (es', om')
→ finite_valid_trace_from_to CE
    eqv_state_is es'
    ((eqv_state_tr ++ emsg_tr) ++
     [{|
        l := el;
        input := iom;
        destination := es';
        output := om'
      |}])
Hesom': (let (si', om') :=
   let (si', om') :=
     transition li (s eqv, iom) in
   (equivocator_state_update (es eqv) 0 si',
    om') in
 (state_update (equivocator_IM IM) es eqv si',
  om')) = (es', om')
Heqes: finite_trace_last eqv_state_is
  (eqv_state_tr ++ emsg_tr) = es
Hes_pr_i: ∀ i : index,
  equivocators_total_state_project IM es i =
  s i
Het: transition el (es, iom) = (es', om')
Hes_pr_eqv: equivocator_state_zero (es eqv) = s eqv
equivocators_total_state_project IM es' = sf
∧ (∃ tr0 : list
             (composite_transition_item
                (equivocator_IM IM)),
     equivocators_total_trace_project IM tr0 =
     tr ++
     [{|
        l := existT eqv li;
        input := iom;
        destination := sf;
        output := oom
      |}]
     ∧ finite_valid_trace_init_to CE eqv_state_is es'
         tr0
       ∧ finite_trace_last_output
           (tr ++
            [{|
               l := existT eqv li;
               input := iom;
               destination := sf;
               output := oom
             |}]) = finite_trace_last_output tr0)
    cbn in Ht.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
seed: message → Prop
constraintX: composite_label IM
→ composite_state IM * option message
  → Prop
CX:= preloaded_vlsm (composite_vlsm IM constraintX)
  seed: VLSM message
constraintE: composite_label (equivocator_IM IM)
→ composite_state (equivocator_IM IM) *
  option message → Prop
CE:= preloaded_vlsm
  (composite_vlsm (equivocator_IM IM) constraintE)
  seed: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hsubsumption: zero_descriptor_constraint_lifting_prop
Hreplayable: replayable_message_prop
is, s: state CX
tr: list transition_item
HtrX1: finite_valid_trace_init_to CX is s tr
iom: option message
iom_si, iom_s: state CX
iom_tr: list transition_item
Heqiom: empty_initial_message_or_final_output CX
  iom_tr iom
HtrX2: finite_valid_trace_init_to CX iom_si iom_s
  iom_tr
sf: state CX
oom: option message
eqv: index
li: label (IM eqv)
Hs: valid_state_prop CX s
Hiom: option_valid_message_prop CX iom
Hv: valid (existT eqv li) (s, iom)
Hc: constraintX (existT eqv li) (s, iom)
Ht: (let (si', om') := transition li (s eqv, iom) in
 (state_update IM s eqv si', om')) = (
sf, oom)
HinclE: VLSM_incl CE
  (preloaded_with_all_messages_vlsm FreeE)
eqv_state_is: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_start_project: equivocators_total_state_project
  IM eqv_state_is = is
eqv_state_s: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_state_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hstate_project: equivocators_total_trace_project IM
  eqv_state_tr = tr
Hstate_trace: finite_valid_trace_init_to CE
  eqv_state_is eqv_state_s eqv_state_tr
eqv_msg_is: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hmsg_start_project: equivocators_total_state_project
  IM eqv_msg_is = iom_si
eqv_msg_s: state
  (free_composite_vlsm (equivocator_IM IM))
eqv_msg_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hmsg_project: equivocators_total_trace_project IM
  eqv_msg_tr = iom_tr
Hmsg_trace: finite_valid_trace_init_to CE eqv_msg_is
  eqv_msg_s eqv_msg_tr
Hfinal_msg: finite_trace_last_output iom_tr =
finite_trace_last_output eqv_msg_tr
Hstate_valid: valid_state_prop CE eqv_state_s
emsg_tr: list transition_item
es: state CE
Hmsg_trace_full_replay: finite_valid_trace_from_to CE
  eqv_state_s es emsg_tr
Hemsg_tr_pr: equivocators_total_trace_project IM
  emsg_tr = []
Hes_pr: equivocators_total_state_project IM es = s
Hbs_iom: sent_except_from (equivocator_IM IM)
  initial_message_prop es iom
Happ: finite_valid_trace_from_to CE eqv_state_is es
  (eqv_state_tr ++ emsg_tr)
el:= existT eqv (ContinueWith 0 li): {i : index & label (equivocator_IM IM i)}
es': state CE
om': option message
Happ_extend: input_valid_transition CE el (
  es, iom) (es', om')
→ finite_valid_trace_from_to CE
    eqv_state_is es'
    ((eqv_state_tr ++ emsg_tr) ++
     [{|
        l := el;
        input := iom;
        destination := es';
        output := om'
      |}])
Hesom': (let (si', om') :=
   let (si', om') :=
     transition li (s eqv, iom) in
   (equivocator_state_update (es eqv) 0 si',
    om') in
 (state_update (equivocator_IM IM) es eqv si',
  om')) = (es', om')
Heqes: finite_trace_last eqv_state_is
  (eqv_state_tr ++ emsg_tr) = es
Hes_pr_i: ∀ i : index,
  equivocators_total_state_project IM es i =
  s i
Het: transition el (es, iom) = (es', om')
Hes_pr_eqv: equivocator_state_zero (es eqv) = s eqv
equivocators_total_state_project IM es' = sf
∧ (∃ tr0 : list
             (composite_transition_item
                (equivocator_IM IM)),
     equivocators_total_trace_project IM tr0 =
     tr ++
     [{|
        l := existT eqv li;
        input := iom;
        destination := sf;
        output := oom
      |}]
     ∧ finite_valid_trace_init_to CE eqv_state_is es'
         tr0
       ∧ finite_trace_last_output
           (tr ++
            [{|
               l := existT eqv li;
               input := iom;
               destination := sf;
               output := oom
             |}]) = finite_trace_last_output tr0)
    destruct (transition _ _ _) as (si', _om) eqn: Hteqv.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
seed: message → Prop
constraintX: composite_label IM
→ composite_state IM * option message
  → Prop
CX:= preloaded_vlsm (composite_vlsm IM constraintX)
  seed: VLSM message
constraintE: composite_label (equivocator_IM IM)
→ composite_state (equivocator_IM IM) *
  option message → Prop
CE:= preloaded_vlsm
  (composite_vlsm (equivocator_IM IM) constraintE)
  seed: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hsubsumption: zero_descriptor_constraint_lifting_prop
Hreplayable: replayable_message_prop
is, s: state CX
tr: list transition_item
HtrX1: finite_valid_trace_init_to CX is s tr
iom: option message
iom_si, iom_s: state CX
iom_tr: list transition_item
Heqiom: empty_initial_message_or_final_output CX
  iom_tr iom
HtrX2: finite_valid_trace_init_to CX iom_si iom_s
  iom_tr
sf: state CX
oom: option message
eqv: index
li: label (IM eqv)
Hs: valid_state_prop CX s
Hiom: option_valid_message_prop CX iom
Hv: valid (existT eqv li) (s, iom)
Hc: constraintX (existT eqv li) (s, iom)
si': state (IM eqv)
_om: option message
Hteqv: transition li (s eqv, iom) = (si', _om)
Ht: (state_update IM s eqv si', _om) = (sf, oom)
HinclE: VLSM_incl CE
  (preloaded_with_all_messages_vlsm FreeE)
eqv_state_is: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_start_project: equivocators_total_state_project
  IM eqv_state_is = is
eqv_state_s: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_state_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hstate_project: equivocators_total_trace_project IM
  eqv_state_tr = tr
Hstate_trace: finite_valid_trace_init_to CE
  eqv_state_is eqv_state_s eqv_state_tr
eqv_msg_is: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hmsg_start_project: equivocators_total_state_project
  IM eqv_msg_is = iom_si
eqv_msg_s: state
  (free_composite_vlsm (equivocator_IM IM))
eqv_msg_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hmsg_project: equivocators_total_trace_project IM
  eqv_msg_tr = iom_tr
Hmsg_trace: finite_valid_trace_init_to CE eqv_msg_is
  eqv_msg_s eqv_msg_tr
Hfinal_msg: finite_trace_last_output iom_tr =
finite_trace_last_output eqv_msg_tr
Hstate_valid: valid_state_prop CE eqv_state_s
emsg_tr: list transition_item
es: state CE
Hmsg_trace_full_replay: finite_valid_trace_from_to CE
  eqv_state_s es emsg_tr
Hemsg_tr_pr: equivocators_total_trace_project IM
  emsg_tr = []
Hes_pr: equivocators_total_state_project IM es = s
Hbs_iom: sent_except_from (equivocator_IM IM)
  initial_message_prop es iom
Happ: finite_valid_trace_from_to CE eqv_state_is es
  (eqv_state_tr ++ emsg_tr)
el:= existT eqv (ContinueWith 0 li): {i : index & label (equivocator_IM IM i)}
es': state CE
om': option message
Happ_extend: input_valid_transition CE el (
  es, iom) (es', om')
→ finite_valid_trace_from_to CE
    eqv_state_is es'
    ((eqv_state_tr ++ emsg_tr) ++
     [{|
        l := el;
        input := iom;
        destination := es';
        output := om'
      |}])
Hesom': (state_update (equivocator_IM IM) es eqv
   (equivocator_state_update (es eqv) 0 si'),
 _om) = (es', om')
Heqes: finite_trace_last eqv_state_is
  (eqv_state_tr ++ emsg_tr) = es
Hes_pr_i: ∀ i : index,
  equivocators_total_state_project IM es i =
  s i
Het: transition el (es, iom) = (es', om')
Hes_pr_eqv: equivocator_state_zero (es eqv) = s eqv
equivocators_total_state_project IM es' = sf
∧ (∃ tr0 : list
             (composite_transition_item
                (equivocator_IM IM)),
     equivocators_total_trace_project IM tr0 =
     tr ++
     [{|
        l := existT eqv li;
        input := iom;
        destination := sf;
        output := oom
      |}]
     ∧ finite_valid_trace_init_to CE eqv_state_is es'
         tr0
       ∧ finite_trace_last_output
           (tr ++
            [{|
               l := existT eqv li;
               input := iom;
               destination := sf;
               output := oom
             |}]) = finite_trace_last_output tr0)
    inversion Ht.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
seed: message → Prop
constraintX: composite_label IM
→ composite_state IM * option message
  → Prop
CX:= preloaded_vlsm (composite_vlsm IM constraintX)
  seed: VLSM message
constraintE: composite_label (equivocator_IM IM)
→ composite_state (equivocator_IM IM) *
  option message → Prop
CE:= preloaded_vlsm
  (composite_vlsm (equivocator_IM IM) constraintE)
  seed: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hsubsumption: zero_descriptor_constraint_lifting_prop
Hreplayable: replayable_message_prop
is, s: state CX
tr: list transition_item
HtrX1: finite_valid_trace_init_to CX is s tr
iom: option message
iom_si, iom_s: state CX
iom_tr: list transition_item
Heqiom: empty_initial_message_or_final_output CX
  iom_tr iom
HtrX2: finite_valid_trace_init_to CX iom_si iom_s
  iom_tr
sf: state CX
oom: option message
eqv: index
li: label (IM eqv)
Hs: valid_state_prop CX s
Hiom: option_valid_message_prop CX iom
Hv: valid (existT eqv li) (s, iom)
Hc: constraintX (existT eqv li) (s, iom)
si': state (IM eqv)
_om: option message
Hteqv: transition li (s eqv, iom) = (si', _om)
Ht: (state_update IM s eqv si', _om) = (sf, oom)
HinclE: VLSM_incl CE
  (preloaded_with_all_messages_vlsm FreeE)
eqv_state_is: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_start_project: equivocators_total_state_project
  IM eqv_state_is = is
eqv_state_s: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_state_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hstate_project: equivocators_total_trace_project IM
  eqv_state_tr = tr
Hstate_trace: finite_valid_trace_init_to CE
  eqv_state_is eqv_state_s eqv_state_tr
eqv_msg_is: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hmsg_start_project: equivocators_total_state_project
  IM eqv_msg_is = iom_si
eqv_msg_s: state
  (free_composite_vlsm (equivocator_IM IM))
eqv_msg_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hmsg_project: equivocators_total_trace_project IM
  eqv_msg_tr = iom_tr
Hmsg_trace: finite_valid_trace_init_to CE eqv_msg_is
  eqv_msg_s eqv_msg_tr
Hfinal_msg: finite_trace_last_output iom_tr =
finite_trace_last_output eqv_msg_tr
Hstate_valid: valid_state_prop CE eqv_state_s
emsg_tr: list transition_item
es: state CE
Hmsg_trace_full_replay: finite_valid_trace_from_to CE
  eqv_state_s es emsg_tr
Hemsg_tr_pr: equivocators_total_trace_project IM
  emsg_tr = []
Hes_pr: equivocators_total_state_project IM es = s
Hbs_iom: sent_except_from (equivocator_IM IM)
  initial_message_prop es iom
Happ: finite_valid_trace_from_to CE eqv_state_is es
  (eqv_state_tr ++ emsg_tr)
el:= existT eqv (ContinueWith 0 li): {i : index & label (equivocator_IM IM i)}
es': state CE
om': option message
Happ_extend: input_valid_transition CE el (
  es, iom) (es', om')
→ finite_valid_trace_from_to CE
    eqv_state_is es'
    ((eqv_state_tr ++ emsg_tr) ++
     [{|
        l := el;
        input := iom;
        destination := es';
        output := om'
      |}])
Hesom': (state_update (equivocator_IM IM) es eqv
   (equivocator_state_update (es eqv) 0 si'),
 _om) = (es', om')
Heqes: finite_trace_last eqv_state_is
  (eqv_state_tr ++ emsg_tr) = es
Hes_pr_i: ∀ i : index,
  equivocators_total_state_project IM es i =
  s i
Het: transition el (es, iom) = (es', om')
Hes_pr_eqv: equivocator_state_zero (es eqv) = s eqv
H2: state_update IM s eqv si' = sf
H3: _om = oom
equivocators_total_state_project IM es' =
state_update IM s eqv si'
∧ (∃ tr0 : list
             (composite_transition_item
                (equivocator_IM IM)),
     equivocators_total_trace_project IM tr0 =
     tr ++
     [{|
        l := existT eqv li;
        input := iom;
        destination := state_update IM s eqv si';
        output := oom
      |}]
     ∧ finite_valid_trace_init_to CE eqv_state_is es'
         tr0
       ∧ finite_trace_last_output
           (tr ++
            [{|
               l := existT eqv li;
               input := iom;
               destination :=
                 state_update IM s eqv si';
               output := oom
             |}]) = finite_trace_last_output tr0) subst sf _om.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
seed: message → Prop
constraintX: composite_label IM
→ composite_state IM * option message
  → Prop
CX:= preloaded_vlsm (composite_vlsm IM constraintX)
  seed: VLSM message
constraintE: composite_label (equivocator_IM IM)
→ composite_state (equivocator_IM IM) *
  option message → Prop
CE:= preloaded_vlsm
  (composite_vlsm (equivocator_IM IM) constraintE)
  seed: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hsubsumption: zero_descriptor_constraint_lifting_prop
Hreplayable: replayable_message_prop
is, s: state CX
tr: list transition_item
HtrX1: finite_valid_trace_init_to CX is s tr
iom: option message
iom_si, iom_s: state CX
iom_tr: list transition_item
Heqiom: empty_initial_message_or_final_output CX
  iom_tr iom
HtrX2: finite_valid_trace_init_to CX iom_si iom_s
  iom_tr
oom: option message
eqv: index
li: label (IM eqv)
Hs: valid_state_prop CX s
Hiom: option_valid_message_prop CX iom
Hv: valid (existT eqv li) (s, iom)
Hc: constraintX (existT eqv li) (s, iom)
si': state (IM eqv)
Ht: (state_update IM s eqv si', oom) =
(state_update IM s eqv si', oom)
Hteqv: transition li (s eqv, iom) = (si', oom)
HinclE: VLSM_incl CE
  (preloaded_with_all_messages_vlsm FreeE)
eqv_state_is: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_start_project: equivocators_total_state_project
  IM eqv_state_is = is
eqv_state_s: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_state_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hstate_project: equivocators_total_trace_project IM
  eqv_state_tr = tr
Hstate_trace: finite_valid_trace_init_to CE
  eqv_state_is eqv_state_s eqv_state_tr
eqv_msg_is: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hmsg_start_project: equivocators_total_state_project
  IM eqv_msg_is = iom_si
eqv_msg_s: state
  (free_composite_vlsm (equivocator_IM IM))
eqv_msg_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hmsg_project: equivocators_total_trace_project IM
  eqv_msg_tr = iom_tr
Hmsg_trace: finite_valid_trace_init_to CE eqv_msg_is
  eqv_msg_s eqv_msg_tr
Hfinal_msg: finite_trace_last_output iom_tr =
finite_trace_last_output eqv_msg_tr
Hstate_valid: valid_state_prop CE eqv_state_s
emsg_tr: list transition_item
es: state CE
Hmsg_trace_full_replay: finite_valid_trace_from_to CE
  eqv_state_s es emsg_tr
Hemsg_tr_pr: equivocators_total_trace_project IM
  emsg_tr = []
Hes_pr: equivocators_total_state_project IM es = s
Hbs_iom: sent_except_from (equivocator_IM IM)
  initial_message_prop es iom
Happ: finite_valid_trace_from_to CE eqv_state_is es
  (eqv_state_tr ++ emsg_tr)
el:= existT eqv (ContinueWith 0 li): {i : index & label (equivocator_IM IM i)}
es': state CE
om': option message
Happ_extend: input_valid_transition CE el (
  es, iom) (es', om')
→ finite_valid_trace_from_to CE
    eqv_state_is es'
    ((eqv_state_tr ++ emsg_tr) ++
     [{|
        l := el;
        input := iom;
        destination := es';
        output := om'
      |}])
Hesom': (state_update (equivocator_IM IM) es eqv
   (equivocator_state_update (es eqv) 0 si'),
 oom) = (es', om')
Heqes: finite_trace_last eqv_state_is
  (eqv_state_tr ++ emsg_tr) = es
Hes_pr_i: ∀ i : index,
  equivocators_total_state_project IM es i =
  s i
Het: transition el (es, iom) = (es', om')
Hes_pr_eqv: equivocator_state_zero (es eqv) = s eqv
equivocators_total_state_project IM es' =
state_update IM s eqv si'
∧ (∃ tr0 : list
             (composite_transition_item
                (equivocator_IM IM)),
     equivocators_total_trace_project IM tr0 =
     tr ++
     [{|
        l := existT eqv li;
        input := iom;
        destination := state_update IM s eqv si';
        output := oom
      |}]
     ∧ finite_valid_trace_init_to CE eqv_state_is es'
         tr0
       ∧ finite_trace_last_output
           (tr ++
            [{|
               l := existT eqv li;
               input := iom;
               destination :=
                 state_update IM s eqv si';
               output := oom
             |}]) = finite_trace_last_output tr0) clear Ht.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
seed: message → Prop
constraintX: composite_label IM
→ composite_state IM * option message
  → Prop
CX:= preloaded_vlsm (composite_vlsm IM constraintX)
  seed: VLSM message
constraintE: composite_label (equivocator_IM IM)
→ composite_state (equivocator_IM IM) *
  option message → Prop
CE:= preloaded_vlsm
  (composite_vlsm (equivocator_IM IM) constraintE)
  seed: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hsubsumption: zero_descriptor_constraint_lifting_prop
Hreplayable: replayable_message_prop
is, s: state CX
tr: list transition_item
HtrX1: finite_valid_trace_init_to CX is s tr
iom: option message
iom_si, iom_s: state CX
iom_tr: list transition_item
Heqiom: empty_initial_message_or_final_output CX
  iom_tr iom
HtrX2: finite_valid_trace_init_to CX iom_si iom_s
  iom_tr
oom: option message
eqv: index
li: label (IM eqv)
Hs: valid_state_prop CX s
Hiom: option_valid_message_prop CX iom
Hv: valid (existT eqv li) (s, iom)
Hc: constraintX (existT eqv li) (s, iom)
si': state (IM eqv)
Hteqv: transition li (s eqv, iom) = (si', oom)
HinclE: VLSM_incl CE
  (preloaded_with_all_messages_vlsm FreeE)
eqv_state_is: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_start_project: equivocators_total_state_project
  IM eqv_state_is = is
eqv_state_s: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_state_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hstate_project: equivocators_total_trace_project IM
  eqv_state_tr = tr
Hstate_trace: finite_valid_trace_init_to CE
  eqv_state_is eqv_state_s eqv_state_tr
eqv_msg_is: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hmsg_start_project: equivocators_total_state_project
  IM eqv_msg_is = iom_si
eqv_msg_s: state
  (free_composite_vlsm (equivocator_IM IM))
eqv_msg_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hmsg_project: equivocators_total_trace_project IM
  eqv_msg_tr = iom_tr
Hmsg_trace: finite_valid_trace_init_to CE eqv_msg_is
  eqv_msg_s eqv_msg_tr
Hfinal_msg: finite_trace_last_output iom_tr =
finite_trace_last_output eqv_msg_tr
Hstate_valid: valid_state_prop CE eqv_state_s
emsg_tr: list transition_item
es: state CE
Hmsg_trace_full_replay: finite_valid_trace_from_to CE
  eqv_state_s es emsg_tr
Hemsg_tr_pr: equivocators_total_trace_project IM
  emsg_tr = []
Hes_pr: equivocators_total_state_project IM es = s
Hbs_iom: sent_except_from (equivocator_IM IM)
  initial_message_prop es iom
Happ: finite_valid_trace_from_to CE eqv_state_is es
  (eqv_state_tr ++ emsg_tr)
el:= existT eqv (ContinueWith 0 li): {i : index & label (equivocator_IM IM i)}
es': state CE
om': option message
Happ_extend: input_valid_transition CE el (
  es, iom) (es', om')
→ finite_valid_trace_from_to CE
    eqv_state_is es'
    ((eqv_state_tr ++ emsg_tr) ++
     [{|
        l := el;
        input := iom;
        destination := es';
        output := om'
      |}])
Hesom': (state_update (equivocator_IM IM) es eqv
   (equivocator_state_update (es eqv) 0 si'),
 oom) = (es', om')
Heqes: finite_trace_last eqv_state_is
  (eqv_state_tr ++ emsg_tr) = es
Hes_pr_i: ∀ i : index,
  equivocators_total_state_project IM es i =
  s i
Het: transition el (es, iom) = (es', om')
Hes_pr_eqv: equivocator_state_zero (es eqv) = s eqv
equivocators_total_state_project IM es' =
state_update IM s eqv si'
∧ (∃ tr0 : list
             (composite_transition_item
                (equivocator_IM IM)),
     equivocators_total_trace_project IM tr0 =
     tr ++
     [{|
        l := existT eqv li;
        input := iom;
        destination := state_update IM s eqv si';
        output := oom
      |}]
     ∧ finite_valid_trace_init_to CE eqv_state_is es'
         tr0
       ∧ finite_trace_last_output
           (tr ++
            [{|
               l := existT eqv li;
               input := iom;
               destination :=
                 state_update IM s eqv si';
               output := oom
             |}]) = finite_trace_last_output tr0)
    inversion Hesom'.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
seed: message → Prop
constraintX: composite_label IM
→ composite_state IM * option message
  → Prop
CX:= preloaded_vlsm (composite_vlsm IM constraintX)
  seed: VLSM message
constraintE: composite_label (equivocator_IM IM)
→ composite_state (equivocator_IM IM) *
  option message → Prop
CE:= preloaded_vlsm
  (composite_vlsm (equivocator_IM IM) constraintE)
  seed: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hsubsumption: zero_descriptor_constraint_lifting_prop
Hreplayable: replayable_message_prop
is, s: state CX
tr: list transition_item
HtrX1: finite_valid_trace_init_to CX is s tr
iom: option message
iom_si, iom_s: state CX
iom_tr: list transition_item
Heqiom: empty_initial_message_or_final_output CX
  iom_tr iom
HtrX2: finite_valid_trace_init_to CX iom_si iom_s
  iom_tr
oom: option message
eqv: index
li: label (IM eqv)
Hs: valid_state_prop CX s
Hiom: option_valid_message_prop CX iom
Hv: valid (existT eqv li) (s, iom)
Hc: constraintX (existT eqv li) (s, iom)
si': state (IM eqv)
Hteqv: transition li (s eqv, iom) = (si', oom)
HinclE: VLSM_incl CE
  (preloaded_with_all_messages_vlsm FreeE)
eqv_state_is: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_start_project: equivocators_total_state_project
  IM eqv_state_is = is
eqv_state_s: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_state_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hstate_project: equivocators_total_trace_project IM
  eqv_state_tr = tr
Hstate_trace: finite_valid_trace_init_to CE
  eqv_state_is eqv_state_s eqv_state_tr
eqv_msg_is: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hmsg_start_project: equivocators_total_state_project
  IM eqv_msg_is = iom_si
eqv_msg_s: state
  (free_composite_vlsm (equivocator_IM IM))
eqv_msg_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hmsg_project: equivocators_total_trace_project IM
  eqv_msg_tr = iom_tr
Hmsg_trace: finite_valid_trace_init_to CE eqv_msg_is
  eqv_msg_s eqv_msg_tr
Hfinal_msg: finite_trace_last_output iom_tr =
finite_trace_last_output eqv_msg_tr
Hstate_valid: valid_state_prop CE eqv_state_s
emsg_tr: list transition_item
es: state CE
Hmsg_trace_full_replay: finite_valid_trace_from_to CE
  eqv_state_s es emsg_tr
Hemsg_tr_pr: equivocators_total_trace_project IM
  emsg_tr = []
Hes_pr: equivocators_total_state_project IM es = s
Hbs_iom: sent_except_from (equivocator_IM IM)
  initial_message_prop es iom
Happ: finite_valid_trace_from_to CE eqv_state_is es
  (eqv_state_tr ++ emsg_tr)
el:= existT eqv (ContinueWith 0 li): {i : index & label (equivocator_IM IM i)}
es': state CE
om': option message
Happ_extend: input_valid_transition CE el (
  es, iom) (es', om')
→ finite_valid_trace_from_to CE
    eqv_state_is es'
    ((eqv_state_tr ++ emsg_tr) ++
     [{|
        l := el;
        input := iom;
        destination := es';
        output := om'
      |}])
Hesom': (state_update (equivocator_IM IM) es eqv
   (equivocator_state_update (es eqv) 0 si'),
 oom) = (es', om')
Heqes: finite_trace_last eqv_state_is
  (eqv_state_tr ++ emsg_tr) = es
Hes_pr_i: ∀ i : index,
  equivocators_total_state_project IM es i =
  s i
Het: transition el (es, iom) = (es', om')
Hes_pr_eqv: equivocator_state_zero (es eqv) = s eqv
H2: state_update (equivocator_IM IM) es eqv
  (equivocator_state_update (es eqv) 0 si') = es'
H3: oom = om'
equivocators_total_state_project IM
  (state_update (equivocator_IM IM) es eqv
     (equivocator_state_update (es eqv) 0 si')) =
state_update IM s eqv si'
∧ (∃ tr0 : list
             (composite_transition_item
                (equivocator_IM IM)),
     equivocators_total_trace_project IM tr0 =
     tr ++
     [{|
        l := existT eqv li;
        input := iom;
        destination := state_update IM s eqv si';
        output := om'
      |}]
     ∧ finite_valid_trace_init_to CE eqv_state_is
         (state_update (equivocator_IM IM) es eqv
            (equivocator_state_update (es eqv) 0 si'))
         tr0
       ∧ finite_trace_last_output
           (tr ++
            [{|
               l := existT eqv li;
               input := iom;
               destination :=
                 state_update IM s eqv si';
               output := om'
             |}]) = finite_trace_last_output tr0) subst es' om'.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
seed: message → Prop
constraintX: composite_label IM
→ composite_state IM * option message
  → Prop
CX:= preloaded_vlsm (composite_vlsm IM constraintX)
  seed: VLSM message
constraintE: composite_label (equivocator_IM IM)
→ composite_state (equivocator_IM IM) *
  option message → Prop
CE:= preloaded_vlsm
  (composite_vlsm (equivocator_IM IM) constraintE)
  seed: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hsubsumption: zero_descriptor_constraint_lifting_prop
Hreplayable: replayable_message_prop
is, s: state CX
tr: list transition_item
HtrX1: finite_valid_trace_init_to CX is s tr
iom: option message
iom_si, iom_s: state CX
iom_tr: list transition_item
Heqiom: empty_initial_message_or_final_output CX
  iom_tr iom
HtrX2: finite_valid_trace_init_to CX iom_si iom_s
  iom_tr
oom: option message
eqv: index
li: label (IM eqv)
Hs: valid_state_prop CX s
Hiom: option_valid_message_prop CX iom
Hv: valid (existT eqv li) (s, iom)
Hc: constraintX (existT eqv li) (s, iom)
si': state (IM eqv)
Hteqv: transition li (s eqv, iom) = (si', oom)
HinclE: VLSM_incl CE
  (preloaded_with_all_messages_vlsm FreeE)
eqv_state_is: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_start_project: equivocators_total_state_project
  IM eqv_state_is = is
eqv_state_s: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_state_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hstate_project: equivocators_total_trace_project IM
  eqv_state_tr = tr
Hstate_trace: finite_valid_trace_init_to CE
  eqv_state_is eqv_state_s eqv_state_tr
eqv_msg_is: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hmsg_start_project: equivocators_total_state_project
  IM eqv_msg_is = iom_si
eqv_msg_s: state
  (free_composite_vlsm (equivocator_IM IM))
eqv_msg_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hmsg_project: equivocators_total_trace_project IM
  eqv_msg_tr = iom_tr
Hmsg_trace: finite_valid_trace_init_to CE eqv_msg_is
  eqv_msg_s eqv_msg_tr
Hfinal_msg: finite_trace_last_output iom_tr =
finite_trace_last_output eqv_msg_tr
Hstate_valid: valid_state_prop CE eqv_state_s
emsg_tr: list transition_item
es: state CE
Hmsg_trace_full_replay: finite_valid_trace_from_to CE
  eqv_state_s es emsg_tr
Hemsg_tr_pr: equivocators_total_trace_project IM
  emsg_tr = []
Hes_pr: equivocators_total_state_project IM es = s
Hbs_iom: sent_except_from (equivocator_IM IM)
  initial_message_prop es iom
Happ: finite_valid_trace_from_to CE eqv_state_is es
  (eqv_state_tr ++ emsg_tr)
el:= existT eqv (ContinueWith 0 li): {i : index & label (equivocator_IM IM i)}
Happ_extend: input_valid_transition CE el (
  es, iom)
  (state_update (equivocator_IM IM) es
     eqv
     (equivocator_state_update 
        (es eqv) 0 si'), oom)
→ finite_valid_trace_from_to CE
    eqv_state_is
    (state_update (equivocator_IM IM) es
       eqv
       (equivocator_state_update
          (es eqv) 0 si'))
    ((eqv_state_tr ++ emsg_tr) ++
     [{|
        l := el;
        input := iom;
        destination :=
          state_update
            (equivocator_IM IM) es eqv
            (equivocator_state_update
               (es eqv) 0 si');
        output := oom
      |}])
Hesom': (state_update (equivocator_IM IM) es eqv
   (equivocator_state_update (es eqv) 0 si'),
 oom) =
(state_update (equivocator_IM IM) es eqv
   (equivocator_state_update (es eqv) 0 si'),
 oom)
Heqes: finite_trace_last eqv_state_is
  (eqv_state_tr ++ emsg_tr) = es
Hes_pr_i: ∀ i : index,
  equivocators_total_state_project IM es i =
  s i
Het: transition el (es, iom) =
(state_update (equivocator_IM IM) es eqv
   (equivocator_state_update (es eqv) 0 si'),
 oom)
Hes_pr_eqv: equivocator_state_zero (es eqv) = s eqv
equivocators_total_state_project IM
  (state_update (equivocator_IM IM) es eqv
     (equivocator_state_update (es eqv) 0 si')) =
state_update IM s eqv si'
∧ (∃ tr0 : list
             (composite_transition_item
                (equivocator_IM IM)),
     equivocators_total_trace_project IM tr0 =
     tr ++
     [{|
        l := existT eqv li;
        input := iom;
        destination := state_update IM s eqv si';
        output := oom
      |}]
     ∧ finite_valid_trace_init_to CE eqv_state_is
         (state_update (equivocator_IM IM) es eqv
            (equivocator_state_update (es eqv) 0 si'))
         tr0
       ∧ finite_trace_last_output
           (tr ++
            [{|
               l := existT eqv li;
               input := iom;
               destination :=
                 state_update IM s eqv si';
               output := oom
             |}]) = finite_trace_last_output tr0) clear Hesom'.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
seed: message → Prop
constraintX: composite_label IM
→ composite_state IM * option message
  → Prop
CX:= preloaded_vlsm (composite_vlsm IM constraintX)
  seed: VLSM message
constraintE: composite_label (equivocator_IM IM)
→ composite_state (equivocator_IM IM) *
  option message → Prop
CE:= preloaded_vlsm
  (composite_vlsm (equivocator_IM IM) constraintE)
  seed: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hsubsumption: zero_descriptor_constraint_lifting_prop
Hreplayable: replayable_message_prop
is, s: state CX
tr: list transition_item
HtrX1: finite_valid_trace_init_to CX is s tr
iom: option message
iom_si, iom_s: state CX
iom_tr: list transition_item
Heqiom: empty_initial_message_or_final_output CX
  iom_tr iom
HtrX2: finite_valid_trace_init_to CX iom_si iom_s
  iom_tr
oom: option message
eqv: index
li: label (IM eqv)
Hs: valid_state_prop CX s
Hiom: option_valid_message_prop CX iom
Hv: valid (existT eqv li) (s, iom)
Hc: constraintX (existT eqv li) (s, iom)
si': state (IM eqv)
Hteqv: transition li (s eqv, iom) = (si', oom)
HinclE: VLSM_incl CE
  (preloaded_with_all_messages_vlsm FreeE)
eqv_state_is: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_start_project: equivocators_total_state_project
  IM eqv_state_is = is
eqv_state_s: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_state_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hstate_project: equivocators_total_trace_project IM
  eqv_state_tr = tr
Hstate_trace: finite_valid_trace_init_to CE
  eqv_state_is eqv_state_s eqv_state_tr
eqv_msg_is: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hmsg_start_project: equivocators_total_state_project
  IM eqv_msg_is = iom_si
eqv_msg_s: state
  (free_composite_vlsm (equivocator_IM IM))
eqv_msg_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hmsg_project: equivocators_total_trace_project IM
  eqv_msg_tr = iom_tr
Hmsg_trace: finite_valid_trace_init_to CE eqv_msg_is
  eqv_msg_s eqv_msg_tr
Hfinal_msg: finite_trace_last_output iom_tr =
finite_trace_last_output eqv_msg_tr
Hstate_valid: valid_state_prop CE eqv_state_s
emsg_tr: list transition_item
es: state CE
Hmsg_trace_full_replay: finite_valid_trace_from_to CE
  eqv_state_s es emsg_tr
Hemsg_tr_pr: equivocators_total_trace_project IM
  emsg_tr = []
Hes_pr: equivocators_total_state_project IM es = s
Hbs_iom: sent_except_from (equivocator_IM IM)
  initial_message_prop es iom
Happ: finite_valid_trace_from_to CE eqv_state_is es
  (eqv_state_tr ++ emsg_tr)
el:= existT eqv (ContinueWith 0 li): {i : index & label (equivocator_IM IM i)}
Happ_extend: input_valid_transition CE el (
  es, iom)
  (state_update (equivocator_IM IM) es
     eqv
     (equivocator_state_update 
        (es eqv) 0 si'), oom)
→ finite_valid_trace_from_to CE
    eqv_state_is
    (state_update (equivocator_IM IM) es
       eqv
       (equivocator_state_update
          (es eqv) 0 si'))
    ((eqv_state_tr ++ emsg_tr) ++
     [{|
        l := el;
        input := iom;
        destination :=
          state_update
            (equivocator_IM IM) es eqv
            (equivocator_state_update
               (es eqv) 0 si');
        output := oom
      |}])
Heqes: finite_trace_last eqv_state_is
  (eqv_state_tr ++ emsg_tr) = es
Hes_pr_i: ∀ i : index,
  equivocators_total_state_project IM es i =
  s i
Het: transition el (es, iom) =
(state_update (equivocator_IM IM) es eqv
   (equivocator_state_update (es eqv) 0 si'),
 oom)
Hes_pr_eqv: equivocator_state_zero (es eqv) = s eqv
equivocators_total_state_project IM
  (state_update (equivocator_IM IM) es eqv
     (equivocator_state_update (es eqv) 0 si')) =
state_update IM s eqv si'
∧ (∃ tr0 : list
             (composite_transition_item
                (equivocator_IM IM)),
     equivocators_total_trace_project IM tr0 =
     tr ++
     [{|
        l := existT eqv li;
        input := iom;
        destination := state_update IM s eqv si';
        output := oom
      |}]
     ∧ finite_valid_trace_init_to CE eqv_state_is
         (state_update (equivocator_IM IM) es eqv
            (equivocator_state_update (es eqv) 0 si'))
         tr0
       ∧ finite_trace_last_output
           (tr ++
            [{|
               l := existT eqv li;
               input := iom;
               destination :=
                 state_update IM s eqv si';
               output := oom
             |}]) = finite_trace_last_output tr0)
    match type of Happ_extend with
    | ?H -> _ => cut H
    end.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
seed: message → Prop
constraintX: composite_label IM
→ composite_state IM * option message
  → Prop
CX:= preloaded_vlsm (composite_vlsm IM constraintX)
  seed: VLSM message
constraintE: composite_label (equivocator_IM IM)
→ composite_state (equivocator_IM IM) *
  option message → Prop
CE:= preloaded_vlsm
  (composite_vlsm (equivocator_IM IM) constraintE)
  seed: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hsubsumption: zero_descriptor_constraint_lifting_prop
Hreplayable: replayable_message_prop
is, s: state CX
tr: list transition_item
HtrX1: finite_valid_trace_init_to CX is s tr
iom: option message
iom_si, iom_s: state CX
iom_tr: list transition_item
Heqiom: empty_initial_message_or_final_output CX
  iom_tr iom
HtrX2: finite_valid_trace_init_to CX iom_si iom_s
  iom_tr
oom: option message
eqv: index
li: label (IM eqv)
Hs: valid_state_prop CX s
Hiom: option_valid_message_prop CX iom
Hv: valid (existT eqv li) (s, iom)
Hc: constraintX (existT eqv li) (s, iom)
si': state (IM eqv)
Hteqv: transition li (s eqv, iom) = (si', oom)
HinclE: VLSM_incl CE
  (preloaded_with_all_messages_vlsm FreeE)
eqv_state_is: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_start_project: equivocators_total_state_project
  IM eqv_state_is = is
eqv_state_s: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_state_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hstate_project: equivocators_total_trace_project IM
  eqv_state_tr = tr
Hstate_trace: finite_valid_trace_init_to CE
  eqv_state_is eqv_state_s eqv_state_tr
eqv_msg_is: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hmsg_start_project: equivocators_total_state_project
  IM eqv_msg_is = iom_si
eqv_msg_s: state
  (free_composite_vlsm (equivocator_IM IM))
eqv_msg_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hmsg_project: equivocators_total_trace_project IM
  eqv_msg_tr = iom_tr
Hmsg_trace: finite_valid_trace_init_to CE eqv_msg_is
  eqv_msg_s eqv_msg_tr
Hfinal_msg: finite_trace_last_output iom_tr =
finite_trace_last_output eqv_msg_tr
Hstate_valid: valid_state_prop CE eqv_state_s
emsg_tr: list transition_item
es: state CE
Hmsg_trace_full_replay: finite_valid_trace_from_to CE
  eqv_state_s es emsg_tr
Hemsg_tr_pr: equivocators_total_trace_project IM
  emsg_tr = []
Hes_pr: equivocators_total_state_project IM es = s
Hbs_iom: sent_except_from (equivocator_IM IM)
  initial_message_prop es iom
Happ: finite_valid_trace_from_to CE eqv_state_is es
  (eqv_state_tr ++ emsg_tr)
el:= existT eqv (ContinueWith 0 li): {i : index & label (equivocator_IM IM i)}
Happ_extend: input_valid_transition CE el (
  es, iom)
  (state_update (equivocator_IM IM) es
     eqv
     (equivocator_state_update 
        (es eqv) 0 si'), oom)
→ finite_valid_trace_from_to CE
    eqv_state_is
    (state_update (equivocator_IM IM) es
       eqv
       (equivocator_state_update
          (es eqv) 0 si'))
    ((eqv_state_tr ++ emsg_tr) ++
     [{|
        l := el;
        input := iom;
        destination :=
          state_update
            (equivocator_IM IM) es eqv
            (equivocator_state_update
               (es eqv) 0 si');
        output := oom
      |}])
Heqes: finite_trace_last eqv_state_is
  (eqv_state_tr ++ emsg_tr) = es
Hes_pr_i: ∀ i : index,
  equivocators_total_state_project IM es i =
  s i
Het: transition el (es, iom) =
(state_update (equivocator_IM IM) es eqv
   (equivocator_state_update (es eqv) 0 si'),
 oom)
Hes_pr_eqv: equivocator_state_zero (es eqv) = s eqv
input_valid_transition CE el (es, iom)
  (state_update (equivocator_IM IM) es eqv
     (equivocator_state_update (es eqv) 0 si'), oom)
→ equivocators_total_state_project IM
    (state_update (equivocator_IM IM) es eqv
       (equivocator_state_update (es eqv) 0 si')) =
  state_update IM s eqv si'
  ∧ (∃ tr0 : list
               (composite_transition_item
                  (equivocator_IM IM)),
       equivocators_total_trace_project IM tr0 =
       tr ++
       [{|
          l := existT eqv li;
          input := iom;
          destination := state_update IM s eqv si';
          output := oom
        |}]
       ∧ finite_valid_trace_init_to CE eqv_state_is
           (state_update (equivocator_IM IM) es eqv
              (equivocator_state_update (es eqv) 0 si'))
           tr0
         ∧ finite_trace_last_output
             (tr ++
              [{|
                 l := existT eqv li;
                 input := iom;
                 destination :=
                   state_update IM s eqv si';
                 output := oom
               |}]) = finite_trace_last_output tr0)
message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
seed: message → Prop
constraintX: composite_label IM
→ composite_state IM * option message
  → Prop
CX:= preloaded_vlsm (composite_vlsm IM constraintX)
  seed: VLSM message
constraintE: composite_label (equivocator_IM IM)
→ composite_state (equivocator_IM IM) *
  option message → Prop
CE:= preloaded_vlsm
  (composite_vlsm (equivocator_IM IM) constraintE)
  seed: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hsubsumption: zero_descriptor_constraint_lifting_prop
Hreplayable: replayable_message_prop
is, s: state CX
tr: list transition_item
HtrX1: finite_valid_trace_init_to CX is s tr
iom: option message
iom_si, iom_s: state CX
iom_tr: list transition_item
Heqiom: empty_initial_message_or_final_output CX
  iom_tr iom
HtrX2: finite_valid_trace_init_to CX iom_si iom_s
  iom_tr
oom: option message
eqv: index
li: label (IM eqv)
Hs: valid_state_prop CX s
Hiom: option_valid_message_prop CX iom
Hv: valid (existT eqv li) (s, iom)
Hc: constraintX (existT eqv li) (s, iom)
si': state (IM eqv)
Hteqv: transition li (s eqv, iom) = (si', oom)
HinclE: VLSM_incl CE
  (preloaded_with_all_messages_vlsm FreeE)
eqv_state_is: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_start_project: equivocators_total_state_project
  IM eqv_state_is = is
eqv_state_s: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_state_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hstate_project: equivocators_total_trace_project IM
  eqv_state_tr = tr
Hstate_trace: finite_valid_trace_init_to CE
  eqv_state_is eqv_state_s eqv_state_tr
eqv_msg_is: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hmsg_start_project: equivocators_total_state_project
  IM eqv_msg_is = iom_si
eqv_msg_s: state
  (free_composite_vlsm (equivocator_IM IM))
eqv_msg_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hmsg_project: equivocators_total_trace_project IM
  eqv_msg_tr = iom_tr
Hmsg_trace: finite_valid_trace_init_to CE eqv_msg_is
  eqv_msg_s eqv_msg_tr
Hfinal_msg: finite_trace_last_output iom_tr =
finite_trace_last_output eqv_msg_tr
Hstate_valid: valid_state_prop CE eqv_state_s
emsg_tr: list transition_item
es: state CE
Hmsg_trace_full_replay: finite_valid_trace_from_to CE
  eqv_state_s es emsg_tr
Hemsg_tr_pr: equivocators_total_trace_project IM
  emsg_tr = []
Hes_pr: equivocators_total_state_project IM es = s
Hbs_iom: sent_except_from (equivocator_IM IM)
  initial_message_prop es iom
Happ: finite_valid_trace_from_to CE eqv_state_is es
  (eqv_state_tr ++ emsg_tr)
el:= existT eqv (ContinueWith 0 li): {i : index & label (equivocator_IM IM i)}
Happ_extend: input_valid_transition CE el (
  es, iom)
  (state_update (equivocator_IM IM) es
     eqv
     (equivocator_state_update 
        (es eqv) 0 si'), oom)
→ finite_valid_trace_from_to CE
    eqv_state_is
    (state_update (equivocator_IM IM) es
       eqv
       (equivocator_state_update
          (es eqv) 0 si'))
    ((eqv_state_tr ++ emsg_tr) ++
     [{|
        l := el;
        input := iom;
        destination :=
          state_update
            (equivocator_IM IM) es eqv
            (equivocator_state_update
               (es eqv) 0 si');
        output := oom
      |}])
Heqes: finite_trace_last eqv_state_is
  (eqv_state_tr ++ emsg_tr) = es
Hes_pr_i: ∀ i : index,
  equivocators_total_state_project IM es i =
  s i
Het: transition el (es, iom) =
(state_update (equivocator_IM IM) es eqv
   (equivocator_state_update (es eqv) 0 si'),
 oom)
Hes_pr_eqv: equivocator_state_zero (es eqv) = s eqv
input_valid_transition CE el (
  es, iom)
  (state_update (equivocator_IM IM) es eqv
     (equivocator_state_update (es eqv) 0 si'), oom)
    {message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
seed: message → Prop
constraintX: composite_label IM
→ composite_state IM * option message
  → Prop
CX:= preloaded_vlsm (composite_vlsm IM constraintX)
  seed: VLSM message
constraintE: composite_label (equivocator_IM IM)
→ composite_state (equivocator_IM IM) *
  option message → Prop
CE:= preloaded_vlsm
  (composite_vlsm (equivocator_IM IM) constraintE)
  seed: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hsubsumption: zero_descriptor_constraint_lifting_prop
Hreplayable: replayable_message_prop
is, s: state CX
tr: list transition_item
HtrX1: finite_valid_trace_init_to CX is s tr
iom: option message
iom_si, iom_s: state CX
iom_tr: list transition_item
Heqiom: empty_initial_message_or_final_output CX
  iom_tr iom
HtrX2: finite_valid_trace_init_to CX iom_si iom_s
  iom_tr
oom: option message
eqv: index
li: label (IM eqv)
Hs: valid_state_prop CX s
Hiom: option_valid_message_prop CX iom
Hv: valid (existT eqv li) (s, iom)
Hc: constraintX (existT eqv li) (s, iom)
si': state (IM eqv)
Hteqv: transition li (s eqv, iom) = (si', oom)
HinclE: VLSM_incl CE
  (preloaded_with_all_messages_vlsm FreeE)
eqv_state_is: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_start_project: equivocators_total_state_project
  IM eqv_state_is = is
eqv_state_s: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_state_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hstate_project: equivocators_total_trace_project IM
  eqv_state_tr = tr
Hstate_trace: finite_valid_trace_init_to CE
  eqv_state_is eqv_state_s eqv_state_tr
eqv_msg_is: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hmsg_start_project: equivocators_total_state_project
  IM eqv_msg_is = iom_si
eqv_msg_s: state
  (free_composite_vlsm (equivocator_IM IM))
eqv_msg_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hmsg_project: equivocators_total_trace_project IM
  eqv_msg_tr = iom_tr
Hmsg_trace: finite_valid_trace_init_to CE eqv_msg_is
  eqv_msg_s eqv_msg_tr
Hfinal_msg: finite_trace_last_output iom_tr =
finite_trace_last_output eqv_msg_tr
Hstate_valid: valid_state_prop CE eqv_state_s
emsg_tr: list transition_item
es: state CE
Hmsg_trace_full_replay: finite_valid_trace_from_to CE
  eqv_state_s es emsg_tr
Hemsg_tr_pr: equivocators_total_trace_project IM
  emsg_tr = []
Hes_pr: equivocators_total_state_project IM es = s
Hbs_iom: sent_except_from (equivocator_IM IM)
  initial_message_prop es iom
Happ: finite_valid_trace_from_to CE eqv_state_is es
  (eqv_state_tr ++ emsg_tr)
el:= existT eqv (ContinueWith 0 li): {i : index & label (equivocator_IM IM i)}
Happ_extend: input_valid_transition CE el (
  es, iom)
  (state_update (equivocator_IM IM) es
     eqv
     (equivocator_state_update 
        (es eqv) 0 si'), oom)
→ finite_valid_trace_from_to CE
    eqv_state_is
    (state_update (equivocator_IM IM) es
       eqv
       (equivocator_state_update
          (es eqv) 0 si'))
    ((eqv_state_tr ++ emsg_tr) ++
     [{|
        l := el;
        input := iom;
        destination :=
          state_update
            (equivocator_IM IM) es eqv
            (equivocator_state_update
               (es eqv) 0 si');
        output := oom
      |}])
Heqes: finite_trace_last eqv_state_is
  (eqv_state_tr ++ emsg_tr) = es
Hes_pr_i: ∀ i : index,
  equivocators_total_state_project IM es i =
  s i
Het: transition el (es, iom) =
(state_update (equivocator_IM IM) es eqv
   (equivocator_state_update (es eqv) 0 si'),
 oom)
Hes_pr_eqv: equivocator_state_zero (es eqv) = s eqv
input_valid_transition CE el (es, iom)
  (state_update (equivocator_IM IM) es eqv
     (equivocator_state_update (es eqv) 0 si'), oom)
→ equivocators_total_state_project IM
    (state_update (equivocator_IM IM) es eqv
       (equivocator_state_update (es eqv) 0 si')) =
  state_update IM s eqv si'
  ∧ (∃ tr0 : list
               (composite_transition_item
                  (equivocator_IM IM)),
       equivocators_total_trace_project IM tr0 =
       tr ++
       [{|
          l := existT eqv li;
          input := iom;
          destination := state_update IM s eqv si';
          output := oom
        |}]
       ∧ finite_valid_trace_init_to CE eqv_state_is
           (state_update (equivocator_IM IM) es eqv
              (equivocator_state_update (es eqv) 0 si'))
           tr0
         ∧ finite_trace_last_output
             (tr ++
              [{|
                 l := existT eqv li;
                 input := iom;
                 destination :=
                   state_update IM s eqv si';
                 output := oom
               |}]) = finite_trace_last_output tr0) intro Hivt.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
seed: message → Prop
constraintX: composite_label IM
→ composite_state IM * option message
  → Prop
CX:= preloaded_vlsm (composite_vlsm IM constraintX)
  seed: VLSM message
constraintE: composite_label (equivocator_IM IM)
→ composite_state (equivocator_IM IM) *
  option message → Prop
CE:= preloaded_vlsm
  (composite_vlsm (equivocator_IM IM) constraintE)
  seed: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hsubsumption: zero_descriptor_constraint_lifting_prop
Hreplayable: replayable_message_prop
is, s: state CX
tr: list transition_item
HtrX1: finite_valid_trace_init_to CX is s tr
iom: option message
iom_si, iom_s: state CX
iom_tr: list transition_item
Heqiom: empty_initial_message_or_final_output CX
  iom_tr iom
HtrX2: finite_valid_trace_init_to CX iom_si iom_s
  iom_tr
oom: option message
eqv: index
li: label (IM eqv)
Hs: valid_state_prop CX s
Hiom: option_valid_message_prop CX iom
Hv: valid (existT eqv li) (s, iom)
Hc: constraintX (existT eqv li) (s, iom)
si': state (IM eqv)
Hteqv: transition li (s eqv, iom) = (si', oom)
HinclE: VLSM_incl CE
  (preloaded_with_all_messages_vlsm FreeE)
eqv_state_is: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_start_project: equivocators_total_state_project
  IM eqv_state_is = is
eqv_state_s: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_state_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hstate_project: equivocators_total_trace_project IM
  eqv_state_tr = tr
Hstate_trace: finite_valid_trace_init_to CE
  eqv_state_is eqv_state_s eqv_state_tr
eqv_msg_is: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hmsg_start_project: equivocators_total_state_project
  IM eqv_msg_is = iom_si
eqv_msg_s: state
  (free_composite_vlsm (equivocator_IM IM))
eqv_msg_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hmsg_project: equivocators_total_trace_project IM
  eqv_msg_tr = iom_tr
Hmsg_trace: finite_valid_trace_init_to CE eqv_msg_is
  eqv_msg_s eqv_msg_tr
Hfinal_msg: finite_trace_last_output iom_tr =
finite_trace_last_output eqv_msg_tr
Hstate_valid: valid_state_prop CE eqv_state_s
emsg_tr: list transition_item
es: state CE
Hmsg_trace_full_replay: finite_valid_trace_from_to CE
  eqv_state_s es emsg_tr
Hemsg_tr_pr: equivocators_total_trace_project IM
  emsg_tr = []
Hes_pr: equivocators_total_state_project IM es = s
Hbs_iom: sent_except_from (equivocator_IM IM)
  initial_message_prop es iom
Happ: finite_valid_trace_from_to CE eqv_state_is es
  (eqv_state_tr ++ emsg_tr)
el:= existT eqv (ContinueWith 0 li): {i : index & label (equivocator_IM IM i)}
Happ_extend: input_valid_transition CE el (
  es, iom)
  (state_update (equivocator_IM IM) es
     eqv
     (equivocator_state_update 
        (es eqv) 0 si'), oom)
→ finite_valid_trace_from_to CE
    eqv_state_is
    (state_update (equivocator_IM IM) es
       eqv
       (equivocator_state_update
          (es eqv) 0 si'))
    ((eqv_state_tr ++ emsg_tr) ++
     [{|
        l := el;
        input := iom;
        destination :=
          state_update
            (equivocator_IM IM) es eqv
            (equivocator_state_update
               (es eqv) 0 si');
        output := oom
      |}])
Heqes: finite_trace_last eqv_state_is
  (eqv_state_tr ++ emsg_tr) = es
Hes_pr_i: ∀ i : index,
  equivocators_total_state_project IM es i =
  s i
Het: transition el (es, iom) =
(state_update (equivocator_IM IM) es eqv
   (equivocator_state_update (es eqv) 0 si'),
 oom)
Hes_pr_eqv: equivocator_state_zero (es eqv) = s eqv
Hivt: input_valid_transition CE el (
  es, iom)
  (state_update (equivocator_IM IM) es eqv
     (equivocator_state_update (es eqv) 0 si'),
   oom)
equivocators_total_state_project IM
  (state_update (equivocator_IM IM) es eqv
     (equivocator_state_update (es eqv) 0 si')) =
state_update IM s eqv si'
∧ (∃ tr0 : list
             (composite_transition_item
                (equivocator_IM IM)),
     equivocators_total_trace_project IM tr0 =
     tr ++
     [{|
        l := existT eqv li;
        input := iom;
        destination := state_update IM s eqv si';
        output := oom
      |}]
     ∧ finite_valid_trace_init_to CE eqv_state_is
         (state_update (equivocator_IM IM) es eqv
            (equivocator_state_update (es eqv) 0 si'))
         tr0
       ∧ finite_trace_last_output
           (tr ++
            [{|
               l := existT eqv li;
               input := iom;
               destination :=
                 state_update IM s eqv si';
               output := oom
             |}]) = finite_trace_last_output tr0)
      specialize (Happ_extend Hivt).message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
seed: message → Prop
constraintX: composite_label IM
→ composite_state IM * option message
  → Prop
CX:= preloaded_vlsm (composite_vlsm IM constraintX)
  seed: VLSM message
constraintE: composite_label (equivocator_IM IM)
→ composite_state (equivocator_IM IM) *
  option message → Prop
CE:= preloaded_vlsm
  (composite_vlsm (equivocator_IM IM) constraintE)
  seed: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hsubsumption: zero_descriptor_constraint_lifting_prop
Hreplayable: replayable_message_prop
is, s: state CX
tr: list transition_item
HtrX1: finite_valid_trace_init_to CX is s tr
iom: option message
iom_si, iom_s: state CX
iom_tr: list transition_item
Heqiom: empty_initial_message_or_final_output CX
  iom_tr iom
HtrX2: finite_valid_trace_init_to CX iom_si iom_s
  iom_tr
oom: option message
eqv: index
li: label (IM eqv)
Hs: valid_state_prop CX s
Hiom: option_valid_message_prop CX iom
Hv: valid (existT eqv li) (s, iom)
Hc: constraintX (existT eqv li) (s, iom)
si': state (IM eqv)
Hteqv: transition li (s eqv, iom) = (si', oom)
HinclE: VLSM_incl CE
  (preloaded_with_all_messages_vlsm FreeE)
eqv_state_is: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_start_project: equivocators_total_state_project
  IM eqv_state_is = is
eqv_state_s: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_state_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hstate_project: equivocators_total_trace_project IM
  eqv_state_tr = tr
Hstate_trace: finite_valid_trace_init_to CE
  eqv_state_is eqv_state_s eqv_state_tr
eqv_msg_is: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hmsg_start_project: equivocators_total_state_project
  IM eqv_msg_is = iom_si
eqv_msg_s: state
  (free_composite_vlsm (equivocator_IM IM))
eqv_msg_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hmsg_project: equivocators_total_trace_project IM
  eqv_msg_tr = iom_tr
Hmsg_trace: finite_valid_trace_init_to CE eqv_msg_is
  eqv_msg_s eqv_msg_tr
Hfinal_msg: finite_trace_last_output iom_tr =
finite_trace_last_output eqv_msg_tr
Hstate_valid: valid_state_prop CE eqv_state_s
emsg_tr: list transition_item
es: state CE
Hmsg_trace_full_replay: finite_valid_trace_from_to CE
  eqv_state_s es emsg_tr
Hemsg_tr_pr: equivocators_total_trace_project IM
  emsg_tr = []
Hes_pr: equivocators_total_state_project IM es = s
Hbs_iom: sent_except_from (equivocator_IM IM)
  initial_message_prop es iom
Happ: finite_valid_trace_from_to CE eqv_state_is es
  (eqv_state_tr ++ emsg_tr)
el:= existT eqv (ContinueWith 0 li): {i : index & label (equivocator_IM IM i)}
Happ_extend: finite_valid_trace_from_to CE
  eqv_state_is
  (state_update (equivocator_IM IM) es
     eqv
     (equivocator_state_update 
        (es eqv) 0 si'))
  ((eqv_state_tr ++ emsg_tr) ++
   [{|
      l := el;
      input := iom;
      destination :=
        state_update (equivocator_IM IM)
          es eqv
          (equivocator_state_update
             (es eqv) 0 si');
      output := oom
    |}])
Heqes: finite_trace_last eqv_state_is
  (eqv_state_tr ++ emsg_tr) = es
Hes_pr_i: ∀ i : index,
  equivocators_total_state_project IM es i =
  s i
Het: transition el (es, iom) =
(state_update (equivocator_IM IM) es eqv
   (equivocator_state_update (es eqv) 0 si'),
 oom)
Hes_pr_eqv: equivocator_state_zero (es eqv) = s eqv
Hivt: input_valid_transition CE el (
  es, iom)
  (state_update (equivocator_IM IM) es eqv
     (equivocator_state_update (es eqv) 0 si'),
   oom)
equivocators_total_state_project IM
  (state_update (equivocator_IM IM) es eqv
     (equivocator_state_update (es eqv) 0 si')) =
state_update IM s eqv si'
∧ (∃ tr0 : list
             (composite_transition_item
                (equivocator_IM IM)),
     equivocators_total_trace_project IM tr0 =
     tr ++
     [{|
        l := existT eqv li;
        input := iom;
        destination := state_update IM s eqv si';
        output := oom
      |}]
     ∧ finite_valid_trace_init_to CE eqv_state_is
         (state_update (equivocator_IM IM) es eqv
            (equivocator_state_update (es eqv) 0 si'))
         tr0
       ∧ finite_trace_last_output
           (tr ++
            [{|
               l := existT eqv li;
               input := iom;
               destination :=
                 state_update IM s eqv si';
               output := oom
             |}]) = finite_trace_last_output tr0)
      match goal with
      |- ?H /\ _ => assert (Hproject : H)
      end.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
seed: message → Prop
constraintX: composite_label IM
→ composite_state IM * option message
  → Prop
CX:= preloaded_vlsm (composite_vlsm IM constraintX)
  seed: VLSM message
constraintE: composite_label (equivocator_IM IM)
→ composite_state (equivocator_IM IM) *
  option message → Prop
CE:= preloaded_vlsm
  (composite_vlsm (equivocator_IM IM) constraintE)
  seed: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hsubsumption: zero_descriptor_constraint_lifting_prop
Hreplayable: replayable_message_prop
is, s: state CX
tr: list transition_item
HtrX1: finite_valid_trace_init_to CX is s tr
iom: option message
iom_si, iom_s: state CX
iom_tr: list transition_item
Heqiom: empty_initial_message_or_final_output CX
  iom_tr iom
HtrX2: finite_valid_trace_init_to CX iom_si iom_s
  iom_tr
oom: option message
eqv: index
li: label (IM eqv)
Hs: valid_state_prop CX s
Hiom: option_valid_message_prop CX iom
Hv: valid (existT eqv li) (s, iom)
Hc: constraintX (existT eqv li) (s, iom)
si': state (IM eqv)
Hteqv: transition li (s eqv, iom) = (si', oom)
HinclE: VLSM_incl CE
  (preloaded_with_all_messages_vlsm FreeE)
eqv_state_is: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_start_project: equivocators_total_state_project
  IM eqv_state_is = is
eqv_state_s: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_state_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hstate_project: equivocators_total_trace_project IM
  eqv_state_tr = tr
Hstate_trace: finite_valid_trace_init_to CE
  eqv_state_is eqv_state_s eqv_state_tr
eqv_msg_is: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hmsg_start_project: equivocators_total_state_project
  IM eqv_msg_is = iom_si
eqv_msg_s: state
  (free_composite_vlsm (equivocator_IM IM))
eqv_msg_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hmsg_project: equivocators_total_trace_project IM
  eqv_msg_tr = iom_tr
Hmsg_trace: finite_valid_trace_init_to CE eqv_msg_is
  eqv_msg_s eqv_msg_tr
Hfinal_msg: finite_trace_last_output iom_tr =
finite_trace_last_output eqv_msg_tr
Hstate_valid: valid_state_prop CE eqv_state_s
emsg_tr: list transition_item
es: state CE
Hmsg_trace_full_replay: finite_valid_trace_from_to CE
  eqv_state_s es emsg_tr
Hemsg_tr_pr: equivocators_total_trace_project IM
  emsg_tr = []
Hes_pr: equivocators_total_state_project IM es = s
Hbs_iom: sent_except_from (equivocator_IM IM)
  initial_message_prop es iom
Happ: finite_valid_trace_from_to CE eqv_state_is es
  (eqv_state_tr ++ emsg_tr)
el:= existT eqv (ContinueWith 0 li): {i : index & label (equivocator_IM IM i)}
Happ_extend: finite_valid_trace_from_to CE
  eqv_state_is
  (state_update (equivocator_IM IM) es
     eqv
     (equivocator_state_update 
        (es eqv) 0 si'))
  ((eqv_state_tr ++ emsg_tr) ++
   [{|
      l := el;
      input := iom;
      destination :=
        state_update (equivocator_IM IM)
          es eqv
          (equivocator_state_update
             (es eqv) 0 si');
      output := oom
    |}])
Heqes: finite_trace_last eqv_state_is
  (eqv_state_tr ++ emsg_tr) = es
Hes_pr_i: ∀ i : index,
  equivocators_total_state_project IM es i =
  s i
Het: transition el (es, iom) =
(state_update (equivocator_IM IM) es eqv
   (equivocator_state_update (es eqv) 0 si'),
 oom)
Hes_pr_eqv: equivocator_state_zero (es eqv) = s eqv
Hivt: input_valid_transition CE el (
  es, iom)
  (state_update (equivocator_IM IM) es eqv
     (equivocator_state_update (es eqv) 0 si'),
   oom)
equivocators_total_state_project IM
  (state_update (equivocator_IM IM) es eqv
     (equivocator_state_update (es eqv) 0 si')) =
state_update IM s eqv si'
message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
seed: message → Prop
constraintX: composite_label IM
→ composite_state IM * option message
  → Prop
CX:= preloaded_vlsm (composite_vlsm IM constraintX)
  seed: VLSM message
constraintE: composite_label (equivocator_IM IM)
→ composite_state (equivocator_IM IM) *
  option message → Prop
CE:= preloaded_vlsm
  (composite_vlsm (equivocator_IM IM) constraintE)
  seed: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hsubsumption: zero_descriptor_constraint_lifting_prop
Hreplayable: replayable_message_prop
is, s: state CX
tr: list transition_item
HtrX1: finite_valid_trace_init_to CX is s tr
iom: option message
iom_si, iom_s: state CX
iom_tr: list transition_item
Heqiom: empty_initial_message_or_final_output CX
  iom_tr iom
HtrX2: finite_valid_trace_init_to CX iom_si iom_s
  iom_tr
oom: option message
eqv: index
li: label (IM eqv)
Hs: valid_state_prop CX s
Hiom: option_valid_message_prop CX iom
Hv: valid (existT eqv li) (s, iom)
Hc: constraintX (existT eqv li) (s, iom)
si': state (IM eqv)
Hteqv: transition li (s eqv, iom) = (si', oom)
HinclE: VLSM_incl CE
  (preloaded_with_all_messages_vlsm FreeE)
eqv_state_is: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_start_project: equivocators_total_state_project
  IM eqv_state_is = is
eqv_state_s: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_state_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hstate_project: equivocators_total_trace_project IM
  eqv_state_tr = tr
Hstate_trace: finite_valid_trace_init_to CE
  eqv_state_is eqv_state_s eqv_state_tr
eqv_msg_is: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hmsg_start_project: equivocators_total_state_project
  IM eqv_msg_is = iom_si
eqv_msg_s: state
  (free_composite_vlsm (equivocator_IM IM))
eqv_msg_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hmsg_project: equivocators_total_trace_project IM
  eqv_msg_tr = iom_tr
Hmsg_trace: finite_valid_trace_init_to CE eqv_msg_is
  eqv_msg_s eqv_msg_tr
Hfinal_msg: finite_trace_last_output iom_tr =
finite_trace_last_output eqv_msg_tr
Hstate_valid: valid_state_prop CE eqv_state_s
emsg_tr: list transition_item
es: state CE
Hmsg_trace_full_replay: finite_valid_trace_from_to CE
  eqv_state_s es emsg_tr
Hemsg_tr_pr: equivocators_total_trace_project IM
  emsg_tr = []
Hes_pr: equivocators_total_state_project IM es = s
Hbs_iom: sent_except_from (equivocator_IM IM)
  initial_message_prop es iom
Happ: finite_valid_trace_from_to CE eqv_state_is es
  (eqv_state_tr ++ emsg_tr)
el:= existT eqv (ContinueWith 0 li): {i : index & label (equivocator_IM IM i)}
Happ_extend: finite_valid_trace_from_to CE
  eqv_state_is
  (state_update (equivocator_IM IM) es
     eqv
     (equivocator_state_update 
        (es eqv) 0 si'))
  ((eqv_state_tr ++ emsg_tr) ++
   [{|
      l := el;
      input := iom;
      destination :=
        state_update (equivocator_IM IM)
          es eqv
          (equivocator_state_update
             (es eqv) 0 si');
      output := oom
    |}])
Heqes: finite_trace_last eqv_state_is
  (eqv_state_tr ++ emsg_tr) = es
Hes_pr_i: ∀ i : index,
  equivocators_total_state_project IM es i =
  s i
Het: transition el (es, iom) =
(state_update (equivocator_IM IM) es eqv
   (equivocator_state_update (es eqv) 0 si'),
 oom)
Hes_pr_eqv: equivocator_state_zero (es eqv) = s eqv
Hivt: input_valid_transition CE el (
  es, iom)
  (state_update (equivocator_IM IM) es eqv
     (equivocator_state_update (es eqv) 0 si'),
   oom)
Hproject: equivocators_total_state_project IM
  (state_update (equivocator_IM IM) es eqv
     (equivocator_state_update 
        (es eqv) 0 si')) =
state_update IM s eqv si'
equivocators_total_state_project IM
  (state_update (equivocator_IM IM) es eqv
     (equivocator_state_update (es eqv) 0 si')) =
state_update IM s eqv si'
∧ (∃ tr0 : list
             (composite_transition_item
                (equivocator_IM IM)),
     equivocators_total_trace_project IM tr0 =
     tr ++
     [{|
        l := existT eqv li;
        input := iom;
        destination := state_update IM s eqv si';
        output := oom
      |}]
     ∧ finite_valid_trace_init_to CE eqv_state_is
         (state_update 
            (equivocator_IM IM) es eqv
            (equivocator_state_update (es eqv) 0 si'))
         tr0
       ∧ finite_trace_last_output
           (tr ++
            [{|
               l := existT eqv li;
               input := iom;
               destination :=
                 state_update IM s eqv si';
               output := oom
             |}]) = finite_trace_last_output tr0)
      {message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
seed: message → Prop
constraintX: composite_label IM
→ composite_state IM * option message
  → Prop
CX:= preloaded_vlsm (composite_vlsm IM constraintX)
  seed: VLSM message
constraintE: composite_label (equivocator_IM IM)
→ composite_state (equivocator_IM IM) *
  option message → Prop
CE:= preloaded_vlsm
  (composite_vlsm (equivocator_IM IM) constraintE)
  seed: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hsubsumption: zero_descriptor_constraint_lifting_prop
Hreplayable: replayable_message_prop
is, s: state CX
tr: list transition_item
HtrX1: finite_valid_trace_init_to CX is s tr
iom: option message
iom_si, iom_s: state CX
iom_tr: list transition_item
Heqiom: empty_initial_message_or_final_output CX
  iom_tr iom
HtrX2: finite_valid_trace_init_to CX iom_si iom_s
  iom_tr
oom: option message
eqv: index
li: label (IM eqv)
Hs: valid_state_prop CX s
Hiom: option_valid_message_prop CX iom
Hv: valid (existT eqv li) (s, iom)
Hc: constraintX (existT eqv li) (s, iom)
si': state (IM eqv)
Hteqv: transition li (s eqv, iom) = (si', oom)
HinclE: VLSM_incl CE
  (preloaded_with_all_messages_vlsm FreeE)
eqv_state_is: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_start_project: equivocators_total_state_project
  IM eqv_state_is = is
eqv_state_s: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_state_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hstate_project: equivocators_total_trace_project IM
  eqv_state_tr = tr
Hstate_trace: finite_valid_trace_init_to CE
  eqv_state_is eqv_state_s eqv_state_tr
eqv_msg_is: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hmsg_start_project: equivocators_total_state_project
  IM eqv_msg_is = iom_si
eqv_msg_s: state
  (free_composite_vlsm (equivocator_IM IM))
eqv_msg_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hmsg_project: equivocators_total_trace_project IM
  eqv_msg_tr = iom_tr
Hmsg_trace: finite_valid_trace_init_to CE eqv_msg_is
  eqv_msg_s eqv_msg_tr
Hfinal_msg: finite_trace_last_output iom_tr =
finite_trace_last_output eqv_msg_tr
Hstate_valid: valid_state_prop CE eqv_state_s
emsg_tr: list transition_item
es: state CE
Hmsg_trace_full_replay: finite_valid_trace_from_to CE
  eqv_state_s es emsg_tr
Hemsg_tr_pr: equivocators_total_trace_project IM
  emsg_tr = []
Hes_pr: equivocators_total_state_project IM es = s
Hbs_iom: sent_except_from (equivocator_IM IM)
  initial_message_prop es iom
Happ: finite_valid_trace_from_to CE eqv_state_is es
  (eqv_state_tr ++ emsg_tr)
el:= existT eqv (ContinueWith 0 li): {i : index & label (equivocator_IM IM i)}
Happ_extend: finite_valid_trace_from_to CE
  eqv_state_is
  (state_update (equivocator_IM IM) es
     eqv
     (equivocator_state_update 
        (es eqv) 0 si'))
  ((eqv_state_tr ++ emsg_tr) ++
   [{|
      l := el;
      input := iom;
      destination :=
        state_update (equivocator_IM IM)
          es eqv
          (equivocator_state_update
             (es eqv) 0 si');
      output := oom
    |}])
Heqes: finite_trace_last eqv_state_is
  (eqv_state_tr ++ emsg_tr) = es
Hes_pr_i: ∀ i : index,
  equivocators_total_state_project IM es i =
  s i
Het: transition el (es, iom) =
(state_update (equivocator_IM IM) es eqv
   (equivocator_state_update (es eqv) 0 si'),
 oom)
Hes_pr_eqv: equivocator_state_zero (es eqv) = s eqv
Hivt: input_valid_transition CE el (
  es, iom)
  (state_update (equivocator_IM IM) es eqv
     (equivocator_state_update (es eqv) 0 si'),
   oom)
equivocators_total_state_project IM
  (state_update (equivocator_IM IM) es eqv
     (equivocator_state_update (es eqv) 0 si')) =
state_update IM s eqv si' apply functional_extensionality_dep.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
seed: message → Prop
constraintX: composite_label IM
→ composite_state IM * option message
  → Prop
CX:= preloaded_vlsm (composite_vlsm IM constraintX)
  seed: VLSM message
constraintE: composite_label (equivocator_IM IM)
→ composite_state (equivocator_IM IM) *
  option message → Prop
CE:= preloaded_vlsm
  (composite_vlsm (equivocator_IM IM) constraintE)
  seed: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hsubsumption: zero_descriptor_constraint_lifting_prop
Hreplayable: replayable_message_prop
is, s: state CX
tr: list transition_item
HtrX1: finite_valid_trace_init_to CX is s tr
iom: option message
iom_si, iom_s: state CX
iom_tr: list transition_item
Heqiom: empty_initial_message_or_final_output CX
  iom_tr iom
HtrX2: finite_valid_trace_init_to CX iom_si iom_s
  iom_tr
oom: option message
eqv: index
li: label (IM eqv)
Hs: valid_state_prop CX s
Hiom: option_valid_message_prop CX iom
Hv: valid (existT eqv li) (s, iom)
Hc: constraintX (existT eqv li) (s, iom)
si': state (IM eqv)
Hteqv: transition li (s eqv, iom) = (si', oom)
HinclE: VLSM_incl CE
  (preloaded_with_all_messages_vlsm FreeE)
eqv_state_is: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_start_project: equivocators_total_state_project
  IM eqv_state_is = is
eqv_state_s: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_state_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hstate_project: equivocators_total_trace_project IM
  eqv_state_tr = tr
Hstate_trace: finite_valid_trace_init_to CE
  eqv_state_is eqv_state_s eqv_state_tr
eqv_msg_is: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hmsg_start_project: equivocators_total_state_project
  IM eqv_msg_is = iom_si
eqv_msg_s: state
  (free_composite_vlsm (equivocator_IM IM))
eqv_msg_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hmsg_project: equivocators_total_trace_project IM
  eqv_msg_tr = iom_tr
Hmsg_trace: finite_valid_trace_init_to CE eqv_msg_is
  eqv_msg_s eqv_msg_tr
Hfinal_msg: finite_trace_last_output iom_tr =
finite_trace_last_output eqv_msg_tr
Hstate_valid: valid_state_prop CE eqv_state_s
emsg_tr: list transition_item
es: state CE
Hmsg_trace_full_replay: finite_valid_trace_from_to CE
  eqv_state_s es emsg_tr
Hemsg_tr_pr: equivocators_total_trace_project IM
  emsg_tr = []
Hes_pr: equivocators_total_state_project IM es = s
Hbs_iom: sent_except_from (equivocator_IM IM)
  initial_message_prop es iom
Happ: finite_valid_trace_from_to CE eqv_state_is es
  (eqv_state_tr ++ emsg_tr)
el:= existT eqv (ContinueWith 0 li): {i : index & label (equivocator_IM IM i)}
Happ_extend: finite_valid_trace_from_to CE
  eqv_state_is
  (state_update (equivocator_IM IM) es
     eqv
     (equivocator_state_update 
        (es eqv) 0 si'))
  ((eqv_state_tr ++ emsg_tr) ++
   [{|
      l := el;
      input := iom;
      destination :=
        state_update (equivocator_IM IM)
          es eqv
          (equivocator_state_update
             (es eqv) 0 si');
      output := oom
    |}])
Heqes: finite_trace_last eqv_state_is
  (eqv_state_tr ++ emsg_tr) = es
Hes_pr_i: ∀ i : index,
  equivocators_total_state_project IM es i =
  s i
Het: transition el (es, iom) =
(state_update (equivocator_IM IM) es eqv
   (equivocator_state_update (es eqv) 0 si'),
 oom)
Hes_pr_eqv: equivocator_state_zero (es eqv) = s eqv
Hivt: input_valid_transition CE el (
  es, iom)
  (state_update (equivocator_IM IM) es eqv
     (equivocator_state_update (es eqv) 0 si'),
   oom)
∀ x : index,
  equivocators_total_state_project IM
    (state_update (equivocator_IM IM) es eqv
       (equivocator_state_update (es eqv) 0 si')) x =
  state_update IM s eqv si' x
        intro i.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
seed: message → Prop
constraintX: composite_label IM
→ composite_state IM * option message
  → Prop
CX:= preloaded_vlsm (composite_vlsm IM constraintX)
  seed: VLSM message
constraintE: composite_label (equivocator_IM IM)
→ composite_state (equivocator_IM IM) *
  option message → Prop
CE:= preloaded_vlsm
  (composite_vlsm (equivocator_IM IM) constraintE)
  seed: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hsubsumption: zero_descriptor_constraint_lifting_prop
Hreplayable: replayable_message_prop
is, s: state CX
tr: list transition_item
HtrX1: finite_valid_trace_init_to CX is s tr
iom: option message
iom_si, iom_s: state CX
iom_tr: list transition_item
Heqiom: empty_initial_message_or_final_output CX
  iom_tr iom
HtrX2: finite_valid_trace_init_to CX iom_si iom_s
  iom_tr
oom: option message
eqv: index
li: label (IM eqv)
Hs: valid_state_prop CX s
Hiom: option_valid_message_prop CX iom
Hv: valid (existT eqv li) (s, iom)
Hc: constraintX (existT eqv li) (s, iom)
si': state (IM eqv)
Hteqv: transition li (s eqv, iom) = (si', oom)
HinclE: VLSM_incl CE
  (preloaded_with_all_messages_vlsm FreeE)
eqv_state_is: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_start_project: equivocators_total_state_project
  IM eqv_state_is = is
eqv_state_s: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_state_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hstate_project: equivocators_total_trace_project IM
  eqv_state_tr = tr
Hstate_trace: finite_valid_trace_init_to CE
  eqv_state_is eqv_state_s eqv_state_tr
eqv_msg_is: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hmsg_start_project: equivocators_total_state_project
  IM eqv_msg_is = iom_si
eqv_msg_s: state
  (free_composite_vlsm (equivocator_IM IM))
eqv_msg_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hmsg_project: equivocators_total_trace_project IM
  eqv_msg_tr = iom_tr
Hmsg_trace: finite_valid_trace_init_to CE eqv_msg_is
  eqv_msg_s eqv_msg_tr
Hfinal_msg: finite_trace_last_output iom_tr =
finite_trace_last_output eqv_msg_tr
Hstate_valid: valid_state_prop CE eqv_state_s
emsg_tr: list transition_item
es: state CE
Hmsg_trace_full_replay: finite_valid_trace_from_to CE
  eqv_state_s es emsg_tr
Hemsg_tr_pr: equivocators_total_trace_project IM
  emsg_tr = []
Hes_pr: equivocators_total_state_project IM es = s
Hbs_iom: sent_except_from (equivocator_IM IM)
  initial_message_prop es iom
Happ: finite_valid_trace_from_to CE eqv_state_is es
  (eqv_state_tr ++ emsg_tr)
el:= existT eqv (ContinueWith 0 li): {i : index & label (equivocator_IM IM i)}
Happ_extend: finite_valid_trace_from_to CE
  eqv_state_is
  (state_update (equivocator_IM IM) es
     eqv
     (equivocator_state_update 
        (es eqv) 0 si'))
  ((eqv_state_tr ++ emsg_tr) ++
   [{|
      l := el;
      input := iom;
      destination :=
        state_update (equivocator_IM IM)
          es eqv
          (equivocator_state_update
             (es eqv) 0 si');
      output := oom
    |}])
Heqes: finite_trace_last eqv_state_is
  (eqv_state_tr ++ emsg_tr) = es
Hes_pr_i: ∀ i : index,
  equivocators_total_state_project IM es i =
  s i
Het: transition el (es, iom) =
(state_update (equivocator_IM IM) es eqv
   (equivocator_state_update (es eqv) 0 si'),
 oom)
Hes_pr_eqv: equivocator_state_zero (es eqv) = s eqv
Hivt: input_valid_transition CE el (
  es, iom)
  (state_update (equivocator_IM IM) es eqv
     (equivocator_state_update (es eqv) 0 si'),
   oom)
i: index
equivocators_total_state_project IM
  (state_update (equivocator_IM IM) es eqv
     (equivocator_state_update (es eqv) 0 si')) i =
state_update IM s eqv si' i
        unfold equivocators_total_state_project at 1.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
seed: message → Prop
constraintX: composite_label IM
→ composite_state IM * option message
  → Prop
CX:= preloaded_vlsm (composite_vlsm IM constraintX)
  seed: VLSM message
constraintE: composite_label (equivocator_IM IM)
→ composite_state (equivocator_IM IM) *
  option message → Prop
CE:= preloaded_vlsm
  (composite_vlsm (equivocator_IM IM) constraintE)
  seed: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hsubsumption: zero_descriptor_constraint_lifting_prop
Hreplayable: replayable_message_prop
is, s: state CX
tr: list transition_item
HtrX1: finite_valid_trace_init_to CX is s tr
iom: option message
iom_si, iom_s: state CX
iom_tr: list transition_item
Heqiom: empty_initial_message_or_final_output CX
  iom_tr iom
HtrX2: finite_valid_trace_init_to CX iom_si iom_s
  iom_tr
oom: option message
eqv: index
li: label (IM eqv)
Hs: valid_state_prop CX s
Hiom: option_valid_message_prop CX iom
Hv: valid (existT eqv li) (s, iom)
Hc: constraintX (existT eqv li) (s, iom)
si': state (IM eqv)
Hteqv: transition li (s eqv, iom) = (si', oom)
HinclE: VLSM_incl CE
  (preloaded_with_all_messages_vlsm FreeE)
eqv_state_is: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_start_project: equivocators_total_state_project
  IM eqv_state_is = is
eqv_state_s: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_state_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hstate_project: equivocators_total_trace_project IM
  eqv_state_tr = tr
Hstate_trace: finite_valid_trace_init_to CE
  eqv_state_is eqv_state_s eqv_state_tr
eqv_msg_is: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hmsg_start_project: equivocators_total_state_project
  IM eqv_msg_is = iom_si
eqv_msg_s: state
  (free_composite_vlsm (equivocator_IM IM))
eqv_msg_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hmsg_project: equivocators_total_trace_project IM
  eqv_msg_tr = iom_tr
Hmsg_trace: finite_valid_trace_init_to CE eqv_msg_is
  eqv_msg_s eqv_msg_tr
Hfinal_msg: finite_trace_last_output iom_tr =
finite_trace_last_output eqv_msg_tr
Hstate_valid: valid_state_prop CE eqv_state_s
emsg_tr: list transition_item
es: state CE
Hmsg_trace_full_replay: finite_valid_trace_from_to CE
  eqv_state_s es emsg_tr
Hemsg_tr_pr: equivocators_total_trace_project IM
  emsg_tr = []
Hes_pr: equivocators_total_state_project IM es = s
Hbs_iom: sent_except_from (equivocator_IM IM)
  initial_message_prop es iom
Happ: finite_valid_trace_from_to CE eqv_state_is es
  (eqv_state_tr ++ emsg_tr)
el:= existT eqv (ContinueWith 0 li): {i : index & label (equivocator_IM IM i)}
Happ_extend: finite_valid_trace_from_to CE
  eqv_state_is
  (state_update (equivocator_IM IM) es
     eqv
     (equivocator_state_update 
        (es eqv) 0 si'))
  ((eqv_state_tr ++ emsg_tr) ++
   [{|
      l := el;
      input := iom;
      destination :=
        state_update (equivocator_IM IM)
          es eqv
          (equivocator_state_update
             (es eqv) 0 si');
      output := oom
    |}])
Heqes: finite_trace_last eqv_state_is
  (eqv_state_tr ++ emsg_tr) = es
Hes_pr_i: ∀ i : index,
  equivocators_total_state_project IM es i =
  s i
Het: transition el (es, iom) =
(state_update (equivocator_IM IM) es eqv
   (equivocator_state_update (es eqv) 0 si'),
 oom)
Hes_pr_eqv: equivocator_state_zero (es eqv) = s eqv
Hivt: input_valid_transition CE el (
  es, iom)
  (state_update (equivocator_IM IM) es eqv
     (equivocator_state_update (es eqv) 0 si'),
   oom)
i: index
equivocators_state_project IM (zero_descriptor IM)
  (state_update (equivocator_IM IM) es eqv
     (equivocator_state_update (es eqv) 0 si')) i =
state_update IM s eqv si' i
        unfold equivocators_state_project.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
seed: message → Prop
constraintX: composite_label IM
→ composite_state IM * option message
  → Prop
CX:= preloaded_vlsm (composite_vlsm IM constraintX)
  seed: VLSM message
constraintE: composite_label (equivocator_IM IM)
→ composite_state (equivocator_IM IM) *
  option message → Prop
CE:= preloaded_vlsm
  (composite_vlsm (equivocator_IM IM) constraintE)
  seed: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hsubsumption: zero_descriptor_constraint_lifting_prop
Hreplayable: replayable_message_prop
is, s: state CX
tr: list transition_item
HtrX1: finite_valid_trace_init_to CX is s tr
iom: option message
iom_si, iom_s: state CX
iom_tr: list transition_item
Heqiom: empty_initial_message_or_final_output CX
  iom_tr iom
HtrX2: finite_valid_trace_init_to CX iom_si iom_s
  iom_tr
oom: option message
eqv: index
li: label (IM eqv)
Hs: valid_state_prop CX s
Hiom: option_valid_message_prop CX iom
Hv: valid (existT eqv li) (s, iom)
Hc: constraintX (existT eqv li) (s, iom)
si': state (IM eqv)
Hteqv: transition li (s eqv, iom) = (si', oom)
HinclE: VLSM_incl CE
  (preloaded_with_all_messages_vlsm FreeE)
eqv_state_is: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_start_project: equivocators_total_state_project
  IM eqv_state_is = is
eqv_state_s: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_state_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hstate_project: equivocators_total_trace_project IM
  eqv_state_tr = tr
Hstate_trace: finite_valid_trace_init_to CE
  eqv_state_is eqv_state_s eqv_state_tr
eqv_msg_is: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hmsg_start_project: equivocators_total_state_project
  IM eqv_msg_is = iom_si
eqv_msg_s: state
  (free_composite_vlsm (equivocator_IM IM))
eqv_msg_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hmsg_project: equivocators_total_trace_project IM
  eqv_msg_tr = iom_tr
Hmsg_trace: finite_valid_trace_init_to CE eqv_msg_is
  eqv_msg_s eqv_msg_tr
Hfinal_msg: finite_trace_last_output iom_tr =
finite_trace_last_output eqv_msg_tr
Hstate_valid: valid_state_prop CE eqv_state_s
emsg_tr: list transition_item
es: state CE
Hmsg_trace_full_replay: finite_valid_trace_from_to CE
  eqv_state_s es emsg_tr
Hemsg_tr_pr: equivocators_total_trace_project IM
  emsg_tr = []
Hes_pr: equivocators_total_state_project IM es = s
Hbs_iom: sent_except_from (equivocator_IM IM)
  initial_message_prop es iom
Happ: finite_valid_trace_from_to CE eqv_state_is es
  (eqv_state_tr ++ emsg_tr)
el:= existT eqv (ContinueWith 0 li): {i : index & label (equivocator_IM IM i)}
Happ_extend: finite_valid_trace_from_to CE
  eqv_state_is
  (state_update (equivocator_IM IM) es
     eqv
     (equivocator_state_update 
        (es eqv) 0 si'))
  ((eqv_state_tr ++ emsg_tr) ++
   [{|
      l := el;
      input := iom;
      destination :=
        state_update (equivocator_IM IM)
          es eqv
          (equivocator_state_update
             (es eqv) 0 si');
      output := oom
    |}])
Heqes: finite_trace_last eqv_state_is
  (eqv_state_tr ++ emsg_tr) = es
Hes_pr_i: ∀ i : index,
  equivocators_total_state_project IM es i =
  s i
Het: transition el (es, iom) =
(state_update (equivocator_IM IM) es eqv
   (equivocator_state_update (es eqv) 0 si'),
 oom)
Hes_pr_eqv: equivocator_state_zero (es eqv) = s eqv
Hivt: input_valid_transition CE el (
  es, iom)
  (state_update (equivocator_IM IM) es eqv
     (equivocator_state_update (es eqv) 0 si'),
   oom)
i: index
equivocator_state_descriptor_project
  (state_update (equivocator_IM IM) es eqv
     (equivocator_state_update (es eqv) 0 si') i)
  (zero_descriptor IM i) = state_update IM s eqv si' i
        by destruct (decide (i = eqv)); subst; state_update_simpl; [| apply Hes_pr_i].
      }message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
seed: message → Prop
constraintX: composite_label IM
→ composite_state IM * option message
  → Prop
CX:= preloaded_vlsm (composite_vlsm IM constraintX)
  seed: VLSM message
constraintE: composite_label (equivocator_IM IM)
→ composite_state (equivocator_IM IM) *
  option message → Prop
CE:= preloaded_vlsm
  (composite_vlsm (equivocator_IM IM) constraintE)
  seed: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hsubsumption: zero_descriptor_constraint_lifting_prop
Hreplayable: replayable_message_prop
is, s: state CX
tr: list transition_item
HtrX1: finite_valid_trace_init_to CX is s tr
iom: option message
iom_si, iom_s: state CX
iom_tr: list transition_item
Heqiom: empty_initial_message_or_final_output CX
  iom_tr iom
HtrX2: finite_valid_trace_init_to CX iom_si iom_s
  iom_tr
oom: option message
eqv: index
li: label (IM eqv)
Hs: valid_state_prop CX s
Hiom: option_valid_message_prop CX iom
Hv: valid (existT eqv li) (s, iom)
Hc: constraintX (existT eqv li) (s, iom)
si': state (IM eqv)
Hteqv: transition li (s eqv, iom) = (si', oom)
HinclE: VLSM_incl CE
  (preloaded_with_all_messages_vlsm FreeE)
eqv_state_is: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_start_project: equivocators_total_state_project
  IM eqv_state_is = is
eqv_state_s: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_state_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hstate_project: equivocators_total_trace_project IM
  eqv_state_tr = tr
Hstate_trace: finite_valid_trace_init_to CE
  eqv_state_is eqv_state_s eqv_state_tr
eqv_msg_is: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hmsg_start_project: equivocators_total_state_project
  IM eqv_msg_is = iom_si
eqv_msg_s: state
  (free_composite_vlsm (equivocator_IM IM))
eqv_msg_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hmsg_project: equivocators_total_trace_project IM
  eqv_msg_tr = iom_tr
Hmsg_trace: finite_valid_trace_init_to CE eqv_msg_is
  eqv_msg_s eqv_msg_tr
Hfinal_msg: finite_trace_last_output iom_tr =
finite_trace_last_output eqv_msg_tr
Hstate_valid: valid_state_prop CE eqv_state_s
emsg_tr: list transition_item
es: state CE
Hmsg_trace_full_replay: finite_valid_trace_from_to CE
  eqv_state_s es emsg_tr
Hemsg_tr_pr: equivocators_total_trace_project IM
  emsg_tr = []
Hes_pr: equivocators_total_state_project IM es = s
Hbs_iom: sent_except_from (equivocator_IM IM)
  initial_message_prop es iom
Happ: finite_valid_trace_from_to CE eqv_state_is es
  (eqv_state_tr ++ emsg_tr)
el:= existT eqv (ContinueWith 0 li): {i : index & label (equivocator_IM IM i)}
Happ_extend: finite_valid_trace_from_to CE
  eqv_state_is
  (state_update (equivocator_IM IM) es
     eqv
     (equivocator_state_update 
        (es eqv) 0 si'))
  ((eqv_state_tr ++ emsg_tr) ++
   [{|
      l := el;
      input := iom;
      destination :=
        state_update (equivocator_IM IM)
          es eqv
          (equivocator_state_update
             (es eqv) 0 si');
      output := oom
    |}])
Heqes: finite_trace_last eqv_state_is
  (eqv_state_tr ++ emsg_tr) = es
Hes_pr_i: ∀ i : index,
  equivocators_total_state_project IM es i =
  s i
Het: transition el (es, iom) =
(state_update (equivocator_IM IM) es eqv
   (equivocator_state_update (es eqv) 0 si'),
 oom)
Hes_pr_eqv: equivocator_state_zero (es eqv) = s eqv
Hivt: input_valid_transition CE el (
  es, iom)
  (state_update (equivocator_IM IM) es eqv
     (equivocator_state_update (es eqv) 0 si'),
   oom)
Hproject: equivocators_total_state_project IM
  (state_update (equivocator_IM IM) es eqv
     (equivocator_state_update 
        (es eqv) 0 si')) =
state_update IM s eqv si'
equivocators_total_state_project IM
  (state_update (equivocator_IM IM) es eqv
     (equivocator_state_update (es eqv) 0 si')) =
state_update IM s eqv si'
∧ (∃ tr0 : list
             (composite_transition_item
                (equivocator_IM IM)),
     equivocators_total_trace_project IM tr0 =
     tr ++
     [{|
        l := existT eqv li;
        input := iom;
        destination := state_update IM s eqv si';
        output := oom
      |}]
     ∧ finite_valid_trace_init_to CE eqv_state_is
         (state_update (equivocator_IM IM) es eqv
            (equivocator_state_update (es eqv) 0 si'))
         tr0
       ∧ finite_trace_last_output
           (tr ++
            [{|
               l := existT eqv li;
               input := iom;
               destination :=
                 state_update IM s eqv si';
               output := oom
             |}]) = finite_trace_last_output tr0)
      split; [done |].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
seed: message → Prop
constraintX: composite_label IM
→ composite_state IM * option message
  → Prop
CX:= preloaded_vlsm (composite_vlsm IM constraintX)
  seed: VLSM message
constraintE: composite_label (equivocator_IM IM)
→ composite_state (equivocator_IM IM) *
  option message → Prop
CE:= preloaded_vlsm
  (composite_vlsm (equivocator_IM IM) constraintE)
  seed: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hsubsumption: zero_descriptor_constraint_lifting_prop
Hreplayable: replayable_message_prop
is, s: state CX
tr: list transition_item
HtrX1: finite_valid_trace_init_to CX is s tr
iom: option message
iom_si, iom_s: state CX
iom_tr: list transition_item
Heqiom: empty_initial_message_or_final_output CX
  iom_tr iom
HtrX2: finite_valid_trace_init_to CX iom_si iom_s
  iom_tr
oom: option message
eqv: index
li: label (IM eqv)
Hs: valid_state_prop CX s
Hiom: option_valid_message_prop CX iom
Hv: valid (existT eqv li) (s, iom)
Hc: constraintX (existT eqv li) (s, iom)
si': state (IM eqv)
Hteqv: transition li (s eqv, iom) = (si', oom)
HinclE: VLSM_incl CE
  (preloaded_with_all_messages_vlsm FreeE)
eqv_state_is: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_start_project: equivocators_total_state_project
  IM eqv_state_is = is
eqv_state_s: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_state_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hstate_project: equivocators_total_trace_project IM
  eqv_state_tr = tr
Hstate_trace: finite_valid_trace_init_to CE
  eqv_state_is eqv_state_s eqv_state_tr
eqv_msg_is: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hmsg_start_project: equivocators_total_state_project
  IM eqv_msg_is = iom_si
eqv_msg_s: state
  (free_composite_vlsm (equivocator_IM IM))
eqv_msg_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hmsg_project: equivocators_total_trace_project IM
  eqv_msg_tr = iom_tr
Hmsg_trace: finite_valid_trace_init_to CE eqv_msg_is
  eqv_msg_s eqv_msg_tr
Hfinal_msg: finite_trace_last_output iom_tr =
finite_trace_last_output eqv_msg_tr
Hstate_valid: valid_state_prop CE eqv_state_s
emsg_tr: list transition_item
es: state CE
Hmsg_trace_full_replay: finite_valid_trace_from_to CE
  eqv_state_s es emsg_tr
Hemsg_tr_pr: equivocators_total_trace_project IM
  emsg_tr = []
Hes_pr: equivocators_total_state_project IM es = s
Hbs_iom: sent_except_from (equivocator_IM IM)
  initial_message_prop es iom
Happ: finite_valid_trace_from_to CE eqv_state_is es
  (eqv_state_tr ++ emsg_tr)
el:= existT eqv (ContinueWith 0 li): {i : index & label (equivocator_IM IM i)}
Happ_extend: finite_valid_trace_from_to CE
  eqv_state_is
  (state_update (equivocator_IM IM) es
     eqv
     (equivocator_state_update 
        (es eqv) 0 si'))
  ((eqv_state_tr ++ emsg_tr) ++
   [{|
      l := el;
      input := iom;
      destination :=
        state_update (equivocator_IM IM)
          es eqv
          (equivocator_state_update
             (es eqv) 0 si');
      output := oom
    |}])
Heqes: finite_trace_last eqv_state_is
  (eqv_state_tr ++ emsg_tr) = es
Hes_pr_i: ∀ i : index,
  equivocators_total_state_project IM es i =
  s i
Het: transition el (es, iom) =
(state_update (equivocator_IM IM) es eqv
   (equivocator_state_update (es eqv) 0 si'),
 oom)
Hes_pr_eqv: equivocator_state_zero (es eqv) = s eqv
Hivt: input_valid_transition CE el (
  es, iom)
  (state_update (equivocator_IM IM) es eqv
     (equivocator_state_update (es eqv) 0 si'),
   oom)
Hproject: equivocators_total_state_project IM
  (state_update (equivocator_IM IM) es eqv
     (equivocator_state_update 
        (es eqv) 0 si')) =
state_update IM s eqv si'
∃ tr0 : list
          (composite_transition_item
             (equivocator_IM IM)),
  equivocators_total_trace_project IM tr0 =
  tr ++
  [{|
     l := existT eqv li;
     input := iom;
     destination := state_update IM s eqv si';
     output := oom
   |}]
  ∧ finite_valid_trace_init_to CE eqv_state_is
      (state_update (equivocator_IM IM) es eqv
         (equivocator_state_update (es eqv) 0 si'))
      tr0
    ∧ finite_trace_last_output
        (tr ++
         [{|
            l := existT eqv li;
            input := iom;
            destination := state_update IM s eqv si';
            output := oom
          |}]) = finite_trace_last_output tr0
      eexists; split; [| split; [split; [done |] |]].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
seed: message → Prop
constraintX: composite_label IM
→ composite_state IM * option message
  → Prop
CX:= preloaded_vlsm (composite_vlsm IM constraintX)
  seed: VLSM message
constraintE: composite_label (equivocator_IM IM)
→ composite_state (equivocator_IM IM) *
  option message → Prop
CE:= preloaded_vlsm
  (composite_vlsm (equivocator_IM IM) constraintE)
  seed: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hsubsumption: zero_descriptor_constraint_lifting_prop
Hreplayable: replayable_message_prop
is, s: state CX
tr: list transition_item
HtrX1: finite_valid_trace_init_to CX is s tr
iom: option message
iom_si, iom_s: state CX
iom_tr: list transition_item
Heqiom: empty_initial_message_or_final_output CX
  iom_tr iom
HtrX2: finite_valid_trace_init_to CX iom_si iom_s
  iom_tr
oom: option message
eqv: index
li: label (IM eqv)
Hs: valid_state_prop CX s
Hiom: option_valid_message_prop CX iom
Hv: valid (existT eqv li) (s, iom)
Hc: constraintX (existT eqv li) (s, iom)
si': state (IM eqv)
Hteqv: transition li (s eqv, iom) = (si', oom)
HinclE: VLSM_incl CE
  (preloaded_with_all_messages_vlsm FreeE)
eqv_state_is: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_start_project: equivocators_total_state_project
  IM eqv_state_is = is
eqv_state_s: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_state_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hstate_project: equivocators_total_trace_project IM
  eqv_state_tr = tr
Hstate_trace: finite_valid_trace_init_to CE
  eqv_state_is eqv_state_s eqv_state_tr
eqv_msg_is: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hmsg_start_project: equivocators_total_state_project
  IM eqv_msg_is = iom_si
eqv_msg_s: state
  (free_composite_vlsm (equivocator_IM IM))
eqv_msg_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hmsg_project: equivocators_total_trace_project IM
  eqv_msg_tr = iom_tr
Hmsg_trace: finite_valid_trace_init_to CE eqv_msg_is
  eqv_msg_s eqv_msg_tr
Hfinal_msg: finite_trace_last_output iom_tr =
finite_trace_last_output eqv_msg_tr
Hstate_valid: valid_state_prop CE eqv_state_s
emsg_tr: list transition_item
es: state CE
Hmsg_trace_full_replay: finite_valid_trace_from_to CE
  eqv_state_s es emsg_tr
Hemsg_tr_pr: equivocators_total_trace_project IM
  emsg_tr = []
Hes_pr: equivocators_total_state_project IM es = s
Hbs_iom: sent_except_from (equivocator_IM IM)
  initial_message_prop es iom
Happ: finite_valid_trace_from_to CE eqv_state_is es
  (eqv_state_tr ++ emsg_tr)
el:= existT eqv (ContinueWith 0 li): {i : index & label (equivocator_IM IM i)}
Happ_extend: finite_valid_trace_from_to CE
  eqv_state_is
  (state_update (equivocator_IM IM) es
     eqv
     (equivocator_state_update 
        (es eqv) 0 si'))
  ((eqv_state_tr ++ emsg_tr) ++
   [{|
      l := el;
      input := iom;
      destination :=
        state_update (equivocator_IM IM)
          es eqv
          (equivocator_state_update
             (es eqv) 0 si');
      output := oom
    |}])
Heqes: finite_trace_last eqv_state_is
  (eqv_state_tr ++ emsg_tr) = es
Hes_pr_i: ∀ i : index,
  equivocators_total_state_project IM es i =
  s i
Het: transition el (es, iom) =
(state_update (equivocator_IM IM) es eqv
   (equivocator_state_update (es eqv) 0 si'),
 oom)
Hes_pr_eqv: equivocator_state_zero (es eqv) = s eqv
Hivt: input_valid_transition CE el (
  es, iom)
  (state_update (equivocator_IM IM) es eqv
     (equivocator_state_update (es eqv) 0 si'),
   oom)
Hproject: equivocators_total_state_project IM
  (state_update (equivocator_IM IM) es eqv
     (equivocator_state_update 
        (es eqv) 0 si')) =
state_update IM s eqv si'
equivocators_total_trace_project IM
  ((eqv_state_tr ++ emsg_tr) ++
   [{|
      l := el;
      input := iom;
      destination :=
        state_update (equivocator_IM IM) es eqv
          (equivocator_state_update (es eqv) 0 si');
      output := oom
    |}]) =
tr ++
[{|
   l := existT eqv li;
   input := iom;
   destination := state_update IM s eqv si';
   output := oom
 |}]
message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
seed: message → Prop
constraintX: composite_label IM
→ composite_state IM * option message
  → Prop
CX:= preloaded_vlsm (composite_vlsm IM constraintX)
  seed: VLSM message
constraintE: composite_label (equivocator_IM IM)
→ composite_state (equivocator_IM IM) *
  option message → Prop
CE:= preloaded_vlsm
  (composite_vlsm (equivocator_IM IM) constraintE)
  seed: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hsubsumption: zero_descriptor_constraint_lifting_prop
Hreplayable: replayable_message_prop
is, s: state CX
tr: list transition_item
HtrX1: finite_valid_trace_init_to CX is s tr
iom: option message
iom_si, iom_s: state CX
iom_tr: list transition_item
Heqiom: empty_initial_message_or_final_output CX
  iom_tr iom
HtrX2: finite_valid_trace_init_to CX iom_si iom_s
  iom_tr
oom: option message
eqv: index
li: label (IM eqv)
Hs: valid_state_prop CX s
Hiom: option_valid_message_prop CX iom
Hv: valid (existT eqv li) (s, iom)
Hc: constraintX (existT eqv li) (s, iom)
si': state (IM eqv)
Hteqv: transition li (s eqv, iom) = (si', oom)
HinclE: VLSM_incl CE
  (preloaded_with_all_messages_vlsm FreeE)
eqv_state_is: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_start_project: equivocators_total_state_project
  IM eqv_state_is = is
eqv_state_s: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_state_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hstate_project: equivocators_total_trace_project IM
  eqv_state_tr = tr
Hstate_trace: finite_valid_trace_init_to CE
  eqv_state_is eqv_state_s eqv_state_tr
eqv_msg_is: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hmsg_start_project: equivocators_total_state_project
  IM eqv_msg_is = iom_si
eqv_msg_s: state
  (free_composite_vlsm (equivocator_IM IM))
eqv_msg_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hmsg_project: equivocators_total_trace_project IM
  eqv_msg_tr = iom_tr
Hmsg_trace: finite_valid_trace_init_to CE eqv_msg_is
  eqv_msg_s eqv_msg_tr
Hfinal_msg: finite_trace_last_output iom_tr =
finite_trace_last_output eqv_msg_tr
Hstate_valid: valid_state_prop CE eqv_state_s
emsg_tr: list transition_item
es: state CE
Hmsg_trace_full_replay: finite_valid_trace_from_to CE
  eqv_state_s es emsg_tr
Hemsg_tr_pr: equivocators_total_trace_project IM
  emsg_tr = []
Hes_pr: equivocators_total_state_project IM es = s
Hbs_iom: sent_except_from (equivocator_IM IM)
  initial_message_prop es iom
Happ: finite_valid_trace_from_to CE eqv_state_is es
  (eqv_state_tr ++ emsg_tr)
el:= existT eqv (ContinueWith 0 li): {i : index & label (equivocator_IM IM i)}
Happ_extend: finite_valid_trace_from_to CE
  eqv_state_is
  (state_update (equivocator_IM IM) es
     eqv
     (equivocator_state_update 
        (es eqv) 0 si'))
  ((eqv_state_tr ++ emsg_tr) ++
   [{|
      l := el;
      input := iom;
      destination :=
        state_update (equivocator_IM IM)
          es eqv
          (equivocator_state_update
             (es eqv) 0 si');
      output := oom
    |}])
Heqes: finite_trace_last eqv_state_is
  (eqv_state_tr ++ emsg_tr) = es
Hes_pr_i: ∀ i : index,
  equivocators_total_state_project IM es i =
  s i
Het: transition el (es, iom) =
(state_update (equivocator_IM IM) es eqv
   (equivocator_state_update (es eqv) 0 si'),
 oom)
Hes_pr_eqv: equivocator_state_zero (es eqv) = s eqv
Hivt: input_valid_transition CE el (
  es, iom)
  (state_update (equivocator_IM IM) es eqv
     (equivocator_state_update (es eqv) 0 si'),
   oom)
Hproject: equivocators_total_state_project IM
  (state_update (equivocator_IM IM) es eqv
     (equivocator_state_update 
        (es eqv) 0 si')) =
state_update IM s eqv si'
initial_state_prop eqv_state_is
message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
seed: message → Prop
constraintX: composite_label IM
→ composite_state IM * option message
  → Prop
CX:= preloaded_vlsm (composite_vlsm IM constraintX)
  seed: VLSM message
constraintE: composite_label (equivocator_IM IM)
→ composite_state (equivocator_IM IM) *
  option message → Prop
CE:= preloaded_vlsm
  (composite_vlsm (equivocator_IM IM) constraintE)
  seed: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hsubsumption: zero_descriptor_constraint_lifting_prop
Hreplayable: replayable_message_prop
is, s: state CX
tr: list transition_item
HtrX1: finite_valid_trace_init_to CX is s tr
iom: option message
iom_si, iom_s: state CX
iom_tr: list transition_item
Heqiom: empty_initial_message_or_final_output CX
  iom_tr iom
HtrX2: finite_valid_trace_init_to CX iom_si iom_s
  iom_tr
oom: option message
eqv: index
li: label (IM eqv)
Hs: valid_state_prop CX s
Hiom: option_valid_message_prop CX iom
Hv: valid (existT eqv li) (s, iom)
Hc: constraintX (existT eqv li) (s, iom)
si': state (IM eqv)
Hteqv: transition li (s eqv, iom) = (si', oom)
HinclE: VLSM_incl CE
  (preloaded_with_all_messages_vlsm FreeE)
eqv_state_is: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_start_project: equivocators_total_state_project
  IM eqv_state_is = is
eqv_state_s: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_state_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hstate_project: equivocators_total_trace_project IM
  eqv_state_tr = tr
Hstate_trace: finite_valid_trace_init_to CE
  eqv_state_is eqv_state_s eqv_state_tr
eqv_msg_is: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hmsg_start_project: equivocators_total_state_project
  IM eqv_msg_is = iom_si
eqv_msg_s: state
  (free_composite_vlsm (equivocator_IM IM))
eqv_msg_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hmsg_project: equivocators_total_trace_project IM
  eqv_msg_tr = iom_tr
Hmsg_trace: finite_valid_trace_init_to CE eqv_msg_is
  eqv_msg_s eqv_msg_tr
Hfinal_msg: finite_trace_last_output iom_tr =
finite_trace_last_output eqv_msg_tr
Hstate_valid: valid_state_prop CE eqv_state_s
emsg_tr: list transition_item
es: state CE
Hmsg_trace_full_replay: finite_valid_trace_from_to CE
  eqv_state_s es emsg_tr
Hemsg_tr_pr: equivocators_total_trace_project IM
  emsg_tr = []
Hes_pr: equivocators_total_state_project IM es = s
Hbs_iom: sent_except_from (equivocator_IM IM)
  initial_message_prop es iom
Happ: finite_valid_trace_from_to CE eqv_state_is es
  (eqv_state_tr ++ emsg_tr)
el:= existT eqv (ContinueWith 0 li): {i : index & label (equivocator_IM IM i)}
Happ_extend: finite_valid_trace_from_to CE
  eqv_state_is
  (state_update (equivocator_IM IM) es
     eqv
     (equivocator_state_update 
        (es eqv) 0 si'))
  ((eqv_state_tr ++ emsg_tr) ++
   [{|
      l := el;
      input := iom;
      destination :=
        state_update (equivocator_IM IM)
          es eqv
          (equivocator_state_update
             (es eqv) 0 si');
      output := oom
    |}])
Heqes: finite_trace_last eqv_state_is
  (eqv_state_tr ++ emsg_tr) = es
Hes_pr_i: ∀ i : index,
  equivocators_total_state_project IM es i =
  s i
Het: transition el (es, iom) =
(state_update (equivocator_IM IM) es eqv
   (equivocator_state_update (es eqv) 0 si'),
 oom)
Hes_pr_eqv: equivocator_state_zero (es eqv) = s eqv
Hivt: input_valid_transition CE el (
  es, iom)
  (state_update (equivocator_IM IM) es eqv
     (equivocator_state_update (es eqv) 0 si'),
   oom)
Hproject: equivocators_total_state_project IM
  (state_update (equivocator_IM IM) es eqv
     (equivocator_state_update 
        (es eqv) 0 si')) =
state_update IM s eqv si'
finite_trace_last_output
  (tr ++
   [{|
      l := existT eqv li;
      input := iom;
      destination := state_update IM s eqv si';
      output := oom
    |}]) =
finite_trace_last_output
  ((eqv_state_tr ++ emsg_tr) ++
   [{|
      l := el;
      input := iom;
      destination :=
        state_update (equivocator_IM IM) es eqv
          (equivocator_state_update (es eqv) 0 si');
      output := oom
    |}])
      -message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
seed: message → Prop
constraintX: composite_label IM
→ composite_state IM * option message
  → Prop
CX:= preloaded_vlsm (composite_vlsm IM constraintX)
  seed: VLSM message
constraintE: composite_label (equivocator_IM IM)
→ composite_state (equivocator_IM IM) *
  option message → Prop
CE:= preloaded_vlsm
  (composite_vlsm (equivocator_IM IM) constraintE)
  seed: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hsubsumption: zero_descriptor_constraint_lifting_prop
Hreplayable: replayable_message_prop
is, s: state CX
tr: list transition_item
HtrX1: finite_valid_trace_init_to CX is s tr
iom: option message
iom_si, iom_s: state CX
iom_tr: list transition_item
Heqiom: empty_initial_message_or_final_output CX
  iom_tr iom
HtrX2: finite_valid_trace_init_to CX iom_si iom_s
  iom_tr
oom: option message
eqv: index
li: label (IM eqv)
Hs: valid_state_prop CX s
Hiom: option_valid_message_prop CX iom
Hv: valid (existT eqv li) (s, iom)
Hc: constraintX (existT eqv li) (s, iom)
si': state (IM eqv)
Hteqv: transition li (s eqv, iom) = (si', oom)
HinclE: VLSM_incl CE
  (preloaded_with_all_messages_vlsm FreeE)
eqv_state_is: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_start_project: equivocators_total_state_project
  IM eqv_state_is = is
eqv_state_s: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_state_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hstate_project: equivocators_total_trace_project IM
  eqv_state_tr = tr
Hstate_trace: finite_valid_trace_init_to CE
  eqv_state_is eqv_state_s eqv_state_tr
eqv_msg_is: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hmsg_start_project: equivocators_total_state_project
  IM eqv_msg_is = iom_si
eqv_msg_s: state
  (free_composite_vlsm (equivocator_IM IM))
eqv_msg_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hmsg_project: equivocators_total_trace_project IM
  eqv_msg_tr = iom_tr
Hmsg_trace: finite_valid_trace_init_to CE eqv_msg_is
  eqv_msg_s eqv_msg_tr
Hfinal_msg: finite_trace_last_output iom_tr =
finite_trace_last_output eqv_msg_tr
Hstate_valid: valid_state_prop CE eqv_state_s
emsg_tr: list transition_item
es: state CE
Hmsg_trace_full_replay: finite_valid_trace_from_to CE
  eqv_state_s es emsg_tr
Hemsg_tr_pr: equivocators_total_trace_project IM
  emsg_tr = []
Hes_pr: equivocators_total_state_project IM es = s
Hbs_iom: sent_except_from (equivocator_IM IM)
  initial_message_prop es iom
Happ: finite_valid_trace_from_to CE eqv_state_is es
  (eqv_state_tr ++ emsg_tr)
el:= existT eqv (ContinueWith 0 li): {i : index & label (equivocator_IM IM i)}
Happ_extend: finite_valid_trace_from_to CE
  eqv_state_is
  (state_update (equivocator_IM IM) es
     eqv
     (equivocator_state_update 
        (es eqv) 0 si'))
  ((eqv_state_tr ++ emsg_tr) ++
   [{|
      l := el;
      input := iom;
      destination :=
        state_update (equivocator_IM IM)
          es eqv
          (equivocator_state_update
             (es eqv) 0 si');
      output := oom
    |}])
Heqes: finite_trace_last eqv_state_is
  (eqv_state_tr ++ emsg_tr) = es
Hes_pr_i: ∀ i : index,
  equivocators_total_state_project IM es i =
  s i
Het: transition el (es, iom) =
(state_update (equivocator_IM IM) es eqv
   (equivocator_state_update (es eqv) 0 si'),
 oom)
Hes_pr_eqv: equivocator_state_zero (es eqv) = s eqv
Hivt: input_valid_transition CE el (
  es, iom)
  (state_update (equivocator_IM IM) es eqv
     (equivocator_state_update (es eqv) 0 si'),
   oom)
Hproject: equivocators_total_state_project IM
  (state_update (equivocator_IM IM) es eqv
     (equivocator_state_update 
        (es eqv) 0 si')) =
state_update IM s eqv si'
equivocators_total_trace_project IM
  ((eqv_state_tr ++ emsg_tr) ++
   [{|
      l := el;
      input := iom;
      destination :=
        state_update (equivocator_IM IM) es eqv
          (equivocator_state_update (es eqv) 0 si');
      output := oom
    |}]) =
tr ++
[{|
   l := existT eqv li;
   input := iom;
   destination := state_update IM s eqv si';
   output := oom
 |}] apply valid_trace_forget_last in Happ_extend.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
seed: message → Prop
constraintX: composite_label IM
→ composite_state IM * option message
  → Prop
CX:= preloaded_vlsm (composite_vlsm IM constraintX)
  seed: VLSM message
constraintE: composite_label (equivocator_IM IM)
→ composite_state (equivocator_IM IM) *
  option message → Prop
CE:= preloaded_vlsm
  (composite_vlsm (equivocator_IM IM) constraintE)
  seed: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hsubsumption: zero_descriptor_constraint_lifting_prop
Hreplayable: replayable_message_prop
is, s: state CX
tr: list transition_item
HtrX1: finite_valid_trace_init_to CX is s tr
iom: option message
iom_si, iom_s: state CX
iom_tr: list transition_item
Heqiom: empty_initial_message_or_final_output CX
  iom_tr iom
HtrX2: finite_valid_trace_init_to CX iom_si iom_s
  iom_tr
oom: option message
eqv: index
li: label (IM eqv)
Hs: valid_state_prop CX s
Hiom: option_valid_message_prop CX iom
Hv: valid (existT eqv li) (s, iom)
Hc: constraintX (existT eqv li) (s, iom)
si': state (IM eqv)
Hteqv: transition li (s eqv, iom) = (si', oom)
HinclE: VLSM_incl CE
  (preloaded_with_all_messages_vlsm FreeE)
eqv_state_is: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_start_project: equivocators_total_state_project
  IM eqv_state_is = is
eqv_state_s: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_state_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hstate_project: equivocators_total_trace_project IM
  eqv_state_tr = tr
Hstate_trace: finite_valid_trace_init_to CE
  eqv_state_is eqv_state_s eqv_state_tr
eqv_msg_is: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hmsg_start_project: equivocators_total_state_project
  IM eqv_msg_is = iom_si
eqv_msg_s: state
  (free_composite_vlsm (equivocator_IM IM))
eqv_msg_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hmsg_project: equivocators_total_trace_project IM
  eqv_msg_tr = iom_tr
Hmsg_trace: finite_valid_trace_init_to CE eqv_msg_is
  eqv_msg_s eqv_msg_tr
Hfinal_msg: finite_trace_last_output iom_tr =
finite_trace_last_output eqv_msg_tr
Hstate_valid: valid_state_prop CE eqv_state_s
emsg_tr: list transition_item
es: state CE
Hmsg_trace_full_replay: finite_valid_trace_from_to CE
  eqv_state_s es emsg_tr
Hemsg_tr_pr: equivocators_total_trace_project IM
  emsg_tr = []
Hes_pr: equivocators_total_state_project IM es = s
Hbs_iom: sent_except_from (equivocator_IM IM)
  initial_message_prop es iom
Happ: finite_valid_trace_from_to CE eqv_state_is es
  (eqv_state_tr ++ emsg_tr)
el:= existT eqv (ContinueWith 0 li): {i : index & label (equivocator_IM IM i)}
Happ_extend: finite_valid_trace_from CE eqv_state_is
  ((eqv_state_tr ++ emsg_tr) ++
   [{|
      l := el;
      input := iom;
      destination :=
        state_update (equivocator_IM IM)
          es eqv
          (equivocator_state_update
             (es eqv) 0 si');
      output := oom
    |}])
Heqes: finite_trace_last eqv_state_is
  (eqv_state_tr ++ emsg_tr) = es
Hes_pr_i: ∀ i : index,
  equivocators_total_state_project IM es i =
  s i
Het: transition el (es, iom) =
(state_update (equivocator_IM IM) es eqv
   (equivocator_state_update (es eqv) 0 si'),
 oom)
Hes_pr_eqv: equivocator_state_zero (es eqv) = s eqv
Hivt: input_valid_transition CE el (
  es, iom)
  (state_update (equivocator_IM IM) es eqv
     (equivocator_state_update (es eqv) 0 si'),
   oom)
Hproject: equivocators_total_state_project IM
  (state_update (equivocator_IM IM) es eqv
     (equivocator_state_update 
        (es eqv) 0 si')) =
state_update IM s eqv si'
equivocators_total_trace_project IM
  ((eqv_state_tr ++ emsg_tr) ++
   [{|
      l := el;
      input := iom;
      destination :=
        state_update (equivocator_IM IM) es eqv
          (equivocator_state_update (es eqv) 0 si');
      output := oom
    |}]) =
tr ++
[{|
   l := existT eqv li;
   input := iom;
   destination := state_update IM s eqv si';
   output := oom
 |}]
        apply (VLSM_incl_finite_valid_trace_from HinclE) in Happ_extend.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
seed: message → Prop
constraintX: composite_label IM
→ composite_state IM * option message
  → Prop
CX:= preloaded_vlsm (composite_vlsm IM constraintX)
  seed: VLSM message
constraintE: composite_label (equivocator_IM IM)
→ composite_state (equivocator_IM IM) *
  option message → Prop
CE:= preloaded_vlsm
  (composite_vlsm (equivocator_IM IM) constraintE)
  seed: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hsubsumption: zero_descriptor_constraint_lifting_prop
Hreplayable: replayable_message_prop
is, s: state CX
tr: list transition_item
HtrX1: finite_valid_trace_init_to CX is s tr
iom: option message
iom_si, iom_s: state CX
iom_tr: list transition_item
Heqiom: empty_initial_message_or_final_output CX
  iom_tr iom
HtrX2: finite_valid_trace_init_to CX iom_si iom_s
  iom_tr
oom: option message
eqv: index
li: label (IM eqv)
Hs: valid_state_prop CX s
Hiom: option_valid_message_prop CX iom
Hv: valid (existT eqv li) (s, iom)
Hc: constraintX (existT eqv li) (s, iom)
si': state (IM eqv)
Hteqv: transition li (s eqv, iom) = (si', oom)
HinclE: VLSM_incl CE
  (preloaded_with_all_messages_vlsm FreeE)
eqv_state_is: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_start_project: equivocators_total_state_project
  IM eqv_state_is = is
eqv_state_s: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_state_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hstate_project: equivocators_total_trace_project IM
  eqv_state_tr = tr
Hstate_trace: finite_valid_trace_init_to CE
  eqv_state_is eqv_state_s eqv_state_tr
eqv_msg_is: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hmsg_start_project: equivocators_total_state_project
  IM eqv_msg_is = iom_si
eqv_msg_s: state
  (free_composite_vlsm (equivocator_IM IM))
eqv_msg_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hmsg_project: equivocators_total_trace_project IM
  eqv_msg_tr = iom_tr
Hmsg_trace: finite_valid_trace_init_to CE eqv_msg_is
  eqv_msg_s eqv_msg_tr
Hfinal_msg: finite_trace_last_output iom_tr =
finite_trace_last_output eqv_msg_tr
Hstate_valid: valid_state_prop CE eqv_state_s
emsg_tr: list transition_item
es: state CE
Hmsg_trace_full_replay: finite_valid_trace_from_to CE
  eqv_state_s es emsg_tr
Hemsg_tr_pr: equivocators_total_trace_project IM
  emsg_tr = []
Hes_pr: equivocators_total_state_project IM es = s
Hbs_iom: sent_except_from (equivocator_IM IM)
  initial_message_prop es iom
Happ: finite_valid_trace_from_to CE eqv_state_is es
  (eqv_state_tr ++ emsg_tr)
el:= existT eqv (ContinueWith 0 li): {i : index & label (equivocator_IM IM i)}
Happ_extend: finite_valid_trace_from
  {|
    vlsm_type := CE;
    vlsm_machine :=
      preloaded_with_all_messages_vlsm
        FreeE
  |} eqv_state_is
  ((eqv_state_tr ++ emsg_tr) ++
   [{|
      l := el;
      input := iom;
      destination :=
        state_update (equivocator_IM IM)
          es eqv
          (equivocator_state_update
             (es eqv) 0 si');
      output := oom
    |}])
Heqes: finite_trace_last eqv_state_is
  (eqv_state_tr ++ emsg_tr) = es
Hes_pr_i: ∀ i : index,
  equivocators_total_state_project IM es i =
  s i
Het: transition el (es, iom) =
(state_update (equivocator_IM IM) es eqv
   (equivocator_state_update (es eqv) 0 si'),
 oom)
Hes_pr_eqv: equivocator_state_zero (es eqv) = s eqv
Hivt: input_valid_transition CE el (
  es, iom)
  (state_update (equivocator_IM IM) es eqv
     (equivocator_state_update (es eqv) 0 si'),
   oom)
Hproject: equivocators_total_state_project IM
  (state_update (equivocator_IM IM) es eqv
     (equivocator_state_update 
        (es eqv) 0 si')) =
state_update IM s eqv si'
equivocators_total_trace_project IM
  ((eqv_state_tr ++ emsg_tr) ++
   [{|
      l := el;
      input := iom;
      destination :=
        state_update (equivocator_IM IM) es eqv
          (equivocator_state_update (es eqv) 0 si');
      output := oom
    |}]) =
tr ++
[{|
   l := existT eqv li;
   input := iom;
   destination := state_update IM s eqv si';
   output := oom
 |}]
        rewrite (equivocators_total_trace_project_app IM)
          by (eexists; done).message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
seed: message → Prop
constraintX: composite_label IM
→ composite_state IM * option message
  → Prop
CX:= preloaded_vlsm (composite_vlsm IM constraintX)
  seed: VLSM message
constraintE: composite_label (equivocator_IM IM)
→ composite_state (equivocator_IM IM) *
  option message → Prop
CE:= preloaded_vlsm
  (composite_vlsm (equivocator_IM IM) constraintE)
  seed: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hsubsumption: zero_descriptor_constraint_lifting_prop
Hreplayable: replayable_message_prop
is, s: state CX
tr: list transition_item
HtrX1: finite_valid_trace_init_to CX is s tr
iom: option message
iom_si, iom_s: state CX
iom_tr: list transition_item
Heqiom: empty_initial_message_or_final_output CX
  iom_tr iom
HtrX2: finite_valid_trace_init_to CX iom_si iom_s
  iom_tr
oom: option message
eqv: index
li: label (IM eqv)
Hs: valid_state_prop CX s
Hiom: option_valid_message_prop CX iom
Hv: valid (existT eqv li) (s, iom)
Hc: constraintX (existT eqv li) (s, iom)
si': state (IM eqv)
Hteqv: transition li (s eqv, iom) = (si', oom)
HinclE: VLSM_incl CE
  (preloaded_with_all_messages_vlsm FreeE)
eqv_state_is: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_start_project: equivocators_total_state_project
  IM eqv_state_is = is
eqv_state_s: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_state_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hstate_project: equivocators_total_trace_project IM
  eqv_state_tr = tr
Hstate_trace: finite_valid_trace_init_to CE
  eqv_state_is eqv_state_s eqv_state_tr
eqv_msg_is: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hmsg_start_project: equivocators_total_state_project
  IM eqv_msg_is = iom_si
eqv_msg_s: state
  (free_composite_vlsm (equivocator_IM IM))
eqv_msg_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hmsg_project: equivocators_total_trace_project IM
  eqv_msg_tr = iom_tr
Hmsg_trace: finite_valid_trace_init_to CE eqv_msg_is
  eqv_msg_s eqv_msg_tr
Hfinal_msg: finite_trace_last_output iom_tr =
finite_trace_last_output eqv_msg_tr
Hstate_valid: valid_state_prop CE eqv_state_s
emsg_tr: list transition_item
es: state CE
Hmsg_trace_full_replay: finite_valid_trace_from_to CE
  eqv_state_s es emsg_tr
Hemsg_tr_pr: equivocators_total_trace_project IM
  emsg_tr = []
Hes_pr: equivocators_total_state_project IM es = s
Hbs_iom: sent_except_from (equivocator_IM IM)
  initial_message_prop es iom
Happ: finite_valid_trace_from_to CE eqv_state_is es
  (eqv_state_tr ++ emsg_tr)
el:= existT eqv (ContinueWith 0 li): {i : index & label (equivocator_IM IM i)}
Happ_extend: finite_valid_trace_from
  {|
    vlsm_type := CE;
    vlsm_machine :=
      preloaded_with_all_messages_vlsm
        FreeE
  |} eqv_state_is
  ((eqv_state_tr ++ emsg_tr) ++
   [{|
      l := el;
      input := iom;
      destination :=
        state_update (equivocator_IM IM)
          es eqv
          (equivocator_state_update
             (es eqv) 0 si');
      output := oom
    |}])
Heqes: finite_trace_last eqv_state_is
  (eqv_state_tr ++ emsg_tr) = es
Hes_pr_i: ∀ i : index,
  equivocators_total_state_project IM es i =
  s i
Het: transition el (es, iom) =
(state_update (equivocator_IM IM) es eqv
   (equivocator_state_update (es eqv) 0 si'),
 oom)
Hes_pr_eqv: equivocator_state_zero (es eqv) = s eqv
Hivt: input_valid_transition CE el (
  es, iom)
  (state_update (equivocator_IM IM) es eqv
     (equivocator_state_update (es eqv) 0 si'),
   oom)
Hproject: equivocators_total_state_project IM
  (state_update (equivocator_IM IM) es eqv
     (equivocator_state_update 
        (es eqv) 0 si')) =
state_update IM s eqv si'
equivocators_total_trace_project IM
  (eqv_state_tr ++ emsg_tr) ++
equivocators_total_trace_project IM
  [{|
     l := el;
     input := iom;
     destination :=
       state_update (equivocator_IM IM) es eqv
         (equivocator_state_update (es eqv) 0 si');
     output := oom
   |}] =
tr ++
[{|
   l := existT eqv li;
   input := iom;
   destination := state_update IM s eqv si';
   output := oom
 |}]
        apply valid_trace_forget_last in Happ.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
seed: message → Prop
constraintX: composite_label IM
→ composite_state IM * option message
  → Prop
CX:= preloaded_vlsm (composite_vlsm IM constraintX)
  seed: VLSM message
constraintE: composite_label (equivocator_IM IM)
→ composite_state (equivocator_IM IM) *
  option message → Prop
CE:= preloaded_vlsm
  (composite_vlsm (equivocator_IM IM) constraintE)
  seed: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hsubsumption: zero_descriptor_constraint_lifting_prop
Hreplayable: replayable_message_prop
is, s: state CX
tr: list transition_item
HtrX1: finite_valid_trace_init_to CX is s tr
iom: option message
iom_si, iom_s: state CX
iom_tr: list transition_item
Heqiom: empty_initial_message_or_final_output CX
  iom_tr iom
HtrX2: finite_valid_trace_init_to CX iom_si iom_s
  iom_tr
oom: option message
eqv: index
li: label (IM eqv)
Hs: valid_state_prop CX s
Hiom: option_valid_message_prop CX iom
Hv: valid (existT eqv li) (s, iom)
Hc: constraintX (existT eqv li) (s, iom)
si': state (IM eqv)
Hteqv: transition li (s eqv, iom) = (si', oom)
HinclE: VLSM_incl CE
  (preloaded_with_all_messages_vlsm FreeE)
eqv_state_is: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_start_project: equivocators_total_state_project
  IM eqv_state_is = is
eqv_state_s: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_state_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hstate_project: equivocators_total_trace_project IM
  eqv_state_tr = tr
Hstate_trace: finite_valid_trace_init_to CE
  eqv_state_is eqv_state_s eqv_state_tr
eqv_msg_is: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hmsg_start_project: equivocators_total_state_project
  IM eqv_msg_is = iom_si
eqv_msg_s: state
  (free_composite_vlsm (equivocator_IM IM))
eqv_msg_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hmsg_project: equivocators_total_trace_project IM
  eqv_msg_tr = iom_tr
Hmsg_trace: finite_valid_trace_init_to CE eqv_msg_is
  eqv_msg_s eqv_msg_tr
Hfinal_msg: finite_trace_last_output iom_tr =
finite_trace_last_output eqv_msg_tr
Hstate_valid: valid_state_prop CE eqv_state_s
emsg_tr: list transition_item
es: state CE
Hmsg_trace_full_replay: finite_valid_trace_from_to CE
  eqv_state_s es emsg_tr
Hemsg_tr_pr: equivocators_total_trace_project IM
  emsg_tr = []
Hes_pr: equivocators_total_state_project IM es = s
Hbs_iom: sent_except_from (equivocator_IM IM)
  initial_message_prop es iom
Happ: finite_valid_trace_from CE eqv_state_is
  (eqv_state_tr ++ emsg_tr)
el:= existT eqv (ContinueWith 0 li): {i : index & label (equivocator_IM IM i)}
Happ_extend: finite_valid_trace_from
  {|
    vlsm_type := CE;
    vlsm_machine :=
      preloaded_with_all_messages_vlsm
        FreeE
  |} eqv_state_is
  ((eqv_state_tr ++ emsg_tr) ++
   [{|
      l := el;
      input := iom;
      destination :=
        state_update (equivocator_IM IM)
          es eqv
          (equivocator_state_update
             (es eqv) 0 si');
      output := oom
    |}])
Heqes: finite_trace_last eqv_state_is
  (eqv_state_tr ++ emsg_tr) = es
Hes_pr_i: ∀ i : index,
  equivocators_total_state_project IM es i =
  s i
Het: transition el (es, iom) =
(state_update (equivocator_IM IM) es eqv
   (equivocator_state_update (es eqv) 0 si'),
 oom)
Hes_pr_eqv: equivocator_state_zero (es eqv) = s eqv
Hivt: input_valid_transition CE el (
  es, iom)
  (state_update (equivocator_IM IM) es eqv
     (equivocator_state_update (es eqv) 0 si'),
   oom)
Hproject: equivocators_total_state_project IM
  (state_update (equivocator_IM IM) es eqv
     (equivocator_state_update 
        (es eqv) 0 si')) =
state_update IM s eqv si'
equivocators_total_trace_project IM
  (eqv_state_tr ++ emsg_tr) ++
equivocators_total_trace_project IM
  [{|
     l := el;
     input := iom;
     destination :=
       state_update (equivocator_IM IM) es eqv
         (equivocator_state_update (es eqv) 0 si');
     output := oom
   |}] =
tr ++
[{|
   l := existT eqv li;
   input := iom;
   destination := state_update IM s eqv si';
   output := oom
 |}]
        apply (VLSM_incl_finite_valid_trace_from HinclE) in Happ.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
seed: message → Prop
constraintX: composite_label IM
→ composite_state IM * option message
  → Prop
CX:= preloaded_vlsm (composite_vlsm IM constraintX)
  seed: VLSM message
constraintE: composite_label (equivocator_IM IM)
→ composite_state (equivocator_IM IM) *
  option message → Prop
CE:= preloaded_vlsm
  (composite_vlsm (equivocator_IM IM) constraintE)
  seed: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hsubsumption: zero_descriptor_constraint_lifting_prop
Hreplayable: replayable_message_prop
is, s: state CX
tr: list transition_item
HtrX1: finite_valid_trace_init_to CX is s tr
iom: option message
iom_si, iom_s: state CX
iom_tr: list transition_item
Heqiom: empty_initial_message_or_final_output CX
  iom_tr iom
HtrX2: finite_valid_trace_init_to CX iom_si iom_s
  iom_tr
oom: option message
eqv: index
li: label (IM eqv)
Hs: valid_state_prop CX s
Hiom: option_valid_message_prop CX iom
Hv: valid (existT eqv li) (s, iom)
Hc: constraintX (existT eqv li) (s, iom)
si': state (IM eqv)
Hteqv: transition li (s eqv, iom) = (si', oom)
HinclE: VLSM_incl CE
  (preloaded_with_all_messages_vlsm FreeE)
eqv_state_is: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_start_project: equivocators_total_state_project
  IM eqv_state_is = is
eqv_state_s: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_state_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hstate_project: equivocators_total_trace_project IM
  eqv_state_tr = tr
Hstate_trace: finite_valid_trace_init_to CE
  eqv_state_is eqv_state_s eqv_state_tr
eqv_msg_is: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hmsg_start_project: equivocators_total_state_project
  IM eqv_msg_is = iom_si
eqv_msg_s: state
  (free_composite_vlsm (equivocator_IM IM))
eqv_msg_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hmsg_project: equivocators_total_trace_project IM
  eqv_msg_tr = iom_tr
Hmsg_trace: finite_valid_trace_init_to CE eqv_msg_is
  eqv_msg_s eqv_msg_tr
Hfinal_msg: finite_trace_last_output iom_tr =
finite_trace_last_output eqv_msg_tr
Hstate_valid: valid_state_prop CE eqv_state_s
emsg_tr: list transition_item
es: state CE
Hmsg_trace_full_replay: finite_valid_trace_from_to CE
  eqv_state_s es emsg_tr
Hemsg_tr_pr: equivocators_total_trace_project IM
  emsg_tr = []
Hes_pr: equivocators_total_state_project IM es = s
Hbs_iom: sent_except_from (equivocator_IM IM)
  initial_message_prop es iom
Happ: finite_valid_trace_from
  {|
    vlsm_type := CE;
    vlsm_machine :=
      preloaded_with_all_messages_vlsm FreeE
  |} eqv_state_is (eqv_state_tr ++ emsg_tr)
el:= existT eqv (ContinueWith 0 li): {i : index & label (equivocator_IM IM i)}
Happ_extend: finite_valid_trace_from
  {|
    vlsm_type := CE;
    vlsm_machine :=
      preloaded_with_all_messages_vlsm
        FreeE
  |} eqv_state_is
  ((eqv_state_tr ++ emsg_tr) ++
   [{|
      l := el;
      input := iom;
      destination :=
        state_update (equivocator_IM IM)
          es eqv
          (equivocator_state_update
             (es eqv) 0 si');
      output := oom
    |}])
Heqes: finite_trace_last eqv_state_is
  (eqv_state_tr ++ emsg_tr) = es
Hes_pr_i: ∀ i : index,
  equivocators_total_state_project IM es i =
  s i
Het: transition el (es, iom) =
(state_update (equivocator_IM IM) es eqv
   (equivocator_state_update (es eqv) 0 si'),
 oom)
Hes_pr_eqv: equivocator_state_zero (es eqv) = s eqv
Hivt: input_valid_transition CE el (
  es, iom)
  (state_update (equivocator_IM IM) es eqv
     (equivocator_state_update (es eqv) 0 si'),
   oom)
Hproject: equivocators_total_state_project IM
  (state_update (equivocator_IM IM) es eqv
     (equivocator_state_update 
        (es eqv) 0 si')) =
state_update IM s eqv si'
equivocators_total_trace_project IM
  (eqv_state_tr ++ emsg_tr) ++
equivocators_total_trace_project IM
  [{|
     l := el;
     input := iom;
     destination :=
       state_update (equivocator_IM IM) es eqv
         (equivocator_state_update (es eqv) 0 si');
     output := oom
   |}] =
tr ++
[{|
   l := existT eqv li;
   input := iom;
   destination := state_update IM s eqv si';
   output := oom
 |}]
        rewrite (equivocators_total_trace_project_app IM)
          by (eexists; done).message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
seed: message → Prop
constraintX: composite_label IM
→ composite_state IM * option message
  → Prop
CX:= preloaded_vlsm (composite_vlsm IM constraintX)
  seed: VLSM message
constraintE: composite_label (equivocator_IM IM)
→ composite_state (equivocator_IM IM) *
  option message → Prop
CE:= preloaded_vlsm
  (composite_vlsm (equivocator_IM IM) constraintE)
  seed: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hsubsumption: zero_descriptor_constraint_lifting_prop
Hreplayable: replayable_message_prop
is, s: state CX
tr: list transition_item
HtrX1: finite_valid_trace_init_to CX is s tr
iom: option message
iom_si, iom_s: state CX
iom_tr: list transition_item
Heqiom: empty_initial_message_or_final_output CX
  iom_tr iom
HtrX2: finite_valid_trace_init_to CX iom_si iom_s
  iom_tr
oom: option message
eqv: index
li: label (IM eqv)
Hs: valid_state_prop CX s
Hiom: option_valid_message_prop CX iom
Hv: valid (existT eqv li) (s, iom)
Hc: constraintX (existT eqv li) (s, iom)
si': state (IM eqv)
Hteqv: transition li (s eqv, iom) = (si', oom)
HinclE: VLSM_incl CE
  (preloaded_with_all_messages_vlsm FreeE)
eqv_state_is: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_start_project: equivocators_total_state_project
  IM eqv_state_is = is
eqv_state_s: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_state_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hstate_project: equivocators_total_trace_project IM
  eqv_state_tr = tr
Hstate_trace: finite_valid_trace_init_to CE
  eqv_state_is eqv_state_s eqv_state_tr
eqv_msg_is: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hmsg_start_project: equivocators_total_state_project
  IM eqv_msg_is = iom_si
eqv_msg_s: state
  (free_composite_vlsm (equivocator_IM IM))
eqv_msg_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hmsg_project: equivocators_total_trace_project IM
  eqv_msg_tr = iom_tr
Hmsg_trace: finite_valid_trace_init_to CE eqv_msg_is
  eqv_msg_s eqv_msg_tr
Hfinal_msg: finite_trace_last_output iom_tr =
finite_trace_last_output eqv_msg_tr
Hstate_valid: valid_state_prop CE eqv_state_s
emsg_tr: list transition_item
es: state CE
Hmsg_trace_full_replay: finite_valid_trace_from_to CE
  eqv_state_s es emsg_tr
Hemsg_tr_pr: equivocators_total_trace_project IM
  emsg_tr = []
Hes_pr: equivocators_total_state_project IM es = s
Hbs_iom: sent_except_from (equivocator_IM IM)
  initial_message_prop es iom
Happ: finite_valid_trace_from
  {|
    vlsm_type := CE;
    vlsm_machine :=
      preloaded_with_all_messages_vlsm FreeE
  |} eqv_state_is (eqv_state_tr ++ emsg_tr)
el:= existT eqv (ContinueWith 0 li): {i : index & label (equivocator_IM IM i)}
Happ_extend: finite_valid_trace_from
  {|
    vlsm_type := CE;
    vlsm_machine :=
      preloaded_with_all_messages_vlsm
        FreeE
  |} eqv_state_is
  ((eqv_state_tr ++ emsg_tr) ++
   [{|
      l := el;
      input := iom;
      destination :=
        state_update (equivocator_IM IM)
          es eqv
          (equivocator_state_update
             (es eqv) 0 si');
      output := oom
    |}])
Heqes: finite_trace_last eqv_state_is
  (eqv_state_tr ++ emsg_tr) = es
Hes_pr_i: ∀ i : index,
  equivocators_total_state_project IM es i =
  s i
Het: transition el (es, iom) =
(state_update (equivocator_IM IM) es eqv
   (equivocator_state_update (es eqv) 0 si'),
 oom)
Hes_pr_eqv: equivocator_state_zero (es eqv) = s eqv
Hivt: input_valid_transition CE el (
  es, iom)
  (state_update (equivocator_IM IM) es eqv
     (equivocator_state_update (es eqv) 0 si'),
   oom)
Hproject: equivocators_total_state_project IM
  (state_update (equivocator_IM IM) es eqv
     (equivocator_state_update 
        (es eqv) 0 si')) =
state_update IM s eqv si'
(equivocators_total_trace_project IM eqv_state_tr ++
 equivocators_total_trace_project IM emsg_tr) ++
equivocators_total_trace_project IM
  [{|
     l := el;
     input := iom;
     destination :=
       state_update (equivocator_IM IM) es eqv
         (equivocator_state_update (es eqv) 0 si');
     output := oom
   |}] =
tr ++
[{|
   l := existT eqv li;
   input := iom;
   destination := state_update IM s eqv si';
   output := oom
 |}]
        rewrite Hemsg_tr_pr.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
seed: message → Prop
constraintX: composite_label IM
→ composite_state IM * option message
  → Prop
CX:= preloaded_vlsm (composite_vlsm IM constraintX)
  seed: VLSM message
constraintE: composite_label (equivocator_IM IM)
→ composite_state (equivocator_IM IM) *
  option message → Prop
CE:= preloaded_vlsm
  (composite_vlsm (equivocator_IM IM) constraintE)
  seed: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hsubsumption: zero_descriptor_constraint_lifting_prop
Hreplayable: replayable_message_prop
is, s: state CX
tr: list transition_item
HtrX1: finite_valid_trace_init_to CX is s tr
iom: option message
iom_si, iom_s: state CX
iom_tr: list transition_item
Heqiom: empty_initial_message_or_final_output CX
  iom_tr iom
HtrX2: finite_valid_trace_init_to CX iom_si iom_s
  iom_tr
oom: option message
eqv: index
li: label (IM eqv)
Hs: valid_state_prop CX s
Hiom: option_valid_message_prop CX iom
Hv: valid (existT eqv li) (s, iom)
Hc: constraintX (existT eqv li) (s, iom)
si': state (IM eqv)
Hteqv: transition li (s eqv, iom) = (si', oom)
HinclE: VLSM_incl CE
  (preloaded_with_all_messages_vlsm FreeE)
eqv_state_is: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_start_project: equivocators_total_state_project
  IM eqv_state_is = is
eqv_state_s: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_state_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hstate_project: equivocators_total_trace_project IM
  eqv_state_tr = tr
Hstate_trace: finite_valid_trace_init_to CE
  eqv_state_is eqv_state_s eqv_state_tr
eqv_msg_is: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hmsg_start_project: equivocators_total_state_project
  IM eqv_msg_is = iom_si
eqv_msg_s: state
  (free_composite_vlsm (equivocator_IM IM))
eqv_msg_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hmsg_project: equivocators_total_trace_project IM
  eqv_msg_tr = iom_tr
Hmsg_trace: finite_valid_trace_init_to CE eqv_msg_is
  eqv_msg_s eqv_msg_tr
Hfinal_msg: finite_trace_last_output iom_tr =
finite_trace_last_output eqv_msg_tr
Hstate_valid: valid_state_prop CE eqv_state_s
emsg_tr: list transition_item
es: state CE
Hmsg_trace_full_replay: finite_valid_trace_from_to CE
  eqv_state_s es emsg_tr
Hemsg_tr_pr: equivocators_total_trace_project IM
  emsg_tr = []
Hes_pr: equivocators_total_state_project IM es = s
Hbs_iom: sent_except_from (equivocator_IM IM)
  initial_message_prop es iom
Happ: finite_valid_trace_from
  {|
    vlsm_type := CE;
    vlsm_machine :=
      preloaded_with_all_messages_vlsm FreeE
  |} eqv_state_is (eqv_state_tr ++ emsg_tr)
el:= existT eqv (ContinueWith 0 li): {i : index & label (equivocator_IM IM i)}
Happ_extend: finite_valid_trace_from
  {|
    vlsm_type := CE;
    vlsm_machine :=
      preloaded_with_all_messages_vlsm
        FreeE
  |} eqv_state_is
  ((eqv_state_tr ++ emsg_tr) ++
   [{|
      l := el;
      input := iom;
      destination :=
        state_update (equivocator_IM IM)
          es eqv
          (equivocator_state_update
             (es eqv) 0 si');
      output := oom
    |}])
Heqes: finite_trace_last eqv_state_is
  (eqv_state_tr ++ emsg_tr) = es
Hes_pr_i: ∀ i : index,
  equivocators_total_state_project IM es i =
  s i
Het: transition el (es, iom) =
(state_update (equivocator_IM IM) es eqv
   (equivocator_state_update (es eqv) 0 si'),
 oom)
Hes_pr_eqv: equivocator_state_zero (es eqv) = s eqv
Hivt: input_valid_transition CE el (
  es, iom)
  (state_update (equivocator_IM IM) es eqv
     (equivocator_state_update (es eqv) 0 si'),
   oom)
Hproject: equivocators_total_state_project IM
  (state_update (equivocator_IM IM) es eqv
     (equivocator_state_update 
        (es eqv) 0 si')) =
state_update IM s eqv si'
(equivocators_total_trace_project IM eqv_state_tr ++
 []) ++
equivocators_total_trace_project IM
  [{|
     l := el;
     input := iom;
     destination :=
       state_update (equivocator_IM IM) es eqv
         (equivocator_state_update (es eqv) 0 si');
     output := oom
   |}] =
tr ++
[{|
   l := existT eqv li;
   input := iom;
   destination := state_update IM s eqv si';
   output := oom
 |}]
        rewrite app_nil_r.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
seed: message → Prop
constraintX: composite_label IM
→ composite_state IM * option message
  → Prop
CX:= preloaded_vlsm (composite_vlsm IM constraintX)
  seed: VLSM message
constraintE: composite_label (equivocator_IM IM)
→ composite_state (equivocator_IM IM) *
  option message → Prop
CE:= preloaded_vlsm
  (composite_vlsm (equivocator_IM IM) constraintE)
  seed: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hsubsumption: zero_descriptor_constraint_lifting_prop
Hreplayable: replayable_message_prop
is, s: state CX
tr: list transition_item
HtrX1: finite_valid_trace_init_to CX is s tr
iom: option message
iom_si, iom_s: state CX
iom_tr: list transition_item
Heqiom: empty_initial_message_or_final_output CX
  iom_tr iom
HtrX2: finite_valid_trace_init_to CX iom_si iom_s
  iom_tr
oom: option message
eqv: index
li: label (IM eqv)
Hs: valid_state_prop CX s
Hiom: option_valid_message_prop CX iom
Hv: valid (existT eqv li) (s, iom)
Hc: constraintX (existT eqv li) (s, iom)
si': state (IM eqv)
Hteqv: transition li (s eqv, iom) = (si', oom)
HinclE: VLSM_incl CE
  (preloaded_with_all_messages_vlsm FreeE)
eqv_state_is: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_start_project: equivocators_total_state_project
  IM eqv_state_is = is
eqv_state_s: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_state_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hstate_project: equivocators_total_trace_project IM
  eqv_state_tr = tr
Hstate_trace: finite_valid_trace_init_to CE
  eqv_state_is eqv_state_s eqv_state_tr
eqv_msg_is: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hmsg_start_project: equivocators_total_state_project
  IM eqv_msg_is = iom_si
eqv_msg_s: state
  (free_composite_vlsm (equivocator_IM IM))
eqv_msg_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hmsg_project: equivocators_total_trace_project IM
  eqv_msg_tr = iom_tr
Hmsg_trace: finite_valid_trace_init_to CE eqv_msg_is
  eqv_msg_s eqv_msg_tr
Hfinal_msg: finite_trace_last_output iom_tr =
finite_trace_last_output eqv_msg_tr
Hstate_valid: valid_state_prop CE eqv_state_s
emsg_tr: list transition_item
es: state CE
Hmsg_trace_full_replay: finite_valid_trace_from_to CE
  eqv_state_s es emsg_tr
Hemsg_tr_pr: equivocators_total_trace_project IM
  emsg_tr = []
Hes_pr: equivocators_total_state_project IM es = s
Hbs_iom: sent_except_from (equivocator_IM IM)
  initial_message_prop es iom
Happ: finite_valid_trace_from
  {|
    vlsm_type := CE;
    vlsm_machine :=
      preloaded_with_all_messages_vlsm FreeE
  |} eqv_state_is (eqv_state_tr ++ emsg_tr)
el:= existT eqv (ContinueWith 0 li): {i : index & label (equivocator_IM IM i)}
Happ_extend: finite_valid_trace_from
  {|
    vlsm_type := CE;
    vlsm_machine :=
      preloaded_with_all_messages_vlsm
        FreeE
  |} eqv_state_is
  ((eqv_state_tr ++ emsg_tr) ++
   [{|
      l := el;
      input := iom;
      destination :=
        state_update (equivocator_IM IM)
          es eqv
          (equivocator_state_update
             (es eqv) 0 si');
      output := oom
    |}])
Heqes: finite_trace_last eqv_state_is
  (eqv_state_tr ++ emsg_tr) = es
Hes_pr_i: ∀ i : index,
  equivocators_total_state_project IM es i =
  s i
Het: transition el (es, iom) =
(state_update (equivocator_IM IM) es eqv
   (equivocator_state_update (es eqv) 0 si'),
 oom)
Hes_pr_eqv: equivocator_state_zero (es eqv) = s eqv
Hivt: input_valid_transition CE el (
  es, iom)
  (state_update (equivocator_IM IM) es eqv
     (equivocator_state_update (es eqv) 0 si'),
   oom)
Hproject: equivocators_total_state_project IM
  (state_update (equivocator_IM IM) es eqv
     (equivocator_state_update 
        (es eqv) 0 si')) =
state_update IM s eqv si'
equivocators_total_trace_project IM eqv_state_tr ++
equivocators_total_trace_project IM
  [{|
     l := el;
     input := iom;
     destination :=
       state_update (equivocator_IM IM) es eqv
         (equivocator_state_update (es eqv) 0 si');
     output := oom
   |}] =
tr ++
[{|
   l := existT eqv li;
   input := iom;
   destination := state_update IM s eqv si';
   output := oom
 |}]
        rewrite Hstate_project.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
seed: message → Prop
constraintX: composite_label IM
→ composite_state IM * option message
  → Prop
CX:= preloaded_vlsm (composite_vlsm IM constraintX)
  seed: VLSM message
constraintE: composite_label (equivocator_IM IM)
→ composite_state (equivocator_IM IM) *
  option message → Prop
CE:= preloaded_vlsm
  (composite_vlsm (equivocator_IM IM) constraintE)
  seed: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hsubsumption: zero_descriptor_constraint_lifting_prop
Hreplayable: replayable_message_prop
is, s: state CX
tr: list transition_item
HtrX1: finite_valid_trace_init_to CX is s tr
iom: option message
iom_si, iom_s: state CX
iom_tr: list transition_item
Heqiom: empty_initial_message_or_final_output CX
  iom_tr iom
HtrX2: finite_valid_trace_init_to CX iom_si iom_s
  iom_tr
oom: option message
eqv: index
li: label (IM eqv)
Hs: valid_state_prop CX s
Hiom: option_valid_message_prop CX iom
Hv: valid (existT eqv li) (s, iom)
Hc: constraintX (existT eqv li) (s, iom)
si': state (IM eqv)
Hteqv: transition li (s eqv, iom) = (si', oom)
HinclE: VLSM_incl CE
  (preloaded_with_all_messages_vlsm FreeE)
eqv_state_is: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_start_project: equivocators_total_state_project
  IM eqv_state_is = is
eqv_state_s: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_state_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hstate_project: equivocators_total_trace_project IM
  eqv_state_tr = tr
Hstate_trace: finite_valid_trace_init_to CE
  eqv_state_is eqv_state_s eqv_state_tr
eqv_msg_is: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hmsg_start_project: equivocators_total_state_project
  IM eqv_msg_is = iom_si
eqv_msg_s: state
  (free_composite_vlsm (equivocator_IM IM))
eqv_msg_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hmsg_project: equivocators_total_trace_project IM
  eqv_msg_tr = iom_tr
Hmsg_trace: finite_valid_trace_init_to CE eqv_msg_is
  eqv_msg_s eqv_msg_tr
Hfinal_msg: finite_trace_last_output iom_tr =
finite_trace_last_output eqv_msg_tr
Hstate_valid: valid_state_prop CE eqv_state_s
emsg_tr: list transition_item
es: state CE
Hmsg_trace_full_replay: finite_valid_trace_from_to CE
  eqv_state_s es emsg_tr
Hemsg_tr_pr: equivocators_total_trace_project IM
  emsg_tr = []
Hes_pr: equivocators_total_state_project IM es = s
Hbs_iom: sent_except_from (equivocator_IM IM)
  initial_message_prop es iom
Happ: finite_valid_trace_from
  {|
    vlsm_type := CE;
    vlsm_machine :=
      preloaded_with_all_messages_vlsm FreeE
  |} eqv_state_is (eqv_state_tr ++ emsg_tr)
el:= existT eqv (ContinueWith 0 li): {i : index & label (equivocator_IM IM i)}
Happ_extend: finite_valid_trace_from
  {|
    vlsm_type := CE;
    vlsm_machine :=
      preloaded_with_all_messages_vlsm
        FreeE
  |} eqv_state_is
  ((eqv_state_tr ++ emsg_tr) ++
   [{|
      l := el;
      input := iom;
      destination :=
        state_update (equivocator_IM IM)
          es eqv
          (equivocator_state_update
             (es eqv) 0 si');
      output := oom
    |}])
Heqes: finite_trace_last eqv_state_is
  (eqv_state_tr ++ emsg_tr) = es
Hes_pr_i: ∀ i : index,
  equivocators_total_state_project IM es i =
  s i
Het: transition el (es, iom) =
(state_update (equivocator_IM IM) es eqv
   (equivocator_state_update (es eqv) 0 si'),
 oom)
Hes_pr_eqv: equivocator_state_zero (es eqv) = s eqv
Hivt: input_valid_transition CE el (
  es, iom)
  (state_update (equivocator_IM IM) es eqv
     (equivocator_state_update (es eqv) 0 si'),
   oom)
Hproject: equivocators_total_state_project IM
  (state_update (equivocator_IM IM) es eqv
     (equivocator_state_update 
        (es eqv) 0 si')) =
state_update IM s eqv si'
tr ++
equivocators_total_trace_project IM
  [{|
     l := el;
     input := iom;
     destination :=
       state_update (equivocator_IM IM) es eqv
         (equivocator_state_update (es eqv) 0 si');
     output := oom
   |}] =
tr ++
[{|
   l := existT eqv li;
   input := iom;
   destination := state_update IM s eqv si';
   output := oom
 |}]
        f_equal.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
seed: message → Prop
constraintX: composite_label IM
→ composite_state IM * option message
  → Prop
CX:= preloaded_vlsm (composite_vlsm IM constraintX)
  seed: VLSM message
constraintE: composite_label (equivocator_IM IM)
→ composite_state (equivocator_IM IM) *
  option message → Prop
CE:= preloaded_vlsm
  (composite_vlsm (equivocator_IM IM) constraintE)
  seed: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hsubsumption: zero_descriptor_constraint_lifting_prop
Hreplayable: replayable_message_prop
is, s: state CX
tr: list transition_item
HtrX1: finite_valid_trace_init_to CX is s tr
iom: option message
iom_si, iom_s: state CX
iom_tr: list transition_item
Heqiom: empty_initial_message_or_final_output CX
  iom_tr iom
HtrX2: finite_valid_trace_init_to CX iom_si iom_s
  iom_tr
oom: option message
eqv: index
li: label (IM eqv)
Hs: valid_state_prop CX s
Hiom: option_valid_message_prop CX iom
Hv: valid (existT eqv li) (s, iom)
Hc: constraintX (existT eqv li) (s, iom)
si': state (IM eqv)
Hteqv: transition li (s eqv, iom) = (si', oom)
HinclE: VLSM_incl CE
  (preloaded_with_all_messages_vlsm FreeE)
eqv_state_is: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_start_project: equivocators_total_state_project
  IM eqv_state_is = is
eqv_state_s: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_state_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hstate_project: equivocators_total_trace_project IM
  eqv_state_tr = tr
Hstate_trace: finite_valid_trace_init_to CE
  eqv_state_is eqv_state_s eqv_state_tr
eqv_msg_is: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hmsg_start_project: equivocators_total_state_project
  IM eqv_msg_is = iom_si
eqv_msg_s: state
  (free_composite_vlsm (equivocator_IM IM))
eqv_msg_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hmsg_project: equivocators_total_trace_project IM
  eqv_msg_tr = iom_tr
Hmsg_trace: finite_valid_trace_init_to CE eqv_msg_is
  eqv_msg_s eqv_msg_tr
Hfinal_msg: finite_trace_last_output iom_tr =
finite_trace_last_output eqv_msg_tr
Hstate_valid: valid_state_prop CE eqv_state_s
emsg_tr: list transition_item
es: state CE
Hmsg_trace_full_replay: finite_valid_trace_from_to CE
  eqv_state_s es emsg_tr
Hemsg_tr_pr: equivocators_total_trace_project IM
  emsg_tr = []
Hes_pr: equivocators_total_state_project IM es = s
Hbs_iom: sent_except_from (equivocator_IM IM)
  initial_message_prop es iom
Happ: finite_valid_trace_from
  {|
    vlsm_type := CE;
    vlsm_machine :=
      preloaded_with_all_messages_vlsm FreeE
  |} eqv_state_is (eqv_state_tr ++ emsg_tr)
el:= existT eqv (ContinueWith 0 li): {i : index & label (equivocator_IM IM i)}
Happ_extend: finite_valid_trace_from
  {|
    vlsm_type := CE;
    vlsm_machine :=
      preloaded_with_all_messages_vlsm
        FreeE
  |} eqv_state_is
  ((eqv_state_tr ++ emsg_tr) ++
   [{|
      l := el;
      input := iom;
      destination :=
        state_update (equivocator_IM IM)
          es eqv
          (equivocator_state_update
             (es eqv) 0 si');
      output := oom
    |}])
Heqes: finite_trace_last eqv_state_is
  (eqv_state_tr ++ emsg_tr) = es
Hes_pr_i: ∀ i : index,
  equivocators_total_state_project IM es i =
  s i
Het: transition el (es, iom) =
(state_update (equivocator_IM IM) es eqv
   (equivocator_state_update (es eqv) 0 si'),
 oom)
Hes_pr_eqv: equivocator_state_zero (es eqv) = s eqv
Hivt: input_valid_transition CE el (
  es, iom)
  (state_update (equivocator_IM IM) es eqv
     (equivocator_state_update (es eqv) 0 si'),
   oom)
Hproject: equivocators_total_state_project IM
  (state_update (equivocator_IM IM) es eqv
     (equivocator_state_update 
        (es eqv) 0 si')) =
state_update IM s eqv si'
equivocators_total_trace_project IM
  [{|
     l := el;
     input := iom;
     destination :=
       state_update (equivocator_IM IM) es eqv
         (equivocator_state_update (es eqv) 0 si');
     output := oom
   |}] =
[{|
   l := existT eqv li;
   input := iom;
   destination := state_update IM s eqv si';
   output := oom
 |}]
        subst el.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
seed: message → Prop
constraintX: composite_label IM
→ composite_state IM * option message
  → Prop
CX:= preloaded_vlsm (composite_vlsm IM constraintX)
  seed: VLSM message
constraintE: composite_label (equivocator_IM IM)
→ composite_state (equivocator_IM IM) *
  option message → Prop
CE:= preloaded_vlsm
  (composite_vlsm (equivocator_IM IM) constraintE)
  seed: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hsubsumption: zero_descriptor_constraint_lifting_prop
Hreplayable: replayable_message_prop
is, s: state CX
tr: list transition_item
HtrX1: finite_valid_trace_init_to CX is s tr
iom: option message
iom_si, iom_s: state CX
iom_tr: list transition_item
Heqiom: empty_initial_message_or_final_output CX
  iom_tr iom
HtrX2: finite_valid_trace_init_to CX iom_si iom_s
  iom_tr
oom: option message
eqv: index
li: label (IM eqv)
Hs: valid_state_prop CX s
Hiom: option_valid_message_prop CX iom
Hv: valid (existT eqv li) (s, iom)
Hc: constraintX (existT eqv li) (s, iom)
si': state (IM eqv)
Hteqv: transition li (s eqv, iom) = (si', oom)
HinclE: VLSM_incl CE
  (preloaded_with_all_messages_vlsm FreeE)
eqv_state_is: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_start_project: equivocators_total_state_project
  IM eqv_state_is = is
eqv_state_s: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_state_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hstate_project: equivocators_total_trace_project IM
  eqv_state_tr = tr
Hstate_trace: finite_valid_trace_init_to CE
  eqv_state_is eqv_state_s eqv_state_tr
eqv_msg_is: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hmsg_start_project: equivocators_total_state_project
  IM eqv_msg_is = iom_si
eqv_msg_s: state
  (free_composite_vlsm (equivocator_IM IM))
eqv_msg_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hmsg_project: equivocators_total_trace_project IM
  eqv_msg_tr = iom_tr
Hmsg_trace: finite_valid_trace_init_to CE eqv_msg_is
  eqv_msg_s eqv_msg_tr
Hfinal_msg: finite_trace_last_output iom_tr =
finite_trace_last_output eqv_msg_tr
Hstate_valid: valid_state_prop CE eqv_state_s
emsg_tr: list transition_item
es: state CE
Hmsg_trace_full_replay: finite_valid_trace_from_to CE
  eqv_state_s es emsg_tr
Hemsg_tr_pr: equivocators_total_trace_project IM
  emsg_tr = []
Hes_pr: equivocators_total_state_project IM es = s
Hbs_iom: sent_except_from (equivocator_IM IM)
  initial_message_prop es iom
Happ: finite_valid_trace_from
  {|
    vlsm_type := CE;
    vlsm_machine :=
      preloaded_with_all_messages_vlsm FreeE
  |} eqv_state_is (eqv_state_tr ++ emsg_tr)
Happ_extend: finite_valid_trace_from
  {|
    vlsm_type := CE;
    vlsm_machine :=
      preloaded_with_all_messages_vlsm
        FreeE
  |} eqv_state_is
  ((eqv_state_tr ++ emsg_tr) ++
   [{|
      l :=
        existT eqv (ContinueWith 0 li);
      input := iom;
      destination :=
        state_update (equivocator_IM IM)
          es eqv
          (equivocator_state_update
             (es eqv) 0 si');
      output := oom
    |}])
Heqes: finite_trace_last eqv_state_is
  (eqv_state_tr ++ emsg_tr) = es
Hes_pr_i: ∀ i : index,
  equivocators_total_state_project IM es i =
  s i
Het: transition (existT eqv (ContinueWith 0 li))
  (es, iom) =
(state_update (equivocator_IM IM) es eqv
   (equivocator_state_update (es eqv) 0 si'),
 oom)
Hes_pr_eqv: equivocator_state_zero (es eqv) = s eqv
Hivt: input_valid_transition CE
  (existT eqv (ContinueWith 0 li)) (
  es, iom)
  (state_update (equivocator_IM IM) es eqv
     (equivocator_state_update (es eqv) 0 si'),
   oom)
Hproject: equivocators_total_state_project IM
  (state_update (equivocator_IM IM) es eqv
     (equivocator_state_update 
        (es eqv) 0 si')) =
state_update IM s eqv si'
equivocators_total_trace_project IM
  [{|
     l := existT eqv (ContinueWith 0 li);
     input := iom;
     destination :=
       state_update (equivocator_IM IM) es eqv
         (equivocator_state_update (es eqv) 0 si');
     output := oom
   |}] =
[{|
   l := existT eqv li;
   input := iom;
   destination := state_update IM s eqv si';
   output := oom
 |}]
        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)
seed: message → Prop
constraintX: composite_label IM
→ composite_state IM * option message
  → Prop
CX:= preloaded_vlsm (composite_vlsm IM constraintX)
  seed: VLSM message
constraintE: composite_label (equivocator_IM IM)
→ composite_state (equivocator_IM IM) *
  option message → Prop
CE:= preloaded_vlsm
  (composite_vlsm (equivocator_IM IM) constraintE)
  seed: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hsubsumption: zero_descriptor_constraint_lifting_prop
Hreplayable: replayable_message_prop
is, s: state CX
tr: list transition_item
HtrX1: finite_valid_trace_init_to CX is s tr
iom: option message
iom_si, iom_s: state CX
iom_tr: list transition_item
Heqiom: empty_initial_message_or_final_output CX
  iom_tr iom
HtrX2: finite_valid_trace_init_to CX iom_si iom_s
  iom_tr
oom: option message
eqv: index
li: label (IM eqv)
Hs: valid_state_prop CX s
Hiom: option_valid_message_prop CX iom
Hv: valid (existT eqv li) (s, iom)
Hc: constraintX (existT eqv li) (s, iom)
si': state (IM eqv)
Hteqv: transition li (s eqv, iom) = (si', oom)
HinclE: VLSM_incl CE
  (preloaded_with_all_messages_vlsm FreeE)
eqv_state_is: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_start_project: equivocators_total_state_project
  IM eqv_state_is = is
eqv_state_s: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_state_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hstate_project: equivocators_total_trace_project IM
  eqv_state_tr = tr
Hstate_trace: finite_valid_trace_init_to CE
  eqv_state_is eqv_state_s eqv_state_tr
eqv_msg_is: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hmsg_start_project: equivocators_total_state_project
  IM eqv_msg_is = iom_si
eqv_msg_s: state
  (free_composite_vlsm (equivocator_IM IM))
eqv_msg_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hmsg_project: equivocators_total_trace_project IM
  eqv_msg_tr = iom_tr
Hmsg_trace: finite_valid_trace_init_to CE eqv_msg_is
  eqv_msg_s eqv_msg_tr
Hfinal_msg: finite_trace_last_output iom_tr =
finite_trace_last_output eqv_msg_tr
Hstate_valid: valid_state_prop CE eqv_state_s
emsg_tr: list transition_item
es: state CE
Hmsg_trace_full_replay: finite_valid_trace_from_to CE
  eqv_state_s es emsg_tr
Hemsg_tr_pr: equivocators_total_trace_project IM
  emsg_tr = []
Hes_pr: equivocators_total_state_project IM es = s
Hbs_iom: sent_except_from (equivocator_IM IM)
  initial_message_prop es iom
Happ: finite_valid_trace_from
  {|
    vlsm_type := CE;
    vlsm_machine :=
      preloaded_with_all_messages_vlsm FreeE
  |} eqv_state_is (eqv_state_tr ++ emsg_tr)
Happ_extend: finite_valid_trace_from
  {|
    vlsm_type := CE;
    vlsm_machine :=
      preloaded_with_all_messages_vlsm
        FreeE
  |} eqv_state_is
  ((eqv_state_tr ++ emsg_tr) ++
   [{|
      l :=
        existT eqv (ContinueWith 0 li);
      input := iom;
      destination :=
        state_update (equivocator_IM IM)
          es eqv
          (equivocator_state_update
             (es eqv) 0 si');
      output := oom
    |}])
Heqes: finite_trace_last eqv_state_is
  (eqv_state_tr ++ emsg_tr) = es
Hes_pr_i: ∀ i : index,
  equivocators_total_state_project IM es i =
  s i
Het: transition (existT eqv (ContinueWith 0 li))
  (es, iom) =
(state_update (equivocator_IM IM) es eqv
   (equivocator_state_update (es eqv) 0 si'),
 oom)
Hes_pr_eqv: equivocator_state_zero (es eqv) = s eqv
Hivt: input_valid_transition CE
  (existT eqv (ContinueWith 0 li)) (
  es, iom)
  (state_update (equivocator_IM IM) es eqv
     (equivocator_state_update (es eqv) 0 si'),
   oom)
Hproject: equivocators_total_state_project IM
  (state_update (equivocator_IM IM) es eqv
     (equivocator_state_update 
        (es eqv) 0 si')) =
state_update IM s eqv si'
from_option fst []
  (equivocators_trace_project IM (zero_descriptor IM)
     [{|
        l := existT eqv (ContinueWith 0 li);
        input := iom;
        destination :=
          state_update (equivocator_IM IM) es eqv
            (equivocator_state_update (es eqv) 0 si');
        output := oom
      |}]) =
[{|
   l := existT eqv li;
   input := iom;
   destination := state_update IM s eqv si';
   output := oom
 |}]
        cbn.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
seed: message → Prop
constraintX: composite_label IM
→ composite_state IM * option message
  → Prop
CX:= preloaded_vlsm (composite_vlsm IM constraintX)
  seed: VLSM message
constraintE: composite_label (equivocator_IM IM)
→ composite_state (equivocator_IM IM) *
  option message → Prop
CE:= preloaded_vlsm
  (composite_vlsm (equivocator_IM IM) constraintE)
  seed: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hsubsumption: zero_descriptor_constraint_lifting_prop
Hreplayable: replayable_message_prop
is, s: state CX
tr: list transition_item
HtrX1: finite_valid_trace_init_to CX is s tr
iom: option message
iom_si, iom_s: state CX
iom_tr: list transition_item
Heqiom: empty_initial_message_or_final_output CX
  iom_tr iom
HtrX2: finite_valid_trace_init_to CX iom_si iom_s
  iom_tr
oom: option message
eqv: index
li: label (IM eqv)
Hs: valid_state_prop CX s
Hiom: option_valid_message_prop CX iom
Hv: valid (existT eqv li) (s, iom)
Hc: constraintX (existT eqv li) (s, iom)
si': state (IM eqv)
Hteqv: transition li (s eqv, iom) = (si', oom)
HinclE: VLSM_incl CE
  (preloaded_with_all_messages_vlsm FreeE)
eqv_state_is: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_start_project: equivocators_total_state_project
  IM eqv_state_is = is
eqv_state_s: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_state_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hstate_project: equivocators_total_trace_project IM
  eqv_state_tr = tr
Hstate_trace: finite_valid_trace_init_to CE
  eqv_state_is eqv_state_s eqv_state_tr
eqv_msg_is: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hmsg_start_project: equivocators_total_state_project
  IM eqv_msg_is = iom_si
eqv_msg_s: state
  (free_composite_vlsm (equivocator_IM IM))
eqv_msg_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hmsg_project: equivocators_total_trace_project IM
  eqv_msg_tr = iom_tr
Hmsg_trace: finite_valid_trace_init_to CE eqv_msg_is
  eqv_msg_s eqv_msg_tr
Hfinal_msg: finite_trace_last_output iom_tr =
finite_trace_last_output eqv_msg_tr
Hstate_valid: valid_state_prop CE eqv_state_s
emsg_tr: list transition_item
es: state CE
Hmsg_trace_full_replay: finite_valid_trace_from_to CE
  eqv_state_s es emsg_tr
Hemsg_tr_pr: equivocators_total_trace_project IM
  emsg_tr = []
Hes_pr: equivocators_total_state_project IM es = s
Hbs_iom: sent_except_from (equivocator_IM IM)
  initial_message_prop es iom
Happ: finite_valid_trace_from
  {|
    vlsm_type := CE;
    vlsm_machine :=
      preloaded_with_all_messages_vlsm FreeE
  |} eqv_state_is (eqv_state_tr ++ emsg_tr)
Happ_extend: finite_valid_trace_from
  {|
    vlsm_type := CE;
    vlsm_machine :=
      preloaded_with_all_messages_vlsm
        FreeE
  |} eqv_state_is
  ((eqv_state_tr ++ emsg_tr) ++
   [{|
      l :=
        existT eqv (ContinueWith 0 li);
      input := iom;
      destination :=
        state_update (equivocator_IM IM)
          es eqv
          (equivocator_state_update
             (es eqv) 0 si');
      output := oom
    |}])
Heqes: finite_trace_last eqv_state_is
  (eqv_state_tr ++ emsg_tr) = es
Hes_pr_i: ∀ i : index,
  equivocators_total_state_project IM es i =
  s i
Het: transition (existT eqv (ContinueWith 0 li))
  (es, iom) =
(state_update (equivocator_IM IM) es eqv
   (equivocator_state_update (es eqv) 0 si'),
 oom)
Hes_pr_eqv: equivocator_state_zero (es eqv) = s eqv
Hivt: input_valid_transition CE
  (existT eqv (ContinueWith 0 li)) (
  es, iom)
  (state_update (equivocator_IM IM) es eqv
     (equivocator_state_update (es eqv) 0 si'),
   oom)
Hproject: equivocators_total_state_project IM
  (state_update (equivocator_IM IM) es eqv
     (equivocator_state_update 
        (es eqv) 0 si')) =
state_update IM s eqv si'
from_option fst []
  match
    equivocators_transition_item_project IM
      (zero_descriptor IM)
      {|
        l := existT eqv (ContinueWith 0 li);
        input := iom;
        destination :=
          state_update (equivocator_IM IM) es eqv
            (equivocator_state_update (es eqv) 0 si');
        output := oom
      |}
  with
  | Some (Some item', odescriptor) =>
      Some ([item'], odescriptor)
  | Some (None, odescriptor) => Some ([], odescriptor)
  | None => None
  end =
[{|
   l := existT eqv li;
   input := iom;
   destination := state_update IM s eqv si';
   output := oom
 |}] 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)
seed: message → Prop
constraintX: composite_label IM
→ composite_state IM * option message
  → Prop
CX:= preloaded_vlsm (composite_vlsm IM constraintX)
  seed: VLSM message
constraintE: composite_label (equivocator_IM IM)
→ composite_state (equivocator_IM IM) *
  option message → Prop
CE:= preloaded_vlsm
  (composite_vlsm (equivocator_IM IM) constraintE)
  seed: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hsubsumption: zero_descriptor_constraint_lifting_prop
Hreplayable: replayable_message_prop
is, s: state CX
tr: list transition_item
HtrX1: finite_valid_trace_init_to CX is s tr
iom: option message
iom_si, iom_s: state CX
iom_tr: list transition_item
Heqiom: empty_initial_message_or_final_output CX
  iom_tr iom
HtrX2: finite_valid_trace_init_to CX iom_si iom_s
  iom_tr
oom: option message
eqv: index
li: label (IM eqv)
Hs: valid_state_prop CX s
Hiom: option_valid_message_prop CX iom
Hv: valid (existT eqv li) (s, iom)
Hc: constraintX (existT eqv li) (s, iom)
si': state (IM eqv)
Hteqv: transition li (s eqv, iom) = (si', oom)
HinclE: VLSM_incl CE
  (preloaded_with_all_messages_vlsm FreeE)
eqv_state_is: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_start_project: equivocators_total_state_project
  IM eqv_state_is = is
eqv_state_s: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_state_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hstate_project: equivocators_total_trace_project IM
  eqv_state_tr = tr
Hstate_trace: finite_valid_trace_init_to CE
  eqv_state_is eqv_state_s eqv_state_tr
eqv_msg_is: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hmsg_start_project: equivocators_total_state_project
  IM eqv_msg_is = iom_si
eqv_msg_s: state
  (free_composite_vlsm (equivocator_IM IM))
eqv_msg_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hmsg_project: equivocators_total_trace_project IM
  eqv_msg_tr = iom_tr
Hmsg_trace: finite_valid_trace_init_to CE eqv_msg_is
  eqv_msg_s eqv_msg_tr
Hfinal_msg: finite_trace_last_output iom_tr =
finite_trace_last_output eqv_msg_tr
Hstate_valid: valid_state_prop CE eqv_state_s
emsg_tr: list transition_item
es: state CE
Hmsg_trace_full_replay: finite_valid_trace_from_to CE
  eqv_state_s es emsg_tr
Hemsg_tr_pr: equivocators_total_trace_project IM
  emsg_tr = []
Hes_pr: equivocators_total_state_project IM es = s
Hbs_iom: sent_except_from (equivocator_IM IM)
  initial_message_prop es iom
Happ: finite_valid_trace_from
  {|
    vlsm_type := CE;
    vlsm_machine :=
      preloaded_with_all_messages_vlsm FreeE
  |} eqv_state_is (eqv_state_tr ++ emsg_tr)
Happ_extend: finite_valid_trace_from
  {|
    vlsm_type := CE;
    vlsm_machine :=
      preloaded_with_all_messages_vlsm
        FreeE
  |} eqv_state_is
  ((eqv_state_tr ++ emsg_tr) ++
   [{|
      l :=
        existT eqv (ContinueWith 0 li);
      input := iom;
      destination :=
        state_update (equivocator_IM IM)
          es eqv
          (equivocator_state_update
             (es eqv) 0 si');
      output := oom
    |}])
Heqes: finite_trace_last eqv_state_is
  (eqv_state_tr ++ emsg_tr) = es
Hes_pr_i: ∀ i : index,
  equivocators_total_state_project IM es i =
  s i
Het: transition (existT eqv (ContinueWith 0 li))
  (es, iom) =
(state_update (equivocator_IM IM) es eqv
   (equivocator_state_update (es eqv) 0 si'),
 oom)
Hes_pr_eqv: equivocator_state_zero (es eqv) = s eqv
Hivt: input_valid_transition CE
  (existT eqv (ContinueWith 0 li)) (
  es, iom)
  (state_update (equivocator_IM IM) es eqv
     (equivocator_state_update (es eqv) 0 si'),
   oom)
Hproject: equivocators_total_state_project IM
  (state_update (equivocator_IM IM) es eqv
     (equivocator_state_update 
        (es eqv) 0 si')) =
state_update IM s eqv si'
from_option fst []
  match
    match
      EquivocatorsProjections.equivocator_vlsm_transition_item_project
        (IM
           (projT1
              (l
                 {|
                   l := existT eqv (ContinueWith 0 li);
                   input := iom;
                   destination :=
                     state_update (equivocator_IM IM)
                       es eqv
                       (equivocator_state_update
                          (es eqv) 0 si');
                   output := oom
                 |})))
        (ProjectionTraces.composite_transition_item_projection
           (equivocator_IM IM)
           {|
             l := existT eqv (ContinueWith 0 li);
             input := iom;
             destination :=
               state_update (equivocator_IM IM) es eqv
                 (equivocator_state_update (es eqv) 0
                    si');
             output := oom
           |})
        (zero_descriptor IM
           (projT1
              (l
                 {|
                   l := existT eqv (ContinueWith 0 li);
                   input := iom;
                   destination :=
                     state_update (equivocator_IM IM)
                       es eqv
                       (equivocator_state_update
                          (es eqv) 0 si');
                   output := oom
                 |})))
    with
    | Some (Some item', deqv') =>
        Some
          (Some
             {|
               l :=
                 existT
                   (projT1
                      (l
                         {|
                           l :=
                             existT eqv
                               (ContinueWith 0 li);
                           input := iom;
                           destination :=
                             state_update
                               (equivocator_IM IM) es
                               eqv
                               (equivocator_state_update
                                  (es eqv) 0 si');
                           output := oom
                         |})) (l item');
               input :=
                 input
                   {|
                     l :=
                       existT eqv (ContinueWith 0 li);
                     input := iom;
                     destination :=
                       state_update
                         (equivocator_IM IM) es eqv
                         (equivocator_state_update
                            (es eqv) 0 si');
                     output := oom
                   |};
               destination :=
                 equivocators_state_project IM
                   (zero_descriptor IM)
                   (destination
                      {|
                        l :=
                          existT eqv
                            (ContinueWith 0 li);
                        input := iom;
                        destination :=
                          state_update
                            (equivocator_IM IM) es eqv
                            (equivocator_state_update
                               (es eqv) 0 si');
                        output := oom
                      |});
               output :=
                 output
                   {|
                     l :=
                       existT eqv (ContinueWith 0 li);
                     input := iom;
                     destination :=
                       state_update
                         (equivocator_IM IM) es eqv
                         (equivocator_state_update
                            (es eqv) 0 si');
                     output := oom
                   |}
             |},
           equivocator_descriptors_update IM
             (zero_descriptor IM)
             (projT1
                (l
                   {|
                     l :=
                       existT eqv (ContinueWith 0 li);
                     input := iom;
                     destination :=
                       state_update
                         (equivocator_IM IM) es eqv
                         (equivocator_state_update
                            (es eqv) 0 si');
                     output := oom
                   |})) deqv')
    | Some (None, deqv') =>
        Some
          (None,
           equivocator_descriptors_update IM
             (zero_descriptor IM)
             (projT1
                (l
                   {|
                     l :=
                       existT eqv (ContinueWith 0 li);
                     input := iom;
                     destination :=
                       state_update
                         (equivocator_IM IM) es eqv
                         (equivocator_state_update
                            (es eqv) 0 si');
                     output := oom
                   |})) deqv')
    | None => None
    end
  with
  | Some (Some item', odescriptor) =>
      Some ([item'], odescriptor)
  | Some (None, odescriptor) => Some ([], odescriptor)
  | None => None
  end =
[{|
   l := existT eqv li;
   input := iom;
   destination := state_update IM s eqv si';
   output := oom
 |}]
        by cbn; rewrite !equivocator_state_project_zero, decide_True
        ; cbn; repeat f_equal.
      -message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
seed: message → Prop
constraintX: composite_label IM
→ composite_state IM * option message
  → Prop
CX:= preloaded_vlsm (composite_vlsm IM constraintX)
  seed: VLSM message
constraintE: composite_label (equivocator_IM IM)
→ composite_state (equivocator_IM IM) *
  option message → Prop
CE:= preloaded_vlsm
  (composite_vlsm (equivocator_IM IM) constraintE)
  seed: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hsubsumption: zero_descriptor_constraint_lifting_prop
Hreplayable: replayable_message_prop
is, s: state CX
tr: list transition_item
HtrX1: finite_valid_trace_init_to CX is s tr
iom: option message
iom_si, iom_s: state CX
iom_tr: list transition_item
Heqiom: empty_initial_message_or_final_output CX
  iom_tr iom
HtrX2: finite_valid_trace_init_to CX iom_si iom_s
  iom_tr
oom: option message
eqv: index
li: label (IM eqv)
Hs: valid_state_prop CX s
Hiom: option_valid_message_prop CX iom
Hv: valid (existT eqv li) (s, iom)
Hc: constraintX (existT eqv li) (s, iom)
si': state (IM eqv)
Hteqv: transition li (s eqv, iom) = (si', oom)
HinclE: VLSM_incl CE
  (preloaded_with_all_messages_vlsm FreeE)
eqv_state_is: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_start_project: equivocators_total_state_project
  IM eqv_state_is = is
eqv_state_s: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_state_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hstate_project: equivocators_total_trace_project IM
  eqv_state_tr = tr
Hstate_trace: finite_valid_trace_init_to CE
  eqv_state_is eqv_state_s eqv_state_tr
eqv_msg_is: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hmsg_start_project: equivocators_total_state_project
  IM eqv_msg_is = iom_si
eqv_msg_s: state
  (free_composite_vlsm (equivocator_IM IM))
eqv_msg_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hmsg_project: equivocators_total_trace_project IM
  eqv_msg_tr = iom_tr
Hmsg_trace: finite_valid_trace_init_to CE eqv_msg_is
  eqv_msg_s eqv_msg_tr
Hfinal_msg: finite_trace_last_output iom_tr =
finite_trace_last_output eqv_msg_tr
Hstate_valid: valid_state_prop CE eqv_state_s
emsg_tr: list transition_item
es: state CE
Hmsg_trace_full_replay: finite_valid_trace_from_to CE
  eqv_state_s es emsg_tr
Hemsg_tr_pr: equivocators_total_trace_project IM
  emsg_tr = []
Hes_pr: equivocators_total_state_project IM es = s
Hbs_iom: sent_except_from (equivocator_IM IM)
  initial_message_prop es iom
Happ: finite_valid_trace_from_to CE eqv_state_is es
  (eqv_state_tr ++ emsg_tr)
el:= existT eqv (ContinueWith 0 li): {i : index & label (equivocator_IM IM i)}
Happ_extend: finite_valid_trace_from_to CE
  eqv_state_is
  (state_update (equivocator_IM IM) es
     eqv
     (equivocator_state_update 
        (es eqv) 0 si'))
  ((eqv_state_tr ++ emsg_tr) ++
   [{|
      l := el;
      input := iom;
      destination :=
        state_update (equivocator_IM IM)
          es eqv
          (equivocator_state_update
             (es eqv) 0 si');
      output := oom
    |}])
Heqes: finite_trace_last eqv_state_is
  (eqv_state_tr ++ emsg_tr) = es
Hes_pr_i: ∀ i : index,
  equivocators_total_state_project IM es i =
  s i
Het: transition el (es, iom) =
(state_update (equivocator_IM IM) es eqv
   (equivocator_state_update (es eqv) 0 si'),
 oom)
Hes_pr_eqv: equivocator_state_zero (es eqv) = s eqv
Hivt: input_valid_transition CE el (
  es, iom)
  (state_update (equivocator_IM IM) es eqv
     (equivocator_state_update (es eqv) 0 si'),
   oom)
Hproject: equivocators_total_state_project IM
  (state_update (equivocator_IM IM) es eqv
     (equivocator_state_update 
        (es eqv) 0 si')) =
state_update IM s eqv si'
initial_state_prop eqv_state_is by apply Hstate_trace.
      -message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
seed: message → Prop
constraintX: composite_label IM
→ composite_state IM * option message
  → Prop
CX:= preloaded_vlsm (composite_vlsm IM constraintX)
  seed: VLSM message
constraintE: composite_label (equivocator_IM IM)
→ composite_state (equivocator_IM IM) *
  option message → Prop
CE:= preloaded_vlsm
  (composite_vlsm (equivocator_IM IM) constraintE)
  seed: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hsubsumption: zero_descriptor_constraint_lifting_prop
Hreplayable: replayable_message_prop
is, s: state CX
tr: list transition_item
HtrX1: finite_valid_trace_init_to CX is s tr
iom: option message
iom_si, iom_s: state CX
iom_tr: list transition_item
Heqiom: empty_initial_message_or_final_output CX
  iom_tr iom
HtrX2: finite_valid_trace_init_to CX iom_si iom_s
  iom_tr
oom: option message
eqv: index
li: label (IM eqv)
Hs: valid_state_prop CX s
Hiom: option_valid_message_prop CX iom
Hv: valid (existT eqv li) (s, iom)
Hc: constraintX (existT eqv li) (s, iom)
si': state (IM eqv)
Hteqv: transition li (s eqv, iom) = (si', oom)
HinclE: VLSM_incl CE
  (preloaded_with_all_messages_vlsm FreeE)
eqv_state_is: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_start_project: equivocators_total_state_project
  IM eqv_state_is = is
eqv_state_s: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_state_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hstate_project: equivocators_total_trace_project IM
  eqv_state_tr = tr
Hstate_trace: finite_valid_trace_init_to CE
  eqv_state_is eqv_state_s eqv_state_tr
eqv_msg_is: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hmsg_start_project: equivocators_total_state_project
  IM eqv_msg_is = iom_si
eqv_msg_s: state
  (free_composite_vlsm (equivocator_IM IM))
eqv_msg_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hmsg_project: equivocators_total_trace_project IM
  eqv_msg_tr = iom_tr
Hmsg_trace: finite_valid_trace_init_to CE eqv_msg_is
  eqv_msg_s eqv_msg_tr
Hfinal_msg: finite_trace_last_output iom_tr =
finite_trace_last_output eqv_msg_tr
Hstate_valid: valid_state_prop CE eqv_state_s
emsg_tr: list transition_item
es: state CE
Hmsg_trace_full_replay: finite_valid_trace_from_to CE
  eqv_state_s es emsg_tr
Hemsg_tr_pr: equivocators_total_trace_project IM
  emsg_tr = []
Hes_pr: equivocators_total_state_project IM es = s
Hbs_iom: sent_except_from (equivocator_IM IM)
  initial_message_prop es iom
Happ: finite_valid_trace_from_to CE eqv_state_is es
  (eqv_state_tr ++ emsg_tr)
el:= existT eqv (ContinueWith 0 li): {i : index & label (equivocator_IM IM i)}
Happ_extend: finite_valid_trace_from_to CE
  eqv_state_is
  (state_update (equivocator_IM IM) es
     eqv
     (equivocator_state_update 
        (es eqv) 0 si'))
  ((eqv_state_tr ++ emsg_tr) ++
   [{|
      l := el;
      input := iom;
      destination :=
        state_update (equivocator_IM IM)
          es eqv
          (equivocator_state_update
             (es eqv) 0 si');
      output := oom
    |}])
Heqes: finite_trace_last eqv_state_is
  (eqv_state_tr ++ emsg_tr) = es
Hes_pr_i: ∀ i : index,
  equivocators_total_state_project IM es i =
  s i
Het: transition el (es, iom) =
(state_update (equivocator_IM IM) es eqv
   (equivocator_state_update (es eqv) 0 si'),
 oom)
Hes_pr_eqv: equivocator_state_zero (es eqv) = s eqv
Hivt: input_valid_transition CE el (
  es, iom)
  (state_update (equivocator_IM IM) es eqv
     (equivocator_state_update (es eqv) 0 si'),
   oom)
Hproject: equivocators_total_state_project IM
  (state_update (equivocator_IM IM) es eqv
     (equivocator_state_update 
        (es eqv) 0 si')) =
state_update IM s eqv si'
finite_trace_last_output
  (tr ++
   [{|
      l := existT eqv li;
      input := iom;
      destination := state_update IM s eqv si';
      output := oom
    |}]) =
finite_trace_last_output
  ((eqv_state_tr ++ emsg_tr) ++
   [{|
      l := el;
      input := iom;
      destination :=
        state_update (equivocator_IM IM) es eqv
          (equivocator_state_update (es eqv) 0 si');
      output := oom
    |}]) by rewrite! finite_trace_last_output_is_last.
    }message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
seed: message → Prop
constraintX: composite_label IM
→ composite_state IM * option message
  → Prop
CX:= preloaded_vlsm (composite_vlsm IM constraintX)
  seed: VLSM message
constraintE: composite_label (equivocator_IM IM)
→ composite_state (equivocator_IM IM) *
  option message → Prop
CE:= preloaded_vlsm
  (composite_vlsm (equivocator_IM IM) constraintE)
  seed: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hsubsumption: zero_descriptor_constraint_lifting_prop
Hreplayable: replayable_message_prop
is, s: state CX
tr: list transition_item
HtrX1: finite_valid_trace_init_to CX is s tr
iom: option message
iom_si, iom_s: state CX
iom_tr: list transition_item
Heqiom: empty_initial_message_or_final_output CX
  iom_tr iom
HtrX2: finite_valid_trace_init_to CX iom_si iom_s
  iom_tr
oom: option message
eqv: index
li: label (IM eqv)
Hs: valid_state_prop CX s
Hiom: option_valid_message_prop CX iom
Hv: valid (existT eqv li) (s, iom)
Hc: constraintX (existT eqv li) (s, iom)
si': state (IM eqv)
Hteqv: transition li (s eqv, iom) = (si', oom)
HinclE: VLSM_incl CE
  (preloaded_with_all_messages_vlsm FreeE)
eqv_state_is: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_start_project: equivocators_total_state_project
  IM eqv_state_is = is
eqv_state_s: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_state_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hstate_project: equivocators_total_trace_project IM
  eqv_state_tr = tr
Hstate_trace: finite_valid_trace_init_to CE
  eqv_state_is eqv_state_s eqv_state_tr
eqv_msg_is: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hmsg_start_project: equivocators_total_state_project
  IM eqv_msg_is = iom_si
eqv_msg_s: state
  (free_composite_vlsm (equivocator_IM IM))
eqv_msg_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hmsg_project: equivocators_total_trace_project IM
  eqv_msg_tr = iom_tr
Hmsg_trace: finite_valid_trace_init_to CE eqv_msg_is
  eqv_msg_s eqv_msg_tr
Hfinal_msg: finite_trace_last_output iom_tr =
finite_trace_last_output eqv_msg_tr
Hstate_valid: valid_state_prop CE eqv_state_s
emsg_tr: list transition_item
es: state CE
Hmsg_trace_full_replay: finite_valid_trace_from_to CE
  eqv_state_s es emsg_tr
Hemsg_tr_pr: equivocators_total_trace_project IM
  emsg_tr = []
Hes_pr: equivocators_total_state_project IM es = s
Hbs_iom: sent_except_from (equivocator_IM IM)
  initial_message_prop es iom
Happ: finite_valid_trace_from_to CE eqv_state_is es
  (eqv_state_tr ++ emsg_tr)
el:= existT eqv (ContinueWith 0 li): {i : index & label (equivocator_IM IM i)}
Happ_extend: input_valid_transition CE el (
  es, iom)
  (state_update (equivocator_IM IM) es
     eqv
     (equivocator_state_update 
        (es eqv) 0 si'), oom)
→ finite_valid_trace_from_to CE
    eqv_state_is
    (state_update (equivocator_IM IM) es
       eqv
       (equivocator_state_update
          (es eqv) 0 si'))
    ((eqv_state_tr ++ emsg_tr) ++
     [{|
        l := el;
        input := iom;
        destination :=
          state_update
            (equivocator_IM IM) es eqv
            (equivocator_state_update
               (es eqv) 0 si');
        output := oom
      |}])
Heqes: finite_trace_last eqv_state_is
  (eqv_state_tr ++ emsg_tr) = es
Hes_pr_i: ∀ i : index,
  equivocators_total_state_project IM es i =
  s i
Het: transition el (es, iom) =
(state_update (equivocator_IM IM) es eqv
   (equivocator_state_update (es eqv) 0 si'),
 oom)
Hes_pr_eqv: equivocator_state_zero (es eqv) = s eqv
input_valid_transition CE el (es, iom)
  (state_update (equivocator_IM IM) es eqv
     (equivocator_state_update (es eqv) 0 si'), oom)
    clear Happ_extend.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
seed: message → Prop
constraintX: composite_label IM
→ composite_state IM * option message
  → Prop
CX:= preloaded_vlsm (composite_vlsm IM constraintX)
  seed: VLSM message
constraintE: composite_label (equivocator_IM IM)
→ composite_state (equivocator_IM IM) *
  option message → Prop
CE:= preloaded_vlsm
  (composite_vlsm (equivocator_IM IM) constraintE)
  seed: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hsubsumption: zero_descriptor_constraint_lifting_prop
Hreplayable: replayable_message_prop
is, s: state CX
tr: list transition_item
HtrX1: finite_valid_trace_init_to CX is s tr
iom: option message
iom_si, iom_s: state CX
iom_tr: list transition_item
Heqiom: empty_initial_message_or_final_output CX
  iom_tr iom
HtrX2: finite_valid_trace_init_to CX iom_si iom_s
  iom_tr
oom: option message
eqv: index
li: label (IM eqv)
Hs: valid_state_prop CX s
Hiom: option_valid_message_prop CX iom
Hv: valid (existT eqv li) (s, iom)
Hc: constraintX (existT eqv li) (s, iom)
si': state (IM eqv)
Hteqv: transition li (s eqv, iom) = (si', oom)
HinclE: VLSM_incl CE
  (preloaded_with_all_messages_vlsm FreeE)
eqv_state_is: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_start_project: equivocators_total_state_project
  IM eqv_state_is = is
eqv_state_s: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_state_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hstate_project: equivocators_total_trace_project IM
  eqv_state_tr = tr
Hstate_trace: finite_valid_trace_init_to CE
  eqv_state_is eqv_state_s eqv_state_tr
eqv_msg_is: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hmsg_start_project: equivocators_total_state_project
  IM eqv_msg_is = iom_si
eqv_msg_s: state
  (free_composite_vlsm (equivocator_IM IM))
eqv_msg_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hmsg_project: equivocators_total_trace_project IM
  eqv_msg_tr = iom_tr
Hmsg_trace: finite_valid_trace_init_to CE eqv_msg_is
  eqv_msg_s eqv_msg_tr
Hfinal_msg: finite_trace_last_output iom_tr =
finite_trace_last_output eqv_msg_tr
Hstate_valid: valid_state_prop CE eqv_state_s
emsg_tr: list transition_item
es: state CE
Hmsg_trace_full_replay: finite_valid_trace_from_to CE
  eqv_state_s es emsg_tr
Hemsg_tr_pr: equivocators_total_trace_project IM
  emsg_tr = []
Hes_pr: equivocators_total_state_project IM es = s
Hbs_iom: sent_except_from (equivocator_IM IM)
  initial_message_prop es iom
Happ: finite_valid_trace_from_to CE eqv_state_is es
  (eqv_state_tr ++ emsg_tr)
el:= existT eqv (ContinueWith 0 li): {i : index & label (equivocator_IM IM i)}
Heqes: finite_trace_last eqv_state_is
  (eqv_state_tr ++ emsg_tr) = es
Hes_pr_i: ∀ i : index,
  equivocators_total_state_project IM es i =
  s i
Het: transition el (es, iom) =
(state_update (equivocator_IM IM) es eqv
   (equivocator_state_update (es eqv) 0 si'),
 oom)
Hes_pr_eqv: equivocator_state_zero (es eqv) = s eqv
input_valid_transition CE el (es, iom)
  (state_update (equivocator_IM IM) es eqv
     (equivocator_state_update (es eqv) 0 si'), oom)
    apply valid_trace_last_pstate in Happ.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
seed: message → Prop
constraintX: composite_label IM
→ composite_state IM * option message
  → Prop
CX:= preloaded_vlsm (composite_vlsm IM constraintX)
  seed: VLSM message
constraintE: composite_label (equivocator_IM IM)
→ composite_state (equivocator_IM IM) *
  option message → Prop
CE:= preloaded_vlsm
  (composite_vlsm (equivocator_IM IM) constraintE)
  seed: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hsubsumption: zero_descriptor_constraint_lifting_prop
Hreplayable: replayable_message_prop
is, s: state CX
tr: list transition_item
HtrX1: finite_valid_trace_init_to CX is s tr
iom: option message
iom_si, iom_s: state CX
iom_tr: list transition_item
Heqiom: empty_initial_message_or_final_output CX
  iom_tr iom
HtrX2: finite_valid_trace_init_to CX iom_si iom_s
  iom_tr
oom: option message
eqv: index
li: label (IM eqv)
Hs: valid_state_prop CX s
Hiom: option_valid_message_prop CX iom
Hv: valid (existT eqv li) (s, iom)
Hc: constraintX (existT eqv li) (s, iom)
si': state (IM eqv)
Hteqv: transition li (s eqv, iom) = (si', oom)
HinclE: VLSM_incl CE
  (preloaded_with_all_messages_vlsm FreeE)
eqv_state_is: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_start_project: equivocators_total_state_project
  IM eqv_state_is = is
eqv_state_s: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_state_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hstate_project: equivocators_total_trace_project IM
  eqv_state_tr = tr
Hstate_trace: finite_valid_trace_init_to CE
  eqv_state_is eqv_state_s eqv_state_tr
eqv_msg_is: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hmsg_start_project: equivocators_total_state_project
  IM eqv_msg_is = iom_si
eqv_msg_s: state
  (free_composite_vlsm (equivocator_IM IM))
eqv_msg_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hmsg_project: equivocators_total_trace_project IM
  eqv_msg_tr = iom_tr
Hmsg_trace: finite_valid_trace_init_to CE eqv_msg_is
  eqv_msg_s eqv_msg_tr
Hfinal_msg: finite_trace_last_output iom_tr =
finite_trace_last_output eqv_msg_tr
Hstate_valid: valid_state_prop CE eqv_state_s
emsg_tr: list transition_item
es: state CE
Hmsg_trace_full_replay: finite_valid_trace_from_to CE
  eqv_state_s es emsg_tr
Hemsg_tr_pr: equivocators_total_trace_project IM
  emsg_tr = []
Hes_pr: equivocators_total_state_project IM es = s
Hbs_iom: sent_except_from (equivocator_IM IM)
  initial_message_prop es iom
Happ: valid_state_prop CE es
el:= existT eqv (ContinueWith 0 li): {i : index & label (equivocator_IM IM i)}
Heqes: finite_trace_last eqv_state_is
  (eqv_state_tr ++ emsg_tr) = es
Hes_pr_i: ∀ i : index,
  equivocators_total_state_project IM es i =
  s i
Het: transition el (es, iom) =
(state_update (equivocator_IM IM) es eqv
   (equivocator_state_update (es eqv) 0 si'),
 oom)
Hes_pr_eqv: equivocator_state_zero (es eqv) = s eqv
input_valid_transition CE el (es, iom)
  (state_update (equivocator_IM IM) es eqv
     (equivocator_state_update (es eqv) 0 si'), oom)
    repeat split; [done | .. | done].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
seed: message → Prop
constraintX: composite_label IM
→ composite_state IM * option message
  → Prop
CX:= preloaded_vlsm (composite_vlsm IM constraintX)
  seed: VLSM message
constraintE: composite_label (equivocator_IM IM)
→ composite_state (equivocator_IM IM) *
  option message → Prop
CE:= preloaded_vlsm
  (composite_vlsm (equivocator_IM IM) constraintE)
  seed: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hsubsumption: zero_descriptor_constraint_lifting_prop
Hreplayable: replayable_message_prop
is, s: state CX
tr: list transition_item
HtrX1: finite_valid_trace_init_to CX is s tr
iom: option message
iom_si, iom_s: state CX
iom_tr: list transition_item
Heqiom: empty_initial_message_or_final_output CX
  iom_tr iom
HtrX2: finite_valid_trace_init_to CX iom_si iom_s
  iom_tr
oom: option message
eqv: index
li: label (IM eqv)
Hs: valid_state_prop CX s
Hiom: option_valid_message_prop CX iom
Hv: valid (existT eqv li) (s, iom)
Hc: constraintX (existT eqv li) (s, iom)
si': state (IM eqv)
Hteqv: transition li (s eqv, iom) = (si', oom)
HinclE: VLSM_incl CE
  (preloaded_with_all_messages_vlsm FreeE)
eqv_state_is: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_start_project: equivocators_total_state_project
  IM eqv_state_is = is
eqv_state_s: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_state_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hstate_project: equivocators_total_trace_project IM
  eqv_state_tr = tr
Hstate_trace: finite_valid_trace_init_to CE
  eqv_state_is eqv_state_s eqv_state_tr
eqv_msg_is: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hmsg_start_project: equivocators_total_state_project
  IM eqv_msg_is = iom_si
eqv_msg_s: state
  (free_composite_vlsm (equivocator_IM IM))
eqv_msg_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hmsg_project: equivocators_total_trace_project IM
  eqv_msg_tr = iom_tr
Hmsg_trace: finite_valid_trace_init_to CE eqv_msg_is
  eqv_msg_s eqv_msg_tr
Hfinal_msg: finite_trace_last_output iom_tr =
finite_trace_last_output eqv_msg_tr
Hstate_valid: valid_state_prop CE eqv_state_s
emsg_tr: list transition_item
es: state CE
Hmsg_trace_full_replay: finite_valid_trace_from_to CE
  eqv_state_s es emsg_tr
Hemsg_tr_pr: equivocators_total_trace_project IM
  emsg_tr = []
Hes_pr: equivocators_total_state_project IM es = s
Hbs_iom: sent_except_from (equivocator_IM IM)
  initial_message_prop es iom
Happ: valid_state_prop CE es
el:= existT eqv (ContinueWith 0 li): {i : index & label (equivocator_IM IM i)}
Heqes: finite_trace_last eqv_state_is
  (eqv_state_tr ++ emsg_tr) = es
Hes_pr_i: ∀ i : index,
  equivocators_total_state_project IM es i =
  s i
Het: transition el (es, iom) =
(state_update (equivocator_IM IM) es eqv
   (equivocator_state_update (es eqv) 0 si'),
 oom)
Hes_pr_eqv: equivocator_state_zero (es eqv) = s eqv
option_valid_message_prop CE iom
message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
seed: message → Prop
constraintX: composite_label IM
→ composite_state IM * option message
  → Prop
CX:= preloaded_vlsm (composite_vlsm IM constraintX)
  seed: VLSM message
constraintE: composite_label (equivocator_IM IM)
→ composite_state (equivocator_IM IM) *
  option message → Prop
CE:= preloaded_vlsm
  (composite_vlsm (equivocator_IM IM) constraintE)
  seed: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hsubsumption: zero_descriptor_constraint_lifting_prop
Hreplayable: replayable_message_prop
is, s: state CX
tr: list transition_item
HtrX1: finite_valid_trace_init_to CX is s tr
iom: option message
iom_si, iom_s: state CX
iom_tr: list transition_item
Heqiom: empty_initial_message_or_final_output CX
  iom_tr iom
HtrX2: finite_valid_trace_init_to CX iom_si iom_s
  iom_tr
oom: option message
eqv: index
li: label (IM eqv)
Hs: valid_state_prop CX s
Hiom: option_valid_message_prop CX iom
Hv: valid (existT eqv li) (s, iom)
Hc: constraintX (existT eqv li) (s, iom)
si': state (IM eqv)
Hteqv: transition li (s eqv, iom) = (si', oom)
HinclE: VLSM_incl CE
  (preloaded_with_all_messages_vlsm FreeE)
eqv_state_is: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_start_project: equivocators_total_state_project
  IM eqv_state_is = is
eqv_state_s: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_state_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hstate_project: equivocators_total_trace_project IM
  eqv_state_tr = tr
Hstate_trace: finite_valid_trace_init_to CE
  eqv_state_is eqv_state_s eqv_state_tr
eqv_msg_is: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hmsg_start_project: equivocators_total_state_project
  IM eqv_msg_is = iom_si
eqv_msg_s: state
  (free_composite_vlsm (equivocator_IM IM))
eqv_msg_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hmsg_project: equivocators_total_trace_project IM
  eqv_msg_tr = iom_tr
Hmsg_trace: finite_valid_trace_init_to CE eqv_msg_is
  eqv_msg_s eqv_msg_tr
Hfinal_msg: finite_trace_last_output iom_tr =
finite_trace_last_output eqv_msg_tr
Hstate_valid: valid_state_prop CE eqv_state_s
emsg_tr: list transition_item
es: state CE
Hmsg_trace_full_replay: finite_valid_trace_from_to CE
  eqv_state_s es emsg_tr
Hemsg_tr_pr: equivocators_total_trace_project IM
  emsg_tr = []
Hes_pr: equivocators_total_state_project IM es = s
Hbs_iom: sent_except_from (equivocator_IM IM)
  initial_message_prop es iom
Happ: valid_state_prop CE es
el:= existT eqv (ContinueWith 0 li): {i : index & label (equivocator_IM IM i)}
Heqes: finite_trace_last eqv_state_is
  (eqv_state_tr ++ emsg_tr) = es
Hes_pr_i: ∀ i : index,
  equivocators_total_state_project IM es i =
  s i
Het: transition el (es, iom) =
(state_update (equivocator_IM IM) es eqv
   (equivocator_state_update (es eqv) 0 si'),
 oom)
Hes_pr_eqv: equivocator_state_zero (es eqv) = s eqv
valid el (es, iom)
message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
seed: message → Prop
constraintX: composite_label IM
→ composite_state IM * option message
  → Prop
CX:= preloaded_vlsm (composite_vlsm IM constraintX)
  seed: VLSM message
constraintE: composite_label (equivocator_IM IM)
→ composite_state (equivocator_IM IM) *
  option message → Prop
CE:= preloaded_vlsm
  (composite_vlsm (equivocator_IM IM) constraintE)
  seed: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hsubsumption: zero_descriptor_constraint_lifting_prop
Hreplayable: replayable_message_prop
is, s: state CX
tr: list transition_item
HtrX1: finite_valid_trace_init_to CX is s tr
iom: option message
iom_si, iom_s: state CX
iom_tr: list transition_item
Heqiom: empty_initial_message_or_final_output CX
  iom_tr iom
HtrX2: finite_valid_trace_init_to CX iom_si iom_s
  iom_tr
oom: option message
eqv: index
li: label (IM eqv)
Hs: valid_state_prop CX s
Hiom: option_valid_message_prop CX iom
Hv: valid (existT eqv li) (s, iom)
Hc: constraintX (existT eqv li) (s, iom)
si': state (IM eqv)
Hteqv: transition li (s eqv, iom) = (si', oom)
HinclE: VLSM_incl CE
  (preloaded_with_all_messages_vlsm FreeE)
eqv_state_is: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_start_project: equivocators_total_state_project
  IM eqv_state_is = is
eqv_state_s: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_state_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hstate_project: equivocators_total_trace_project IM
  eqv_state_tr = tr
Hstate_trace: finite_valid_trace_init_to CE
  eqv_state_is eqv_state_s eqv_state_tr
eqv_msg_is: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hmsg_start_project: equivocators_total_state_project
  IM eqv_msg_is = iom_si
eqv_msg_s: state
  (free_composite_vlsm (equivocator_IM IM))
eqv_msg_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hmsg_project: equivocators_total_trace_project IM
  eqv_msg_tr = iom_tr
Hmsg_trace: finite_valid_trace_init_to CE eqv_msg_is
  eqv_msg_s eqv_msg_tr
Hfinal_msg: finite_trace_last_output iom_tr =
finite_trace_last_output eqv_msg_tr
Hstate_valid: valid_state_prop CE eqv_state_s
emsg_tr: list transition_item
es: state CE
Hmsg_trace_full_replay: finite_valid_trace_from_to CE
  eqv_state_s es emsg_tr
Hemsg_tr_pr: equivocators_total_trace_project IM
  emsg_tr = []
Hes_pr: equivocators_total_state_project IM es = s
Hbs_iom: sent_except_from (equivocator_IM IM)
  initial_message_prop es iom
Happ: valid_state_prop CE es
el:= existT eqv (ContinueWith 0 li): {i : index & label (equivocator_IM IM i)}
Heqes: finite_trace_last eqv_state_is
  (eqv_state_tr ++ emsg_tr) = es
Hes_pr_i: ∀ i : index,
  equivocators_total_state_project IM es i =
  s i
Het: transition el (es, iom) =
(state_update (equivocator_IM IM) es eqv
   (equivocator_state_update (es eqv) 0 si'),
 oom)
Hes_pr_eqv: equivocator_state_zero (es eqv) = s eqv
constraintE el (es, iom)
    +message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
seed: message → Prop
constraintX: composite_label IM
→ composite_state IM * option message
  → Prop
CX:= preloaded_vlsm (composite_vlsm IM constraintX)
  seed: VLSM message
constraintE: composite_label (equivocator_IM IM)
→ composite_state (equivocator_IM IM) *
  option message → Prop
CE:= preloaded_vlsm
  (composite_vlsm (equivocator_IM IM) constraintE)
  seed: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hsubsumption: zero_descriptor_constraint_lifting_prop
Hreplayable: replayable_message_prop
is, s: state CX
tr: list transition_item
HtrX1: finite_valid_trace_init_to CX is s tr
iom: option message
iom_si, iom_s: state CX
iom_tr: list transition_item
Heqiom: empty_initial_message_or_final_output CX
  iom_tr iom
HtrX2: finite_valid_trace_init_to CX iom_si iom_s
  iom_tr
oom: option message
eqv: index
li: label (IM eqv)
Hs: valid_state_prop CX s
Hiom: option_valid_message_prop CX iom
Hv: valid (existT eqv li) (s, iom)
Hc: constraintX (existT eqv li) (s, iom)
si': state (IM eqv)
Hteqv: transition li (s eqv, iom) = (si', oom)
HinclE: VLSM_incl CE
  (preloaded_with_all_messages_vlsm FreeE)
eqv_state_is: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_start_project: equivocators_total_state_project
  IM eqv_state_is = is
eqv_state_s: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_state_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hstate_project: equivocators_total_trace_project IM
  eqv_state_tr = tr
Hstate_trace: finite_valid_trace_init_to CE
  eqv_state_is eqv_state_s eqv_state_tr
eqv_msg_is: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hmsg_start_project: equivocators_total_state_project
  IM eqv_msg_is = iom_si
eqv_msg_s: state
  (free_composite_vlsm (equivocator_IM IM))
eqv_msg_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hmsg_project: equivocators_total_trace_project IM
  eqv_msg_tr = iom_tr
Hmsg_trace: finite_valid_trace_init_to CE eqv_msg_is
  eqv_msg_s eqv_msg_tr
Hfinal_msg: finite_trace_last_output iom_tr =
finite_trace_last_output eqv_msg_tr
Hstate_valid: valid_state_prop CE eqv_state_s
emsg_tr: list transition_item
es: state CE
Hmsg_trace_full_replay: finite_valid_trace_from_to CE
  eqv_state_s es emsg_tr
Hemsg_tr_pr: equivocators_total_trace_project IM
  emsg_tr = []
Hes_pr: equivocators_total_state_project IM es = s
Hbs_iom: sent_except_from (equivocator_IM IM)
  initial_message_prop es iom
Happ: valid_state_prop CE es
el:= existT eqv (ContinueWith 0 li): {i : index & label (equivocator_IM IM i)}
Heqes: finite_trace_last eqv_state_is
  (eqv_state_tr ++ emsg_tr) = es
Hes_pr_i: ∀ i : index,
  equivocators_total_state_project IM es i =
  s i
Het: transition el (es, iom) =
(state_update (equivocator_IM IM) es eqv
   (equivocator_state_update (es eqv) 0 si'),
 oom)
Hes_pr_eqv: equivocator_state_zero (es eqv) = s eqv
option_valid_message_prop CE iom destruct iom as [im |]; [| by apply option_valid_message_None].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
seed: message → Prop
constraintX: composite_label IM
→ composite_state IM * option message
  → Prop
CX:= preloaded_vlsm (composite_vlsm IM constraintX)
  seed: VLSM message
constraintE: composite_label (equivocator_IM IM)
→ composite_state (equivocator_IM IM) *
  option message → Prop
CE:= preloaded_vlsm
  (composite_vlsm (equivocator_IM IM) constraintE)
  seed: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hsubsumption: zero_descriptor_constraint_lifting_prop
Hreplayable: replayable_message_prop
is, s: state CX
tr: list transition_item
HtrX1: finite_valid_trace_init_to CX is s tr
im: message
iom_si, iom_s: state CX
iom_tr: list transition_item
Heqiom: empty_initial_message_or_final_output CX
  iom_tr (Some im)
HtrX2: finite_valid_trace_init_to CX iom_si iom_s
  iom_tr
oom: option message
eqv: index
li: label (IM eqv)
Hs: valid_state_prop CX s
Hiom: option_valid_message_prop CX (Some im)
Hv: valid (existT eqv li) (s, Some im)
Hc: constraintX (existT eqv li) (s, Some im)
si': state (IM eqv)
Hteqv: transition li (s eqv, Some im) = (si', oom)
HinclE: VLSM_incl CE
  (preloaded_with_all_messages_vlsm FreeE)
eqv_state_is: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_start_project: equivocators_total_state_project
  IM eqv_state_is = is
eqv_state_s: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_state_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hstate_project: equivocators_total_trace_project IM
  eqv_state_tr = tr
Hstate_trace: finite_valid_trace_init_to CE
  eqv_state_is eqv_state_s eqv_state_tr
eqv_msg_is: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hmsg_start_project: equivocators_total_state_project
  IM eqv_msg_is = iom_si
eqv_msg_s: state
  (free_composite_vlsm (equivocator_IM IM))
eqv_msg_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hmsg_project: equivocators_total_trace_project IM
  eqv_msg_tr = iom_tr
Hmsg_trace: finite_valid_trace_init_to CE eqv_msg_is
  eqv_msg_s eqv_msg_tr
Hfinal_msg: finite_trace_last_output iom_tr =
finite_trace_last_output eqv_msg_tr
Hstate_valid: valid_state_prop CE eqv_state_s
emsg_tr: list transition_item
es: state CE
Hmsg_trace_full_replay: finite_valid_trace_from_to CE
  eqv_state_s es emsg_tr
Hemsg_tr_pr: equivocators_total_trace_project IM
  emsg_tr = []
Hes_pr: equivocators_total_state_project IM es = s
Hbs_iom: sent_except_from (equivocator_IM IM)
  initial_message_prop es 
  (Some im)
Happ: valid_state_prop CE es
el:= existT eqv (ContinueWith 0 li): {i : index & label (equivocator_IM IM i)}
Heqes: finite_trace_last eqv_state_is
  (eqv_state_tr ++ emsg_tr) = es
Hes_pr_i: ∀ i : index,
  equivocators_total_state_project IM es i =
  s i
Het: transition el (es, Some im) =
(state_update (equivocator_IM IM) es eqv
   (equivocator_state_update (es eqv) 0 si'),
 oom)
Hes_pr_eqv: equivocator_state_zero (es eqv) = s eqv
option_valid_message_prop CE (Some im)
      destruct Hbs_iom as [Hbs_iom | Hseeded].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
seed: message → Prop
constraintX: composite_label IM
→ composite_state IM * option message
  → Prop
CX:= preloaded_vlsm (composite_vlsm IM constraintX)
  seed: VLSM message
constraintE: composite_label (equivocator_IM IM)
→ composite_state (equivocator_IM IM) *
  option message → Prop
CE:= preloaded_vlsm
  (composite_vlsm (equivocator_IM IM) constraintE)
  seed: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hsubsumption: zero_descriptor_constraint_lifting_prop
Hreplayable: replayable_message_prop
is, s: state CX
tr: list transition_item
HtrX1: finite_valid_trace_init_to CX is s tr
im: message
iom_si, iom_s: state CX
iom_tr: list transition_item
Heqiom: empty_initial_message_or_final_output CX
  iom_tr (Some im)
HtrX2: finite_valid_trace_init_to CX iom_si iom_s
  iom_tr
oom: option message
eqv: index
li: label (IM eqv)
Hs: valid_state_prop CX s
Hiom: option_valid_message_prop CX (Some im)
Hv: valid (existT eqv li) (s, Some im)
Hc: constraintX (existT eqv li) (s, Some im)
si': state (IM eqv)
Hteqv: transition li (s eqv, Some im) = (si', oom)
HinclE: VLSM_incl CE
  (preloaded_with_all_messages_vlsm FreeE)
eqv_state_is: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_start_project: equivocators_total_state_project
  IM eqv_state_is = is
eqv_state_s: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_state_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hstate_project: equivocators_total_trace_project IM
  eqv_state_tr = tr
Hstate_trace: finite_valid_trace_init_to CE
  eqv_state_is eqv_state_s eqv_state_tr
eqv_msg_is: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hmsg_start_project: equivocators_total_state_project
  IM eqv_msg_is = iom_si
eqv_msg_s: state
  (free_composite_vlsm (equivocator_IM IM))
eqv_msg_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hmsg_project: equivocators_total_trace_project IM
  eqv_msg_tr = iom_tr
Hmsg_trace: finite_valid_trace_init_to CE eqv_msg_is
  eqv_msg_s eqv_msg_tr
Hfinal_msg: finite_trace_last_output iom_tr =
finite_trace_last_output eqv_msg_tr
Hstate_valid: valid_state_prop CE eqv_state_s
emsg_tr: list transition_item
es: state CE
Hmsg_trace_full_replay: finite_valid_trace_from_to CE
  eqv_state_s es emsg_tr
Hemsg_tr_pr: equivocators_total_trace_project IM
  emsg_tr = []
Hes_pr: equivocators_total_state_project IM es = s
Hbs_iom: composite_has_been_sent 
  (equivocator_IM IM) es im
Happ: valid_state_prop CE es
el:= existT eqv (ContinueWith 0 li): {i : index & label (equivocator_IM IM i)}
Heqes: finite_trace_last eqv_state_is
  (eqv_state_tr ++ emsg_tr) = es
Hes_pr_i: ∀ i : index,
  equivocators_total_state_project IM es i =
  s i
Het: transition el (es, Some im) =
(state_update (equivocator_IM IM) es eqv
   (equivocator_state_update (es eqv) 0 si'),
 oom)
Hes_pr_eqv: equivocator_state_zero (es eqv) = s eqv
option_valid_message_prop CE (Some im)
message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
seed: message → Prop
constraintX: composite_label IM
→ composite_state IM * option message
  → Prop
CX:= preloaded_vlsm (composite_vlsm IM constraintX)
  seed: VLSM message
constraintE: composite_label (equivocator_IM IM)
→ composite_state (equivocator_IM IM) *
  option message → Prop
CE:= preloaded_vlsm
  (composite_vlsm (equivocator_IM IM) constraintE)
  seed: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hsubsumption: zero_descriptor_constraint_lifting_prop
Hreplayable: replayable_message_prop
is, s: state CX
tr: list transition_item
HtrX1: finite_valid_trace_init_to CX is s tr
im: message
iom_si, iom_s: state CX
iom_tr: list transition_item
Heqiom: empty_initial_message_or_final_output CX
  iom_tr (Some im)
HtrX2: finite_valid_trace_init_to CX iom_si iom_s
  iom_tr
oom: option message
eqv: index
li: label (IM eqv)
Hs: valid_state_prop CX s
Hiom: option_valid_message_prop CX (Some im)
Hv: valid (existT eqv li) (s, Some im)
Hc: constraintX (existT eqv li) (s, Some im)
si': state (IM eqv)
Hteqv: transition li (s eqv, Some im) = (si', oom)
HinclE: VLSM_incl CE
  (preloaded_with_all_messages_vlsm FreeE)
eqv_state_is: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_start_project: equivocators_total_state_project
  IM eqv_state_is = is
eqv_state_s: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_state_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hstate_project: equivocators_total_trace_project IM
  eqv_state_tr = tr
Hstate_trace: finite_valid_trace_init_to CE
  eqv_state_is eqv_state_s eqv_state_tr
eqv_msg_is: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hmsg_start_project: equivocators_total_state_project
  IM eqv_msg_is = iom_si
eqv_msg_s: state
  (free_composite_vlsm (equivocator_IM IM))
eqv_msg_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hmsg_project: equivocators_total_trace_project IM
  eqv_msg_tr = iom_tr
Hmsg_trace: finite_valid_trace_init_to CE eqv_msg_is
  eqv_msg_s eqv_msg_tr
Hfinal_msg: finite_trace_last_output iom_tr =
finite_trace_last_output eqv_msg_tr
Hstate_valid: valid_state_prop CE eqv_state_s
emsg_tr: list transition_item
es: state CE
Hmsg_trace_full_replay: finite_valid_trace_from_to CE
  eqv_state_s es emsg_tr
Hemsg_tr_pr: equivocators_total_trace_project IM
  emsg_tr = []
Hes_pr: equivocators_total_state_project IM es = s
Hseeded: initial_message_prop im
Happ: valid_state_prop CE es
el:= existT eqv (ContinueWith 0 li): {i : index & label (equivocator_IM IM i)}
Heqes: finite_trace_last eqv_state_is
  (eqv_state_tr ++ emsg_tr) = es
Hes_pr_i: ∀ i : index,
  equivocators_total_state_project IM es i =
  s i
Het: transition el (es, Some im) =
(state_update (equivocator_IM IM) es eqv
   (equivocator_state_update (es eqv) 0 si'),
 oom)
Hes_pr_eqv: equivocator_state_zero (es eqv) = s eqv
option_valid_message_prop CE (Some im)
      *message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
seed: message → Prop
constraintX: composite_label IM
→ composite_state IM * option message
  → Prop
CX:= preloaded_vlsm (composite_vlsm IM constraintX)
  seed: VLSM message
constraintE: composite_label (equivocator_IM IM)
→ composite_state (equivocator_IM IM) *
  option message → Prop
CE:= preloaded_vlsm
  (composite_vlsm (equivocator_IM IM) constraintE)
  seed: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hsubsumption: zero_descriptor_constraint_lifting_prop
Hreplayable: replayable_message_prop
is, s: state CX
tr: list transition_item
HtrX1: finite_valid_trace_init_to CX is s tr
im: message
iom_si, iom_s: state CX
iom_tr: list transition_item
Heqiom: empty_initial_message_or_final_output CX
  iom_tr (Some im)
HtrX2: finite_valid_trace_init_to CX iom_si iom_s
  iom_tr
oom: option message
eqv: index
li: label (IM eqv)
Hs: valid_state_prop CX s
Hiom: option_valid_message_prop CX (Some im)
Hv: valid (existT eqv li) (s, Some im)
Hc: constraintX (existT eqv li) (s, Some im)
si': state (IM eqv)
Hteqv: transition li (s eqv, Some im) = (si', oom)
HinclE: VLSM_incl CE
  (preloaded_with_all_messages_vlsm FreeE)
eqv_state_is: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_start_project: equivocators_total_state_project
  IM eqv_state_is = is
eqv_state_s: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_state_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hstate_project: equivocators_total_trace_project IM
  eqv_state_tr = tr
Hstate_trace: finite_valid_trace_init_to CE
  eqv_state_is eqv_state_s eqv_state_tr
eqv_msg_is: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hmsg_start_project: equivocators_total_state_project
  IM eqv_msg_is = iom_si
eqv_msg_s: state
  (free_composite_vlsm (equivocator_IM IM))
eqv_msg_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hmsg_project: equivocators_total_trace_project IM
  eqv_msg_tr = iom_tr
Hmsg_trace: finite_valid_trace_init_to CE eqv_msg_is
  eqv_msg_s eqv_msg_tr
Hfinal_msg: finite_trace_last_output iom_tr =
finite_trace_last_output eqv_msg_tr
Hstate_valid: valid_state_prop CE eqv_state_s
emsg_tr: list transition_item
es: state CE
Hmsg_trace_full_replay: finite_valid_trace_from_to CE
  eqv_state_s es emsg_tr
Hemsg_tr_pr: equivocators_total_trace_project IM
  emsg_tr = []
Hes_pr: equivocators_total_state_project IM es = s
Hbs_iom: composite_has_been_sent 
  (equivocator_IM IM) es im
Happ: valid_state_prop CE es
el:= existT eqv (ContinueWith 0 li): {i : index & label (equivocator_IM IM i)}
Heqes: finite_trace_last eqv_state_is
  (eqv_state_tr ++ emsg_tr) = es
Hes_pr_i: ∀ i : index,
  equivocators_total_state_project IM es i =
  s i
Het: transition el (es, Some im) =
(state_update (equivocator_IM IM) es eqv
   (equivocator_state_update (es eqv) 0 si'),
 oom)
Hes_pr_eqv: equivocator_state_zero (es eqv) = s eqv
option_valid_message_prop CE (Some im) by apply (preloaded_composite_sent_valid (equivocator_IM IM) _ _ _ Happ _ Hbs_iom).
      *message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
seed: message → Prop
constraintX: composite_label IM
→ composite_state IM * option message
  → Prop
CX:= preloaded_vlsm (composite_vlsm IM constraintX)
  seed: VLSM message
constraintE: composite_label (equivocator_IM IM)
→ composite_state (equivocator_IM IM) *
  option message → Prop
CE:= preloaded_vlsm
  (composite_vlsm (equivocator_IM IM) constraintE)
  seed: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hsubsumption: zero_descriptor_constraint_lifting_prop
Hreplayable: replayable_message_prop
is, s: state CX
tr: list transition_item
HtrX1: finite_valid_trace_init_to CX is s tr
im: message
iom_si, iom_s: state CX
iom_tr: list transition_item
Heqiom: empty_initial_message_or_final_output CX
  iom_tr (Some im)
HtrX2: finite_valid_trace_init_to CX iom_si iom_s
  iom_tr
oom: option message
eqv: index
li: label (IM eqv)
Hs: valid_state_prop CX s
Hiom: option_valid_message_prop CX (Some im)
Hv: valid (existT eqv li) (s, Some im)
Hc: constraintX (existT eqv li) (s, Some im)
si': state (IM eqv)
Hteqv: transition li (s eqv, Some im) = (si', oom)
HinclE: VLSM_incl CE
  (preloaded_with_all_messages_vlsm FreeE)
eqv_state_is: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_start_project: equivocators_total_state_project
  IM eqv_state_is = is
eqv_state_s: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_state_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hstate_project: equivocators_total_trace_project IM
  eqv_state_tr = tr
Hstate_trace: finite_valid_trace_init_to CE
  eqv_state_is eqv_state_s eqv_state_tr
eqv_msg_is: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hmsg_start_project: equivocators_total_state_project
  IM eqv_msg_is = iom_si
eqv_msg_s: state
  (free_composite_vlsm (equivocator_IM IM))
eqv_msg_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hmsg_project: equivocators_total_trace_project IM
  eqv_msg_tr = iom_tr
Hmsg_trace: finite_valid_trace_init_to CE eqv_msg_is
  eqv_msg_s eqv_msg_tr
Hfinal_msg: finite_trace_last_output iom_tr =
finite_trace_last_output eqv_msg_tr
Hstate_valid: valid_state_prop CE eqv_state_s
emsg_tr: list transition_item
es: state CE
Hmsg_trace_full_replay: finite_valid_trace_from_to CE
  eqv_state_s es emsg_tr
Hemsg_tr_pr: equivocators_total_trace_project IM
  emsg_tr = []
Hes_pr: equivocators_total_state_project IM es = s
Hseeded: initial_message_prop im
Happ: valid_state_prop CE es
el:= existT eqv (ContinueWith 0 li): {i : index & label (equivocator_IM IM i)}
Heqes: finite_trace_last eqv_state_is
  (eqv_state_tr ++ emsg_tr) = es
Hes_pr_i: ∀ i : index,
  equivocators_total_state_project IM es i =
  s i
Het: transition el (es, Some im) =
(state_update (equivocator_IM IM) es eqv
   (equivocator_state_update (es eqv) 0 si'),
 oom)
Hes_pr_eqv: equivocator_state_zero (es eqv) = s eqv
option_valid_message_prop CE (Some im) by apply initial_message_is_valid.
    +message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
seed: message → Prop
constraintX: composite_label IM
→ composite_state IM * option message
  → Prop
CX:= preloaded_vlsm (composite_vlsm IM constraintX)
  seed: VLSM message
constraintE: composite_label (equivocator_IM IM)
→ composite_state (equivocator_IM IM) *
  option message → Prop
CE:= preloaded_vlsm
  (composite_vlsm (equivocator_IM IM) constraintE)
  seed: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hsubsumption: zero_descriptor_constraint_lifting_prop
Hreplayable: replayable_message_prop
is, s: state CX
tr: list transition_item
HtrX1: finite_valid_trace_init_to CX is s tr
iom: option message
iom_si, iom_s: state CX
iom_tr: list transition_item
Heqiom: empty_initial_message_or_final_output CX
  iom_tr iom
HtrX2: finite_valid_trace_init_to CX iom_si iom_s
  iom_tr
oom: option message
eqv: index
li: label (IM eqv)
Hs: valid_state_prop CX s
Hiom: option_valid_message_prop CX iom
Hv: valid (existT eqv li) (s, iom)
Hc: constraintX (existT eqv li) (s, iom)
si': state (IM eqv)
Hteqv: transition li (s eqv, iom) = (si', oom)
HinclE: VLSM_incl CE
  (preloaded_with_all_messages_vlsm FreeE)
eqv_state_is: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_start_project: equivocators_total_state_project
  IM eqv_state_is = is
eqv_state_s: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_state_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hstate_project: equivocators_total_trace_project IM
  eqv_state_tr = tr
Hstate_trace: finite_valid_trace_init_to CE
  eqv_state_is eqv_state_s eqv_state_tr
eqv_msg_is: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hmsg_start_project: equivocators_total_state_project
  IM eqv_msg_is = iom_si
eqv_msg_s: state
  (free_composite_vlsm (equivocator_IM IM))
eqv_msg_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hmsg_project: equivocators_total_trace_project IM
  eqv_msg_tr = iom_tr
Hmsg_trace: finite_valid_trace_init_to CE eqv_msg_is
  eqv_msg_s eqv_msg_tr
Hfinal_msg: finite_trace_last_output iom_tr =
finite_trace_last_output eqv_msg_tr
Hstate_valid: valid_state_prop CE eqv_state_s
emsg_tr: list transition_item
es: state CE
Hmsg_trace_full_replay: finite_valid_trace_from_to CE
  eqv_state_s es emsg_tr
Hemsg_tr_pr: equivocators_total_trace_project IM
  emsg_tr = []
Hes_pr: equivocators_total_state_project IM es = s
Hbs_iom: sent_except_from (equivocator_IM IM)
  initial_message_prop es iom
Happ: valid_state_prop CE es
el:= existT eqv (ContinueWith 0 li): {i : index & label (equivocator_IM IM i)}
Heqes: finite_trace_last eqv_state_is
  (eqv_state_tr ++ emsg_tr) = es
Hes_pr_i: ∀ i : index,
  equivocators_total_state_project IM es i =
  s i
Het: transition el (es, iom) =
(state_update (equivocator_IM IM) es eqv
   (equivocator_state_update (es eqv) 0 si'),
 oom)
Hes_pr_eqv: equivocator_state_zero (es eqv) = s eqv
valid el (es, iom) by subst el; cbn; rewrite equivocator_state_project_zero, Hes_pr_eqv.
    +message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
seed: message → Prop
constraintX: composite_label IM
→ composite_state IM * option message
  → Prop
CX:= preloaded_vlsm (composite_vlsm IM constraintX)
  seed: VLSM message
constraintE: composite_label (equivocator_IM IM)
→ composite_state (equivocator_IM IM) *
  option message → Prop
CE:= preloaded_vlsm
  (composite_vlsm (equivocator_IM IM) constraintE)
  seed: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
Hsubsumption: zero_descriptor_constraint_lifting_prop
Hreplayable: replayable_message_prop
is, s: state CX
tr: list transition_item
HtrX1: finite_valid_trace_init_to CX is s tr
iom: option message
iom_si, iom_s: state CX
iom_tr: list transition_item
Heqiom: empty_initial_message_or_final_output CX
  iom_tr iom
HtrX2: finite_valid_trace_init_to CX iom_si iom_s
  iom_tr
oom: option message
eqv: index
li: label (IM eqv)
Hs: valid_state_prop CX s
Hiom: option_valid_message_prop CX iom
Hv: valid (existT eqv li) (s, iom)
Hc: constraintX (existT eqv li) (s, iom)
si': state (IM eqv)
Hteqv: transition li (s eqv, iom) = (si', oom)
HinclE: VLSM_incl CE
  (preloaded_with_all_messages_vlsm FreeE)
eqv_state_is: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_start_project: equivocators_total_state_project
  IM eqv_state_is = is
eqv_state_s: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_state_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hstate_project: equivocators_total_trace_project IM
  eqv_state_tr = tr
Hstate_trace: finite_valid_trace_init_to CE
  eqv_state_is eqv_state_s eqv_state_tr
eqv_msg_is: state
  (free_composite_vlsm
     (equivocator_IM IM))
Hmsg_start_project: equivocators_total_state_project
  IM eqv_msg_is = iom_si
eqv_msg_s: state
  (free_composite_vlsm (equivocator_IM IM))
eqv_msg_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Hmsg_project: equivocators_total_trace_project IM
  eqv_msg_tr = iom_tr
Hmsg_trace: finite_valid_trace_init_to CE eqv_msg_is
  eqv_msg_s eqv_msg_tr
Hfinal_msg: finite_trace_last_output iom_tr =
finite_trace_last_output eqv_msg_tr
Hstate_valid: valid_state_prop CE eqv_state_s
emsg_tr: list transition_item
es: state CE
Hmsg_trace_full_replay: finite_valid_trace_from_to CE
  eqv_state_s es emsg_tr
Hemsg_tr_pr: equivocators_total_trace_project IM
  emsg_tr = []
Hes_pr: equivocators_total_state_project IM es = s
Hbs_iom: sent_except_from (equivocator_IM IM)
  initial_message_prop es iom
Happ: valid_state_prop CE es
el:= existT eqv (ContinueWith 0 li): {i : index & label (equivocator_IM IM i)}
Heqes: finite_trace_last eqv_state_is
  (eqv_state_tr ++ emsg_tr) = es
Hes_pr_i: ∀ i : index,
  equivocators_total_state_project IM es i =
  s i
Het: transition el (es, iom) =
(state_update (equivocator_IM IM) es eqv
   (equivocator_state_update (es eqv) 0 si'),
 oom)
Hes_pr_eqv: equivocator_state_zero (es eqv) = s eqv
constraintE el (es, iom) by apply Hsubsumption.
Qed.

End sec_generalized_constraints.

VLSM Equivocators Simulating Free Composite

In this section we show that the composition of equivocators with no-message equivocation can simulate the free composition of regular components.

To allow reusing this result in the proof for simulating fixed equivocation, we first prove an intermediate result, where both the composite VLSMs are preloaded with the same set of messages.

Section sec_seeded_all_equivocating.

Context
  {message : Type}
  `{finite.Finite index}
  (IM : index -> VLSM message)
  `{forall i : index, HasBeenSentCapability (IM i)}
  (FreeE := free_composite_vlsm (equivocator_IM IM))
  (seed : message -> Prop)
  (SeededXE : VLSM message :=
    composite_no_equivocation_vlsm_with_preloaded (equivocator_IM IM) (free_constraint _) seed)
  (Free := free_composite_vlsm IM)
  (SeededFree := preloaded_vlsm Free seed)
  .

Since replayed_trace_from was defined for a subset of the equivocators, we here define a specialized version of it when the set of all equivocators is used.

We then restate some of the lemmas concerning replayed_trace_from for the new definition.

Definition all_equivocating_replayed_trace_from
  (full_replay_state : composite_state (equivocator_IM IM))
  (is : composite_state (equivocator_IM IM))
  (tr : list (composite_transition_item (equivocator_IM IM)))
  : list (composite_transition_item (equivocator_IM IM))
  :=
  let
    Hproj := sub_composition_all_embedding (equivocator_IM IM)
              (equivocators_no_equivocations_constraint IM)
  in
    replayed_trace_from IM (enum index)
      full_replay_state
      (composite_state_sub_projection (equivocator_IM IM) (enum index) is)
      (VLSM_embedding_finite_trace_project Hproj tr).

Lemma replayed_trace_from_valid_equivocating
  (full_replay_state : composite_state (equivocator_IM IM))
  (Hfull_replay_state : valid_state_prop SeededXE full_replay_state)
  (is : composite_state (equivocator_IM IM))
  (tr : list (composite_transition_item (equivocator_IM IM)))
  (Htr : finite_valid_trace SeededXE is tr)
  : finite_valid_trace_from SeededXE
      full_replay_state (all_equivocating_replayed_trace_from full_replay_state 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
seed: message → Prop
SeededXE:= composite_no_equivocation_vlsm_with_preloaded
  (equivocator_IM IM)
  (free_constraint (equivocator_IM IM)) seed: VLSM message
Free:= free_composite_vlsm IM: VLSM message
SeededFree:= preloaded_vlsm Free seed: VLSM message
full_replay_state: composite_state
  (equivocator_IM IM)
Hfull_replay_state: valid_state_prop SeededXE
  full_replay_state
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace SeededXE is tr
finite_valid_trace_from SeededXE full_replay_state
  (all_equivocating_replayed_trace_from
     full_replay_state is 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
seed: message → Prop
SeededXE:= composite_no_equivocation_vlsm_with_preloaded
  (equivocator_IM IM)
  (free_constraint (equivocator_IM IM)) seed: VLSM message
Free:= free_composite_vlsm IM: VLSM message
SeededFree:= preloaded_vlsm Free seed: VLSM message
full_replay_state: composite_state
  (equivocator_IM IM)
Hfull_replay_state: valid_state_prop SeededXE
  full_replay_state
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace SeededXE is tr
finite_valid_trace_from SeededXE full_replay_state
  (all_equivocating_replayed_trace_from
     full_replay_state is tr)
  apply
    (sub_replayed_trace_from_valid_equivocating IM seed
      (enum index) _ Hfull_replay_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
seed: message → Prop
SeededXE:= composite_no_equivocation_vlsm_with_preloaded
  (equivocator_IM IM)
  (free_constraint (equivocator_IM IM)) seed: VLSM message
Free:= free_composite_vlsm IM: VLSM message
SeededFree:= preloaded_vlsm Free seed: VLSM message
full_replay_state: composite_state
  (equivocator_IM IM)
Hfull_replay_state: valid_state_prop SeededXE
  full_replay_state
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace SeededXE is tr
finite_valid_trace
  (seeded_equivocators_no_equivocation_vlsm IM
     (enum index) seed)
  (composite_state_sub_projection (equivocator_IM IM)
     (enum index) is)
  (VLSM_embedding_finite_trace_project
     (sub_composition_all_embedding
        (equivocator_IM IM)
        (equivocators_no_equivocations_constraint IM))
     tr)
  pose (Hproj := preloaded_sub_composition_all_embedding (equivocator_IM IM)
    (no_equivocations_additional_constraint_with_preloaded (equivocator_IM IM)
    (free_constraint _) seed) seed).message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
seed: message → Prop
SeededXE:= composite_no_equivocation_vlsm_with_preloaded
  (equivocator_IM IM)
  (free_constraint (equivocator_IM IM)) seed: VLSM message
Free:= free_composite_vlsm IM: VLSM message
SeededFree:= preloaded_vlsm Free seed: VLSM message
full_replay_state: composite_state
  (equivocator_IM IM)
Hfull_replay_state: valid_state_prop SeededXE
  full_replay_state
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace SeededXE is tr
Hproj:= preloaded_sub_composition_all_embedding
  (equivocator_IM IM)
  (no_equivocations_additional_constraint_with_preloaded
     (equivocator_IM IM)
     (free_constraint (equivocator_IM IM)) seed)
  seed: VLSM_embedding
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM IM)
           (free_constraint
              (equivocator_IM IM)) seed)) seed)
  (preloaded_vlsm
     (composite_vlsm
        (sub_IM (equivocator_IM IM)
           (enum index))
        (free_sub_free_constraint
           (equivocator_IM IM)
           (no_equivocations_additional_constraint_with_preloaded
              (equivocator_IM IM)
              (free_constraint
                 (equivocator_IM IM)) seed)))
     seed)
  (free_sub_free_label (equivocator_IM IM))
  (composite_state_sub_projection
     (equivocator_IM IM) 
     (enum index))
finite_valid_trace
  (seeded_equivocators_no_equivocation_vlsm IM
     (enum index) seed)
  (composite_state_sub_projection (equivocator_IM IM)
     (enum index) is)
  (VLSM_embedding_finite_trace_project
     (sub_composition_all_embedding
        (equivocator_IM IM)
        (equivocators_no_equivocations_constraint IM))
     tr)
  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
seed: message → Prop
SeededXE:= composite_no_equivocation_vlsm_with_preloaded
  (equivocator_IM IM)
  (free_constraint (equivocator_IM IM)) seed: VLSM message
Free:= free_composite_vlsm IM: VLSM message
SeededFree:= preloaded_vlsm Free seed: VLSM message
full_replay_state: composite_state
  (equivocator_IM IM)
Hfull_replay_state: valid_state_prop SeededXE
  full_replay_state
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Hproj:= preloaded_sub_composition_all_embedding
  (equivocator_IM IM)
  (no_equivocations_additional_constraint_with_preloaded
     (equivocator_IM IM)
     (free_constraint (equivocator_IM IM)) seed)
  seed: VLSM_embedding
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM IM)
           (free_constraint
              (equivocator_IM IM)) seed)) seed)
  (preloaded_vlsm
     (composite_vlsm
        (sub_IM (equivocator_IM IM)
           (enum index))
        (free_sub_free_constraint
           (equivocator_IM IM)
           (no_equivocations_additional_constraint_with_preloaded
              (equivocator_IM IM)
              (free_constraint
                 (equivocator_IM IM)) seed)))
     seed)
  (free_sub_free_label (equivocator_IM IM))
  (composite_state_sub_projection
     (equivocator_IM IM) 
     (enum index))
Htr: finite_valid_trace
  (preloaded_vlsm
     (composite_vlsm
        (sub_IM (equivocator_IM IM) (enum index))
        (free_sub_free_constraint
           (equivocator_IM IM)
           (no_equivocations_additional_constraint_with_preloaded
              (equivocator_IM IM)
              (free_constraint
                 (equivocator_IM IM)) seed)))
     seed)
  (composite_state_sub_projection
     (equivocator_IM IM) 
     (enum index) is)
  (VLSM_embedding_finite_trace_project Hproj tr)
finite_valid_trace
  (seeded_equivocators_no_equivocation_vlsm IM
     (enum index) seed)
  (composite_state_sub_projection (equivocator_IM IM)
     (enum index) is)
  (VLSM_embedding_finite_trace_project
     (sub_composition_all_embedding
        (equivocator_IM IM)
        (equivocators_no_equivocations_constraint IM))
     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
seed: message → Prop
SeededXE:= composite_no_equivocation_vlsm_with_preloaded
  (equivocator_IM IM)
  (free_constraint (equivocator_IM IM)) seed: VLSM message
Free:= free_composite_vlsm IM: VLSM message
SeededFree:= preloaded_vlsm Free seed: VLSM message
full_replay_state: composite_state
  (equivocator_IM IM)
Hfull_replay_state: valid_state_prop SeededXE
  full_replay_state
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Hproj:= preloaded_sub_composition_all_embedding
  (equivocator_IM IM)
  (no_equivocations_additional_constraint_with_preloaded
     (equivocator_IM IM)
     (free_constraint (equivocator_IM IM)) seed)
  seed: VLSM_embedding
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM IM)
           (free_constraint
              (equivocator_IM IM)) seed)) seed)
  (preloaded_vlsm
     (composite_vlsm
        (sub_IM (equivocator_IM IM)
           (enum index))
        (free_sub_free_constraint
           (equivocator_IM IM)
           (no_equivocations_additional_constraint_with_preloaded
              (equivocator_IM IM)
              (free_constraint
                 (equivocator_IM IM)) seed)))
     seed)
  (free_sub_free_label (equivocator_IM IM))
  (composite_state_sub_projection
     (equivocator_IM IM) 
     (enum index))
finite_valid_trace
  (preloaded_vlsm
     (composite_vlsm
        (sub_IM (equivocator_IM IM) (enum index))
        (free_sub_free_constraint (equivocator_IM IM)
           (no_equivocations_additional_constraint_with_preloaded
              (equivocator_IM IM)
              (free_constraint (equivocator_IM IM))
              seed))) seed)
  (composite_state_sub_projection (equivocator_IM IM)
     (enum index) is)
  (VLSM_embedding_finite_trace_project Hproj tr)
→ finite_valid_trace
    (seeded_equivocators_no_equivocation_vlsm IM
       (enum index) seed)
    (composite_state_sub_projection
       (equivocator_IM IM) (enum index) is)
    (VLSM_embedding_finite_trace_project
       (sub_composition_all_embedding
          (equivocator_IM IM)
          (equivocators_no_equivocations_constraint IM))
       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
seed: message → Prop
SeededXE:= composite_no_equivocation_vlsm_with_preloaded
  (equivocator_IM IM)
  (free_constraint (equivocator_IM IM)) seed: VLSM message
Free:= free_composite_vlsm IM: VLSM message
SeededFree:= preloaded_vlsm Free seed: VLSM message
full_replay_state: composite_state
  (equivocator_IM IM)
Hfull_replay_state: valid_state_prop SeededXE
  full_replay_state
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Hproj:= preloaded_sub_composition_all_embedding
  (equivocator_IM IM)
  (no_equivocations_additional_constraint_with_preloaded
     (equivocator_IM IM)
     (free_constraint (equivocator_IM IM)) seed)
  seed: VLSM_embedding
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM IM)
           (free_constraint
              (equivocator_IM IM)) seed)) seed)
  (preloaded_vlsm
     (composite_vlsm
        (sub_IM (equivocator_IM IM)
           (enum index))
        (free_sub_free_constraint
           (equivocator_IM IM)
           (no_equivocations_additional_constraint_with_preloaded
              (equivocator_IM IM)
              (free_constraint
                 (equivocator_IM IM)) seed)))
     seed)
  (free_sub_free_label (equivocator_IM IM))
  (composite_state_sub_projection
     (equivocator_IM IM) 
     (enum index))
VLSM_incl_part
  (preloaded_vlsm_machine
     (composite_vlsm
        (sub_IM (equivocator_IM IM) (enum index))
        (free_sub_free_constraint (equivocator_IM IM)
           (no_equivocations_additional_constraint_with_preloaded
              (equivocator_IM IM)
              (free_constraint (equivocator_IM IM))
              seed))) seed)
  (preloaded_vlsm_machine
     (composite_vlsm
        (sub_IM (equivocator_IM IM) (enum index))
        (no_equivocations_additional_constraint_with_preloaded
           (sub_IM (equivocator_IM IM) (enum index))
           (free_constraint
              (sub_IM (equivocator_IM IM) (enum index)))
           seed)) seed)
  apply basic_VLSM_incl_preloaded_with; [done | | by intro..].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
seed: message → Prop
SeededXE:= composite_no_equivocation_vlsm_with_preloaded
  (equivocator_IM IM)
  (free_constraint (equivocator_IM IM)) seed: VLSM message
Free:= free_composite_vlsm IM: VLSM message
SeededFree:= preloaded_vlsm Free seed: VLSM message
full_replay_state: composite_state
  (equivocator_IM IM)
Hfull_replay_state: valid_state_prop SeededXE
  full_replay_state
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Hproj:= preloaded_sub_composition_all_embedding
  (equivocator_IM IM)
  (no_equivocations_additional_constraint_with_preloaded
     (equivocator_IM IM)
     (free_constraint (equivocator_IM IM)) seed)
  seed: VLSM_embedding
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM IM)
           (free_constraint
              (equivocator_IM IM)) seed)) seed)
  (preloaded_vlsm
     (composite_vlsm
        (sub_IM (equivocator_IM IM)
           (enum index))
        (free_sub_free_constraint
           (equivocator_IM IM)
           (no_equivocations_additional_constraint_with_preloaded
              (equivocator_IM IM)
              (free_constraint
                 (equivocator_IM IM)) seed)))
     seed)
  (free_sub_free_label (equivocator_IM IM))
  (composite_state_sub_projection
     (equivocator_IM IM) 
     (enum index))
strong_incl_valid_preservation
  (constrained_vlsm_machine
     (free_composite_vlsm
        (sub_IM (equivocator_IM IM) (enum index)))
     (free_sub_free_constraint (equivocator_IM IM)
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM IM)
           (free_constraint (equivocator_IM IM)) seed)))
  (constrained_vlsm_machine
     (free_composite_vlsm
        (sub_IM (equivocator_IM IM) (enum index)))
     (no_equivocations_additional_constraint_with_preloaded
        (sub_IM (equivocator_IM IM) (enum index))
        (free_constraint
           (sub_IM (equivocator_IM IM) (enum index)))
        seed))
  intros l s om (Hv & Hc & _).message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
seed: message → Prop
SeededXE:= composite_no_equivocation_vlsm_with_preloaded
  (equivocator_IM IM)
  (free_constraint (equivocator_IM IM)) seed: VLSM message
Free:= free_composite_vlsm IM: VLSM message
SeededFree:= preloaded_vlsm Free seed: VLSM message
full_replay_state: composite_state
  (equivocator_IM IM)
Hfull_replay_state: valid_state_prop SeededXE
  full_replay_state
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Hproj:= preloaded_sub_composition_all_embedding
  (equivocator_IM IM)
  (no_equivocations_additional_constraint_with_preloaded
     (equivocator_IM IM)
     (free_constraint (equivocator_IM IM)) seed)
  seed: VLSM_embedding
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM IM)
           (free_constraint
              (equivocator_IM IM)) seed)) seed)
  (preloaded_vlsm
     (composite_vlsm
        (sub_IM (equivocator_IM IM)
           (enum index))
        (free_sub_free_constraint
           (equivocator_IM IM)
           (no_equivocations_additional_constraint_with_preloaded
              (equivocator_IM IM)
              (free_constraint
                 (equivocator_IM IM)) seed)))
     seed)
  (free_sub_free_label (equivocator_IM IM))
  (composite_state_sub_projection
     (equivocator_IM IM) 
     (enum index))
l: label
  {|
    vlsm_type :=
      {|
        state :=
          composite_state
            (sub_IM (equivocator_IM IM)
               (enum index));
        label :=
          composite_label
            (sub_IM (equivocator_IM IM)
               (enum index))
      |};
    vlsm_machine :=
      constrained_vlsm_machine
        (free_composite_vlsm
           (sub_IM (equivocator_IM IM)
              (enum index)))
        (free_sub_free_constraint
           (equivocator_IM IM)
           (no_equivocations_additional_constraint_with_preloaded
              (equivocator_IM IM)
              (free_constraint (equivocator_IM IM))
              seed))
  |}
s: state
  {|
    vlsm_type :=
      {|
        state :=
          composite_state
            (sub_IM (equivocator_IM IM)
               (enum index));
        label :=
          composite_label
            (sub_IM (equivocator_IM IM)
               (enum index))
      |};
    vlsm_machine :=
      constrained_vlsm_machine
        (free_composite_vlsm
           (sub_IM (equivocator_IM IM)
              (enum index)))
        (free_sub_free_constraint
           (equivocator_IM IM)
           (no_equivocations_additional_constraint_with_preloaded
              (equivocator_IM IM)
              (free_constraint (equivocator_IM IM))
              seed))
  |}
om: option message
Hv: valid l (s, om)
Hc: composite_no_equivocations_except_from
  (equivocator_IM IM) seed
  (lift_sub_label (equivocator_IM IM)
     (enum index) l)
  (free_sub_free_state (equivocator_IM IM) s, om)
valid (id l) (id s, om)
  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
seed: message → Prop
SeededXE:= composite_no_equivocation_vlsm_with_preloaded
  (equivocator_IM IM)
  (free_constraint (equivocator_IM IM)) seed: VLSM message
Free:= free_composite_vlsm IM: VLSM message
SeededFree:= preloaded_vlsm Free seed: VLSM message
full_replay_state: composite_state
  (equivocator_IM IM)
Hfull_replay_state: valid_state_prop SeededXE
  full_replay_state
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Hproj:= preloaded_sub_composition_all_embedding
  (equivocator_IM IM)
  (no_equivocations_additional_constraint_with_preloaded
     (equivocator_IM IM)
     (free_constraint (equivocator_IM IM)) seed)
  seed: VLSM_embedding
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM IM)
           (free_constraint
              (equivocator_IM IM)) seed)) seed)
  (preloaded_vlsm
     (composite_vlsm
        (sub_IM (equivocator_IM IM)
           (enum index))
        (free_sub_free_constraint
           (equivocator_IM IM)
           (no_equivocations_additional_constraint_with_preloaded
              (equivocator_IM IM)
              (free_constraint
                 (equivocator_IM IM)) seed)))
     seed)
  (free_sub_free_label (equivocator_IM IM))
  (composite_state_sub_projection
     (equivocator_IM IM) 
     (enum index))
l: label
  {|
    vlsm_type :=
      {|
        state :=
          composite_state
            (sub_IM (equivocator_IM IM)
               (enum index));
        label :=
          composite_label
            (sub_IM (equivocator_IM IM)
               (enum index))
      |};
    vlsm_machine :=
      constrained_vlsm_machine
        (free_composite_vlsm
           (sub_IM (equivocator_IM IM)
              (enum index)))
        (free_sub_free_constraint
           (equivocator_IM IM)
           (no_equivocations_additional_constraint_with_preloaded
              (equivocator_IM IM)
              (free_constraint (equivocator_IM IM))
              seed))
  |}
s: state
  {|
    vlsm_type :=
      {|
        state :=
          composite_state
            (sub_IM (equivocator_IM IM)
               (enum index));
        label :=
          composite_label
            (sub_IM (equivocator_IM IM)
               (enum index))
      |};
    vlsm_machine :=
      constrained_vlsm_machine
        (free_composite_vlsm
           (sub_IM (equivocator_IM IM)
              (enum index)))
        (free_sub_free_constraint
           (equivocator_IM IM)
           (no_equivocations_additional_constraint_with_preloaded
              (equivocator_IM IM)
              (free_constraint (equivocator_IM IM))
              seed))
  |}
om: option message
Hv: valid l (s, om)
Hc: composite_no_equivocations_except_from
  (equivocator_IM IM) seed
  (lift_sub_label (equivocator_IM IM)
     (enum index) l)
  (free_sub_free_state (equivocator_IM IM) s, om)
no_equivocations_additional_constraint_with_preloaded
  (sub_IM (equivocator_IM IM) (enum index))
  (free_constraint
     (sub_IM (equivocator_IM IM) (enum index))) seed
  (id l) (id s, om)
  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
seed: message → Prop
SeededXE:= composite_no_equivocation_vlsm_with_preloaded
  (equivocator_IM IM)
  (free_constraint (equivocator_IM IM)) seed: VLSM message
Free:= free_composite_vlsm IM: VLSM message
SeededFree:= preloaded_vlsm Free seed: VLSM message
full_replay_state: composite_state
  (equivocator_IM IM)
Hfull_replay_state: valid_state_prop SeededXE
  full_replay_state
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Hproj:= preloaded_sub_composition_all_embedding
  (equivocator_IM IM)
  (no_equivocations_additional_constraint_with_preloaded
     (equivocator_IM IM)
     (free_constraint (equivocator_IM IM)) seed)
  seed: VLSM_embedding
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM IM)
           (free_constraint
              (equivocator_IM IM)) seed)) seed)
  (preloaded_vlsm
     (composite_vlsm
        (sub_IM (equivocator_IM IM)
           (enum index))
        (free_sub_free_constraint
           (equivocator_IM IM)
           (no_equivocations_additional_constraint_with_preloaded
              (equivocator_IM IM)
              (free_constraint
                 (equivocator_IM IM)) seed)))
     seed)
  (free_sub_free_label (equivocator_IM IM))
  (composite_state_sub_projection
     (equivocator_IM IM) 
     (enum index))
l: label
  {|
    vlsm_type :=
      {|
        state :=
          composite_state
            (sub_IM (equivocator_IM IM)
               (enum index));
        label :=
          composite_label
            (sub_IM (equivocator_IM IM)
               (enum index))
      |};
    vlsm_machine :=
      constrained_vlsm_machine
        (free_composite_vlsm
           (sub_IM (equivocator_IM IM)
              (enum index)))
        (free_sub_free_constraint
           (equivocator_IM IM)
           (no_equivocations_additional_constraint_with_preloaded
              (equivocator_IM IM)
              (free_constraint (equivocator_IM IM))
              seed))
  |}
s: state
  {|
    vlsm_type :=
      {|
        state :=
          composite_state
            (sub_IM (equivocator_IM IM)
               (enum index));
        label :=
          composite_label
            (sub_IM (equivocator_IM IM)
               (enum index))
      |};
    vlsm_machine :=
      constrained_vlsm_machine
        (free_composite_vlsm
           (sub_IM (equivocator_IM IM)
              (enum index)))
        (free_sub_free_constraint
           (equivocator_IM IM)
           (no_equivocations_additional_constraint_with_preloaded
              (equivocator_IM IM)
              (free_constraint (equivocator_IM IM))
              seed))
  |}
om: option message
Hv: valid l (s, om)
Hc: composite_no_equivocations_except_from
  (equivocator_IM IM) seed
  (lift_sub_label (equivocator_IM IM)
     (enum index) l)
  (free_sub_free_state (equivocator_IM IM) s, om)
composite_no_equivocations_except_from
  (sub_IM (equivocator_IM IM) (enum index)) seed
  (id l) (id s, om)
  destruct om; [| 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
seed: message → Prop
SeededXE:= composite_no_equivocation_vlsm_with_preloaded
  (equivocator_IM IM)
  (free_constraint (equivocator_IM IM)) seed: VLSM message
Free:= free_composite_vlsm IM: VLSM message
SeededFree:= preloaded_vlsm Free seed: VLSM message
full_replay_state: composite_state
  (equivocator_IM IM)
Hfull_replay_state: valid_state_prop SeededXE
  full_replay_state
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Hproj:= preloaded_sub_composition_all_embedding
  (equivocator_IM IM)
  (no_equivocations_additional_constraint_with_preloaded
     (equivocator_IM IM)
     (free_constraint (equivocator_IM IM)) seed)
  seed: VLSM_embedding
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM IM)
           (free_constraint
              (equivocator_IM IM)) seed)) seed)
  (preloaded_vlsm
     (composite_vlsm
        (sub_IM (equivocator_IM IM)
           (enum index))
        (free_sub_free_constraint
           (equivocator_IM IM)
           (no_equivocations_additional_constraint_with_preloaded
              (equivocator_IM IM)
              (free_constraint
                 (equivocator_IM IM)) seed)))
     seed)
  (free_sub_free_label (equivocator_IM IM))
  (composite_state_sub_projection
     (equivocator_IM IM) 
     (enum index))
l: label
  {|
    vlsm_type :=
      {|
        state :=
          composite_state
            (sub_IM (equivocator_IM IM)
               (enum index));
        label :=
          composite_label
            (sub_IM (equivocator_IM IM)
               (enum index))
      |};
    vlsm_machine :=
      constrained_vlsm_machine
        (free_composite_vlsm
           (sub_IM (equivocator_IM IM)
              (enum index)))
        (free_sub_free_constraint
           (equivocator_IM IM)
           (no_equivocations_additional_constraint_with_preloaded
              (equivocator_IM IM)
              (free_constraint (equivocator_IM IM))
              seed))
  |}
s: state
  {|
    vlsm_type :=
      {|
        state :=
          composite_state
            (sub_IM (equivocator_IM IM)
               (enum index));
        label :=
          composite_label
            (sub_IM (equivocator_IM IM)
               (enum index))
      |};
    vlsm_machine :=
      constrained_vlsm_machine
        (free_composite_vlsm
           (sub_IM (equivocator_IM IM)
              (enum index)))
        (free_sub_free_constraint
           (equivocator_IM IM)
           (no_equivocations_additional_constraint_with_preloaded
              (equivocator_IM IM)
              (free_constraint (equivocator_IM IM))
              seed))
  |}
m: message
Hv: valid l (s, Some m)
Hc: composite_no_equivocations_except_from
  (equivocator_IM IM) seed
  (lift_sub_label (equivocator_IM IM)
     (enum index) l)
  (free_sub_free_state (equivocator_IM IM) s,
   Some m)
composite_no_equivocations_except_from
  (sub_IM (equivocator_IM IM) (enum index)) seed
  (id l) (id s, Some m)
  destruct Hc as [Hc | Hc]; [| by 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
seed: message → Prop
SeededXE:= composite_no_equivocation_vlsm_with_preloaded
  (equivocator_IM IM)
  (free_constraint (equivocator_IM IM)) seed: VLSM message
Free:= free_composite_vlsm IM: VLSM message
SeededFree:= preloaded_vlsm Free seed: VLSM message
full_replay_state: composite_state
  (equivocator_IM IM)
Hfull_replay_state: valid_state_prop SeededXE
  full_replay_state
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Hproj:= preloaded_sub_composition_all_embedding
  (equivocator_IM IM)
  (no_equivocations_additional_constraint_with_preloaded
     (equivocator_IM IM)
     (free_constraint (equivocator_IM IM)) seed)
  seed: VLSM_embedding
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM IM)
           (free_constraint
              (equivocator_IM IM)) seed)) seed)
  (preloaded_vlsm
     (composite_vlsm
        (sub_IM (equivocator_IM IM)
           (enum index))
        (free_sub_free_constraint
           (equivocator_IM IM)
           (no_equivocations_additional_constraint_with_preloaded
              (equivocator_IM IM)
              (free_constraint
                 (equivocator_IM IM)) seed)))
     seed)
  (free_sub_free_label (equivocator_IM IM))
  (composite_state_sub_projection
     (equivocator_IM IM) 
     (enum index))
l: label
  {|
    vlsm_type :=
      {|
        state :=
          composite_state
            (sub_IM (equivocator_IM IM)
               (enum index));
        label :=
          composite_label
            (sub_IM (equivocator_IM IM)
               (enum index))
      |};
    vlsm_machine :=
      constrained_vlsm_machine
        (free_composite_vlsm
           (sub_IM (equivocator_IM IM)
              (enum index)))
        (free_sub_free_constraint
           (equivocator_IM IM)
           (no_equivocations_additional_constraint_with_preloaded
              (equivocator_IM IM)
              (free_constraint (equivocator_IM IM))
              seed))
  |}
s: state
  {|
    vlsm_type :=
      {|
        state :=
          composite_state
            (sub_IM (equivocator_IM IM)
               (enum index));
        label :=
          composite_label
            (sub_IM (equivocator_IM IM)
               (enum index))
      |};
    vlsm_machine :=
      constrained_vlsm_machine
        (free_composite_vlsm
           (sub_IM (equivocator_IM IM)
              (enum index)))
        (free_sub_free_constraint
           (equivocator_IM IM)
           (no_equivocations_additional_constraint_with_preloaded
              (equivocator_IM IM)
              (free_constraint (equivocator_IM IM))
              seed))
  |}
m: message
Hv: valid l (s, Some m)
Hc: composite_has_been_sent 
  (equivocator_IM IM)
  (free_sub_free_state (equivocator_IM IM) s,
   Some m).1 m
composite_no_equivocations_except_from
  (sub_IM (equivocator_IM IM) (enum index)) seed
  (id l) (id s, Some m)
  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
seed: message → Prop
SeededXE:= composite_no_equivocation_vlsm_with_preloaded
  (equivocator_IM IM)
  (free_constraint (equivocator_IM IM)) seed: VLSM message
Free:= free_composite_vlsm IM: VLSM message
SeededFree:= preloaded_vlsm Free seed: VLSM message
full_replay_state: composite_state
  (equivocator_IM IM)
Hfull_replay_state: valid_state_prop SeededXE
  full_replay_state
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Hproj:= preloaded_sub_composition_all_embedding
  (equivocator_IM IM)
  (no_equivocations_additional_constraint_with_preloaded
     (equivocator_IM IM)
     (free_constraint (equivocator_IM IM)) seed)
  seed: VLSM_embedding
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM IM)
           (free_constraint
              (equivocator_IM IM)) seed)) seed)
  (preloaded_vlsm
     (composite_vlsm
        (sub_IM (equivocator_IM IM)
           (enum index))
        (free_sub_free_constraint
           (equivocator_IM IM)
           (no_equivocations_additional_constraint_with_preloaded
              (equivocator_IM IM)
              (free_constraint
                 (equivocator_IM IM)) seed)))
     seed)
  (free_sub_free_label (equivocator_IM IM))
  (composite_state_sub_projection
     (equivocator_IM IM) 
     (enum index))
l: label
  {|
    vlsm_type :=
      {|
        state :=
          composite_state
            (sub_IM (equivocator_IM IM)
               (enum index));
        label :=
          composite_label
            (sub_IM (equivocator_IM IM)
               (enum index))
      |};
    vlsm_machine :=
      constrained_vlsm_machine
        (free_composite_vlsm
           (sub_IM (equivocator_IM IM)
              (enum index)))
        (free_sub_free_constraint
           (equivocator_IM IM)
           (no_equivocations_additional_constraint_with_preloaded
              (equivocator_IM IM)
              (free_constraint (equivocator_IM IM))
              seed))
  |}
s: state
  {|
    vlsm_type :=
      {|
        state :=
          composite_state
            (sub_IM (equivocator_IM IM)
               (enum index));
        label :=
          composite_label
            (sub_IM (equivocator_IM IM)
               (enum index))
      |};
    vlsm_machine :=
      constrained_vlsm_machine
        (free_composite_vlsm
           (sub_IM (equivocator_IM IM)
              (enum index)))
        (free_sub_free_constraint
           (equivocator_IM IM)
           (no_equivocations_additional_constraint_with_preloaded
              (equivocator_IM IM)
              (free_constraint (equivocator_IM IM))
              seed))
  |}
m: message
Hv: valid l (s, Some m)
Hc: composite_has_been_sent 
  (equivocator_IM IM)
  (free_sub_free_state (equivocator_IM IM) s,
   Some m).1 m
composite_has_been_sent
  (sub_IM (equivocator_IM IM) (enum index))
  (id s, Some m).1 m
  destruct Hc as [i Hsent].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
seed: message → Prop
SeededXE:= composite_no_equivocation_vlsm_with_preloaded
  (equivocator_IM IM)
  (free_constraint (equivocator_IM IM)) seed: VLSM message
Free:= free_composite_vlsm IM: VLSM message
SeededFree:= preloaded_vlsm Free seed: VLSM message
full_replay_state: composite_state
  (equivocator_IM IM)
Hfull_replay_state: valid_state_prop SeededXE
  full_replay_state
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Hproj:= preloaded_sub_composition_all_embedding
  (equivocator_IM IM)
  (no_equivocations_additional_constraint_with_preloaded
     (equivocator_IM IM)
     (free_constraint (equivocator_IM IM)) seed)
  seed: VLSM_embedding
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM IM)
           (free_constraint
              (equivocator_IM IM)) seed)) seed)
  (preloaded_vlsm
     (composite_vlsm
        (sub_IM (equivocator_IM IM)
           (enum index))
        (free_sub_free_constraint
           (equivocator_IM IM)
           (no_equivocations_additional_constraint_with_preloaded
              (equivocator_IM IM)
              (free_constraint
                 (equivocator_IM IM)) seed)))
     seed)
  (free_sub_free_label (equivocator_IM IM))
  (composite_state_sub_projection
     (equivocator_IM IM) 
     (enum index))
l: label
  {|
    vlsm_type :=
      {|
        state :=
          composite_state
            (sub_IM (equivocator_IM IM)
               (enum index));
        label :=
          composite_label
            (sub_IM (equivocator_IM IM)
               (enum index))
      |};
    vlsm_machine :=
      constrained_vlsm_machine
        (free_composite_vlsm
           (sub_IM (equivocator_IM IM)
              (enum index)))
        (free_sub_free_constraint
           (equivocator_IM IM)
           (no_equivocations_additional_constraint_with_preloaded
              (equivocator_IM IM)
              (free_constraint (equivocator_IM IM))
              seed))
  |}
s: state
  {|
    vlsm_type :=
      {|
        state :=
          composite_state
            (sub_IM (equivocator_IM IM)
               (enum index));
        label :=
          composite_label
            (sub_IM (equivocator_IM IM)
               (enum index))
      |};
    vlsm_machine :=
      constrained_vlsm_machine
        (free_composite_vlsm
           (sub_IM (equivocator_IM IM)
              (enum index)))
        (free_sub_free_constraint
           (equivocator_IM IM)
           (no_equivocations_additional_constraint_with_preloaded
              (equivocator_IM IM)
              (free_constraint (equivocator_IM IM))
              seed))
  |}
m: message
Hv: valid l (s, Some m)
i: index
Hsent: has_been_sent (equivocator_IM IM i)
  ((free_sub_free_state (equivocator_IM IM) s,
    Some m).1 i) m
composite_has_been_sent
  (sub_IM (equivocator_IM IM) (enum index))
  (id s, Some m).1 m
  by exists (free_sub_free_index i).
Qed.

Specializing the generalized_equivocators_finite_valid_trace_init_to_rev result using the free_constraint for the composition of regular components and the no-message-equivocation constraint for the composition of equivocators.

Lemma seeded_equivocators_finite_valid_trace_init_to_rev
  (no_initial_messages_in_IM : no_initial_messages_in_IM_prop IM)
  isX sX trX
  (HtrX : finite_valid_trace_init_to SeededFree isX sX trX)
  : exists is, equivocators_total_state_project IM is = isX /\
    exists s, equivocators_total_state_project IM s = sX /\
    exists tr, equivocators_total_trace_project IM tr = trX /\
    finite_valid_trace_init_to SeededXE is s tr /\
    finite_trace_last_output trX = finite_trace_last_output 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
seed: message → Prop
SeededXE:= composite_no_equivocation_vlsm_with_preloaded
  (equivocator_IM IM)
  (free_constraint (equivocator_IM IM)) seed: VLSM message
Free:= free_composite_vlsm IM: VLSM message
SeededFree:= preloaded_vlsm Free seed: VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
isX, sX: state SeededFree
trX: list transition_item
HtrX: finite_valid_trace_init_to SeededFree isX sX
  trX
∃ is : state (free_composite_vlsm (equivocator_IM IM)),
  equivocators_total_state_project IM is = isX
  ∧ (∃ s : state
             (free_composite_vlsm (equivocator_IM IM)),
       equivocators_total_state_project IM s = sX
       ∧ (∃ tr : list
                   (composite_transition_item
                      (equivocator_IM IM)),
            equivocators_total_trace_project IM tr =
            trX
            ∧ finite_valid_trace_init_to SeededXE is s
                tr
              ∧ finite_trace_last_output trX =
                finite_trace_last_output 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
seed: message → Prop
SeededXE:= composite_no_equivocation_vlsm_with_preloaded
  (equivocator_IM IM)
  (free_constraint (equivocator_IM IM)) seed: VLSM message
Free:= free_composite_vlsm IM: VLSM message
SeededFree:= preloaded_vlsm Free seed: VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
isX, sX: state SeededFree
trX: list transition_item
HtrX: finite_valid_trace_init_to SeededFree isX sX
  trX
∃ is : state (free_composite_vlsm (equivocator_IM IM)),
  equivocators_total_state_project IM is = isX
  ∧ (∃ s : state
             (free_composite_vlsm (equivocator_IM IM)),
       equivocators_total_state_project IM s = sX
       ∧ (∃ tr : list
                   (composite_transition_item
                      (equivocator_IM IM)),
            equivocators_total_trace_project IM tr =
            trX
            ∧ finite_valid_trace_init_to SeededXE is s
                tr
              ∧ finite_trace_last_output trX =
                finite_trace_last_output tr))
  apply (generalized_equivocators_finite_valid_trace_init_to_rev IM)
    with (constraintX := free_constraint _); cycle 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
seed: message → Prop
SeededXE:= composite_no_equivocation_vlsm_with_preloaded
  (equivocator_IM IM)
  (free_constraint (equivocator_IM IM)) seed: VLSM message
Free:= free_composite_vlsm IM: VLSM message
SeededFree:= preloaded_vlsm Free seed: VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
isX, sX: state SeededFree
trX: list transition_item
HtrX: finite_valid_trace_init_to SeededFree isX sX
  trX
finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm IM (free_constraint IM)) seed)
  isX 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
seed: message → Prop
SeededXE:= composite_no_equivocation_vlsm_with_preloaded
  (equivocator_IM IM)
  (free_constraint (equivocator_IM IM)) seed: VLSM message
Free:= free_composite_vlsm IM: VLSM message
SeededFree:= preloaded_vlsm Free seed: VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
isX, sX: state SeededFree
trX: list transition_item
HtrX: finite_valid_trace_init_to SeededFree isX sX
  trX
zero_descriptor_constraint_lifting_prop IM seed
  (no_equivocations_additional_constraint_with_preloaded
     (equivocator_IM IM)
     (free_constraint (equivocator_IM IM)) seed)
message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
seed: message → Prop
SeededXE:= composite_no_equivocation_vlsm_with_preloaded
  (equivocator_IM IM)
  (free_constraint (equivocator_IM IM)) seed: VLSM message
Free:= free_composite_vlsm IM: VLSM message
SeededFree:= preloaded_vlsm Free seed: VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
isX, sX: state SeededFree
trX: list transition_item
HtrX: finite_valid_trace_init_to SeededFree isX sX
  trX
replayable_message_prop IM seed 
  (free_constraint IM)
  (no_equivocations_additional_constraint_with_preloaded
     (equivocator_IM IM)
     (free_constraint (equivocator_IM IM)) seed)
  -message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
seed: message → Prop
SeededXE:= composite_no_equivocation_vlsm_with_preloaded
  (equivocator_IM IM)
  (free_constraint (equivocator_IM IM)) seed: VLSM message
Free:= free_composite_vlsm IM: VLSM message
SeededFree:= preloaded_vlsm Free seed: VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
isX, sX: state SeededFree
trX: list transition_item
HtrX: finite_valid_trace_init_to SeededFree isX sX
  trX
finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm IM (free_constraint IM)) seed)
  isX sX trX apply VLSM_incl_finite_valid_trace_init_to; [| by apply 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
seed: message → Prop
SeededXE:= composite_no_equivocation_vlsm_with_preloaded
  (equivocator_IM IM)
  (free_constraint (equivocator_IM IM)) seed: VLSM message
Free:= free_composite_vlsm IM: VLSM message
SeededFree:= preloaded_vlsm Free seed: VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
isX, sX: state SeededFree
trX: list transition_item
HtrX: finite_valid_trace_init_to SeededFree isX sX
  trX
VLSM_incl_part (preloaded_vlsm_machine Free seed)
  (preloaded_vlsm_machine
     (composite_vlsm IM (free_constraint IM)) seed)
    by apply preloaded_free_composite_vlsm_spec.
  -message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
seed: message → Prop
SeededXE:= composite_no_equivocation_vlsm_with_preloaded
  (equivocator_IM IM)
  (free_constraint (equivocator_IM IM)) seed: VLSM message
Free:= free_composite_vlsm IM: VLSM message
SeededFree:= preloaded_vlsm Free seed: VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
isX, sX: state SeededFree
trX: list transition_item
HtrX: finite_valid_trace_init_to SeededFree isX sX
  trX
zero_descriptor_constraint_lifting_prop IM seed
  (no_equivocations_additional_constraint_with_preloaded
     (equivocator_IM IM)
     (free_constraint (equivocator_IM IM)) seed) intro; 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
seed: message → Prop
SeededXE:= composite_no_equivocation_vlsm_with_preloaded
  (equivocator_IM IM)
  (free_constraint (equivocator_IM IM)) seed: VLSM message
Free:= free_composite_vlsm IM: VLSM message
SeededFree:= preloaded_vlsm Free seed: VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
isX, sX: state SeededFree
trX: list transition_item
HtrX: finite_valid_trace_init_to SeededFree isX sX
  trX
es: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM IM)
           (free_constraint (equivocator_IM IM))
           seed)) seed)
Hes: valid_state_prop
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM IM)
           (free_constraint (equivocator_IM IM))
           seed)) seed) es
om: option message
Hom: sent_except_from (equivocator_IM IM)
  initial_message_prop es om
eqv: index
li: label (IM eqv)
no_equivocations_additional_constraint_with_preloaded
  (equivocator_IM IM)
  (free_constraint (equivocator_IM IM)) seed
  (existT eqv (ContinueWith 0 li)) (es, om) 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
seed: message → Prop
SeededXE:= composite_no_equivocation_vlsm_with_preloaded
  (equivocator_IM IM)
  (free_constraint (equivocator_IM IM)) seed: VLSM message
Free:= free_composite_vlsm IM: VLSM message
SeededFree:= preloaded_vlsm Free seed: VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
isX, sX: state SeededFree
trX: list transition_item
HtrX: finite_valid_trace_init_to SeededFree isX sX
  trX
es: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM IM)
           (free_constraint (equivocator_IM IM))
           seed)) seed)
Hes: valid_state_prop
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM IM)
           (free_constraint (equivocator_IM IM))
           seed)) seed) es
om: option message
Hom: sent_except_from (equivocator_IM IM)
  initial_message_prop es om
eqv: index
li: label (IM eqv)
composite_no_equivocations_except_from
  (equivocator_IM IM) seed
  (existT eqv (ContinueWith 0 li)) (es, om)
    destruct om; [| 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
seed: message → Prop
SeededXE:= composite_no_equivocation_vlsm_with_preloaded
  (equivocator_IM IM)
  (free_constraint (equivocator_IM IM)) seed: VLSM message
Free:= free_composite_vlsm IM: VLSM message
SeededFree:= preloaded_vlsm Free seed: VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
isX, sX: state SeededFree
trX: list transition_item
HtrX: finite_valid_trace_init_to SeededFree isX sX
  trX
es: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM IM)
           (free_constraint (equivocator_IM IM))
           seed)) seed)
Hes: valid_state_prop
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM IM)
           (free_constraint (equivocator_IM IM))
           seed)) seed) es
m: message
Hom: sent_except_from (equivocator_IM IM)
  initial_message_prop es 
  (Some m)
eqv: index
li: label (IM eqv)
composite_no_equivocations_except_from
  (equivocator_IM IM) seed
  (existT eqv (ContinueWith 0 li)) (es, Some m)
    destruct Hom as [Hsent | Hinitial]; [by 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
seed: message → Prop
SeededXE:= composite_no_equivocation_vlsm_with_preloaded
  (equivocator_IM IM)
  (free_constraint (equivocator_IM IM)) seed: VLSM message
Free:= free_composite_vlsm IM: VLSM message
SeededFree:= preloaded_vlsm Free seed: VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
isX, sX: state SeededFree
trX: list transition_item
HtrX: finite_valid_trace_init_to SeededFree isX sX
  trX
es: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM IM)
           (free_constraint (equivocator_IM IM))
           seed)) seed)
Hes: valid_state_prop
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM IM)
           (free_constraint (equivocator_IM IM))
           seed)) seed) es
m: message
Hinitial: initial_message_prop m
eqv: index
li: label (IM eqv)
composite_no_equivocations_except_from
  (equivocator_IM IM) seed
  (existT eqv (ContinueWith 0 li)) (es, Some m)
    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
seed: message → Prop
SeededXE:= composite_no_equivocation_vlsm_with_preloaded
  (equivocator_IM IM)
  (free_constraint (equivocator_IM IM)) seed: VLSM message
Free:= free_composite_vlsm IM: VLSM message
SeededFree:= preloaded_vlsm Free seed: VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
isX, sX: state SeededFree
trX: list transition_item
HtrX: finite_valid_trace_init_to SeededFree isX sX
  trX
es: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM IM)
           (free_constraint (equivocator_IM IM))
           seed)) seed)
Hes: valid_state_prop
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM IM)
           (free_constraint (equivocator_IM IM))
           seed)) seed) es
m: message
Hinitial: initial_message_prop m
eqv: index
li: label (IM eqv)
seed m
    destruct Hinitial as [[i [[mi Hmi] Him]] | Hseeded]; [exfalso | 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
seed: message → Prop
SeededXE:= composite_no_equivocation_vlsm_with_preloaded
  (equivocator_IM IM)
  (free_constraint (equivocator_IM IM)) seed: VLSM message
Free:= free_composite_vlsm IM: VLSM message
SeededFree:= preloaded_vlsm Free seed: VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
isX, sX: state SeededFree
trX: list transition_item
HtrX: finite_valid_trace_init_to SeededFree isX sX
  trX
es: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM IM)
           (free_constraint (equivocator_IM IM))
           seed)) seed)
Hes: valid_state_prop
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM IM)
           (free_constraint (equivocator_IM IM))
           seed)) seed) es
m: message
i: index
mi: message
Hmi: initial_message_prop mi
Him: `(mi ↾ Hmi) = m
eqv: index
li: label (IM eqv)
False
    by elim (no_initial_messages_in_IM i mi).
  -message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
seed: message → Prop
SeededXE:= composite_no_equivocation_vlsm_with_preloaded
  (equivocator_IM IM)
  (free_constraint (equivocator_IM IM)) seed: VLSM message
Free:= free_composite_vlsm IM: VLSM message
SeededFree:= preloaded_vlsm Free seed: VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
isX, sX: state SeededFree
trX: list transition_item
HtrX: finite_valid_trace_init_to SeededFree isX sX
  trX
replayable_message_prop IM seed (free_constraint IM)
  (no_equivocations_additional_constraint_with_preloaded
     (equivocator_IM IM)
     (free_constraint (equivocator_IM IM)) seed) clear isX sX trX 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
seed: message → Prop
SeededXE:= composite_no_equivocation_vlsm_with_preloaded
  (equivocator_IM IM)
  (free_constraint (equivocator_IM IM)) seed: VLSM message
Free:= free_composite_vlsm IM: VLSM message
SeededFree:= preloaded_vlsm Free seed: VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
replayable_message_prop IM seed (free_constraint IM)
  (no_equivocations_additional_constraint_with_preloaded
     (equivocator_IM IM)
     (free_constraint (equivocator_IM IM)) seed)
    intro; 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
seed: message → Prop
SeededXE:= composite_no_equivocation_vlsm_with_preloaded
  (equivocator_IM IM)
  (free_constraint (equivocator_IM IM)) seed: VLSM message
Free:= free_composite_vlsm IM: VLSM message
SeededFree:= preloaded_vlsm Free seed: VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
si, s: state
  (preloaded_vlsm
     (composite_vlsm IM (free_constraint IM))
     seed)
tr: list transition_item
HtrX: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm IM (free_constraint IM))
     seed) si s tr
eqv_state_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM IM)
           (free_constraint
              (equivocator_IM IM)) seed))
     seed)
Hstate_valid: valid_state_prop
  (preloaded_vlsm
     (composite_vlsm
        (equivocator_IM IM)
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM IM)
           (free_constraint
              (equivocator_IM IM)) seed))
     seed) eqv_state_s
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_msg_is, eqv_msg_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM IM)
           (free_constraint
              (equivocator_IM IM)) seed))
     seed)
eqv_msg_tr: list transition_item
Hmsg_trace: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM IM)
           (free_constraint
              (equivocator_IM IM)) seed))
     seed) eqv_msg_is eqv_msg_s
  eqv_msg_tr
iom: option message
Hfinal_msg: last_in_trace_except_from
  initial_message_prop eqv_msg_tr iom
l: composite_label IM
HcX: free_constraint IM l (s, iom)
∃ (eqv_msg_tr : list transition_item) (lst_msg_tr : 
                                       state
                                         (preloaded_vlsm
                                            (composite_vlsm
                                               (equivocator_IM
                                                 IM)
                                               (no_equivocations_additional_constraint_with_preloaded
                                                 (equivocator_IM
                                                 IM)
                                                 (free_constraint
                                                 (equivocator_IM
                                                 IM))
                                                 seed))
                                            seed)),
  finite_valid_trace_from_to
    (preloaded_vlsm
       (composite_vlsm (equivocator_IM IM)
          (no_equivocations_additional_constraint_with_preloaded
             (equivocator_IM IM)
             (free_constraint (equivocator_IM IM))
             seed)) seed) eqv_state_s lst_msg_tr
    eqv_msg_tr
  ∧ equivocators_total_trace_project IM eqv_msg_tr =
    []
    ∧ equivocators_total_state_project IM lst_msg_tr =
      s
      ∧ sent_except_from (equivocator_IM IM)
          initial_message_prop lst_msg_tr iom
    destruct iom as [im |]; [| by exists [], eqv_state_s; split; 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
seed: message → Prop
SeededXE:= composite_no_equivocation_vlsm_with_preloaded
  (equivocator_IM IM)
  (free_constraint (equivocator_IM IM)) seed: VLSM message
Free:= free_composite_vlsm IM: VLSM message
SeededFree:= preloaded_vlsm Free seed: VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
si, s: state
  (preloaded_vlsm
     (composite_vlsm IM (free_constraint IM))
     seed)
tr: list transition_item
HtrX: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm IM (free_constraint IM))
     seed) si s tr
eqv_state_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM IM)
           (free_constraint
              (equivocator_IM IM)) seed))
     seed)
Hstate_valid: valid_state_prop
  (preloaded_vlsm
     (composite_vlsm
        (equivocator_IM IM)
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM IM)
           (free_constraint
              (equivocator_IM IM)) seed))
     seed) eqv_state_s
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_msg_is, eqv_msg_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM IM)
           (free_constraint
              (equivocator_IM IM)) seed))
     seed)
eqv_msg_tr: list transition_item
Hmsg_trace: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM IM)
           (free_constraint
              (equivocator_IM IM)) seed))
     seed) eqv_msg_is eqv_msg_s
  eqv_msg_tr
im: message
Hfinal_msg: last_in_trace_except_from
  initial_message_prop eqv_msg_tr
  (Some im)
l: composite_label IM
HcX: free_constraint IM l (s, Some im)
∃ (eqv_msg_tr : list transition_item) (lst_msg_tr : 
                                       state
                                         (preloaded_vlsm
                                            (composite_vlsm
                                               (equivocator_IM
                                                 IM)
                                               (no_equivocations_additional_constraint_with_preloaded
                                                 (equivocator_IM
                                                 IM)
                                                 (free_constraint
                                                 (equivocator_IM
                                                 IM))
                                                 seed))
                                            seed)),
  finite_valid_trace_from_to
    (preloaded_vlsm
       (composite_vlsm (equivocator_IM IM)
          (no_equivocations_additional_constraint_with_preloaded
             (equivocator_IM IM)
             (free_constraint (equivocator_IM IM))
             seed)) seed) eqv_state_s lst_msg_tr
    eqv_msg_tr
  ∧ equivocators_total_trace_project IM eqv_msg_tr =
    []
    ∧ equivocators_total_state_project IM lst_msg_tr =
      s
      ∧ sent_except_from (equivocator_IM IM)
          initial_message_prop lst_msg_tr (Some im)
    specialize (NoEquivocation.seeded_no_equivocation_incl_preloaded
      (equivocator_IM IM) (free_constraint _) seed) as HinclE.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
seed: message → Prop
SeededXE:= composite_no_equivocation_vlsm_with_preloaded
  (equivocator_IM IM)
  (free_constraint (equivocator_IM IM)) seed: VLSM message
Free:= free_composite_vlsm IM: VLSM message
SeededFree:= preloaded_vlsm Free seed: VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
si, s: state
  (preloaded_vlsm
     (composite_vlsm IM (free_constraint IM))
     seed)
tr: list transition_item
HtrX: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm IM (free_constraint IM))
     seed) si s tr
eqv_state_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM IM)
           (free_constraint
              (equivocator_IM IM)) seed))
     seed)
Hstate_valid: valid_state_prop
  (preloaded_vlsm
     (composite_vlsm
        (equivocator_IM IM)
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM IM)
           (free_constraint
              (equivocator_IM IM)) seed))
     seed) eqv_state_s
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_msg_is, eqv_msg_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM IM)
           (free_constraint
              (equivocator_IM IM)) seed))
     seed)
eqv_msg_tr: list transition_item
Hmsg_trace: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM IM)
           (free_constraint
              (equivocator_IM IM)) seed))
     seed) eqv_msg_is eqv_msg_s
  eqv_msg_tr
im: message
Hfinal_msg: last_in_trace_except_from
  initial_message_prop eqv_msg_tr
  (Some im)
l: composite_label IM
HcX: free_constraint IM l (s, Some im)
HinclE: VLSM_incl
  (composite_no_equivocation_vlsm_with_preloaded
     (equivocator_IM IM)
     (free_constraint (equivocator_IM IM))
     seed)
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
∃ (eqv_msg_tr : list transition_item) (lst_msg_tr : 
                                       state
                                         (preloaded_vlsm
                                            (composite_vlsm
                                               (equivocator_IM
                                                 IM)
                                               (no_equivocations_additional_constraint_with_preloaded
                                                 (equivocator_IM
                                                 IM)
                                                 (free_constraint
                                                 (equivocator_IM
                                                 IM))
                                                 seed))
                                            seed)),
  finite_valid_trace_from_to
    (preloaded_vlsm
       (composite_vlsm (equivocator_IM IM)
          (no_equivocations_additional_constraint_with_preloaded
             (equivocator_IM IM)
             (free_constraint (equivocator_IM IM))
             seed)) seed) eqv_state_s lst_msg_tr
    eqv_msg_tr
  ∧ equivocators_total_trace_project IM eqv_msg_tr =
    []
    ∧ equivocators_total_state_project IM lst_msg_tr =
      s
      ∧ sent_except_from (equivocator_IM IM)
          initial_message_prop lst_msg_tr (Some im)
    apply valid_trace_forget_last in Hmsg_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
seed: message → Prop
SeededXE:= composite_no_equivocation_vlsm_with_preloaded
  (equivocator_IM IM)
  (free_constraint (equivocator_IM IM)) seed: VLSM message
Free:= free_composite_vlsm IM: VLSM message
SeededFree:= preloaded_vlsm Free seed: VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
si, s: state
  (preloaded_vlsm
     (composite_vlsm IM (free_constraint IM))
     seed)
tr: list transition_item
HtrX: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm IM (free_constraint IM))
     seed) si s tr
eqv_state_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM IM)
           (free_constraint
              (equivocator_IM IM)) seed))
     seed)
Hstate_valid: valid_state_prop
  (preloaded_vlsm
     (composite_vlsm
        (equivocator_IM IM)
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM IM)
           (free_constraint
              (equivocator_IM IM)) seed))
     seed) eqv_state_s
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_msg_is, eqv_msg_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM IM)
           (free_constraint
              (equivocator_IM IM)) seed))
     seed)
eqv_msg_tr: list transition_item
Hmsg_trace: finite_valid_trace
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM IM)
           (free_constraint
              (equivocator_IM IM)) seed))
     seed) eqv_msg_is eqv_msg_tr
im: message
Hfinal_msg: last_in_trace_except_from
  initial_message_prop eqv_msg_tr
  (Some im)
l: composite_label IM
HcX: free_constraint IM l (s, Some im)
HinclE: VLSM_incl
  (composite_no_equivocation_vlsm_with_preloaded
     (equivocator_IM IM)
     (free_constraint (equivocator_IM IM))
     seed)
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
∃ (eqv_msg_tr : list transition_item) (lst_msg_tr : 
                                       state
                                         (preloaded_vlsm
                                            (composite_vlsm
                                               (equivocator_IM
                                                 IM)
                                               (no_equivocations_additional_constraint_with_preloaded
                                                 (equivocator_IM
                                                 IM)
                                                 (free_constraint
                                                 (equivocator_IM
                                                 IM))
                                                 seed))
                                            seed)),
  finite_valid_trace_from_to
    (preloaded_vlsm
       (composite_vlsm (equivocator_IM IM)
          (no_equivocations_additional_constraint_with_preloaded
             (equivocator_IM IM)
             (free_constraint (equivocator_IM IM))
             seed)) seed) eqv_state_s lst_msg_tr
    eqv_msg_tr
  ∧ equivocators_total_trace_project IM eqv_msg_tr =
    []
    ∧ equivocators_total_state_project IM lst_msg_tr =
      s
      ∧ sent_except_from (equivocator_IM IM)
          initial_message_prop lst_msg_tr (Some im)
    specialize
      (replayed_trace_from_valid_equivocating
       _ Hstate_valid _ _ Hmsg_trace)
      as Hmsg_trace_full_replay.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
seed: message → Prop
SeededXE:= composite_no_equivocation_vlsm_with_preloaded
  (equivocator_IM IM)
  (free_constraint (equivocator_IM IM)) seed: VLSM message
Free:= free_composite_vlsm IM: VLSM message
SeededFree:= preloaded_vlsm Free seed: VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
si, s: state
  (preloaded_vlsm
     (composite_vlsm IM (free_constraint IM))
     seed)
tr: list transition_item
HtrX: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm IM (free_constraint IM))
     seed) si s tr
eqv_state_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM IM)
           (free_constraint
              (equivocator_IM IM)) seed))
     seed)
Hstate_valid: valid_state_prop
  (preloaded_vlsm
     (composite_vlsm
        (equivocator_IM IM)
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM IM)
           (free_constraint
              (equivocator_IM IM)) seed))
     seed) eqv_state_s
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_msg_is, eqv_msg_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM IM)
           (free_constraint
              (equivocator_IM IM)) seed))
     seed)
eqv_msg_tr: list transition_item
Hmsg_trace: finite_valid_trace
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM IM)
           (free_constraint
              (equivocator_IM IM)) seed))
     seed) eqv_msg_is eqv_msg_tr
im: message
Hfinal_msg: last_in_trace_except_from
  initial_message_prop eqv_msg_tr
  (Some im)
l: composite_label IM
HcX: free_constraint IM l (s, Some im)
HinclE: VLSM_incl
  (composite_no_equivocation_vlsm_with_preloaded
     (equivocator_IM IM)
     (free_constraint (equivocator_IM IM))
     seed)
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
Hmsg_trace_full_replay: finite_valid_trace_from
  SeededXE eqv_state_s
  (all_equivocating_replayed_trace_from
     eqv_state_s eqv_msg_is
     eqv_msg_tr)
∃ (eqv_msg_tr : list transition_item) (lst_msg_tr : 
                                       state
                                         (preloaded_vlsm
                                            (composite_vlsm
                                               (equivocator_IM
                                                 IM)
                                               (no_equivocations_additional_constraint_with_preloaded
                                                 (equivocator_IM
                                                 IM)
                                                 (free_constraint
                                                 (equivocator_IM
                                                 IM))
                                                 seed))
                                            seed)),
  finite_valid_trace_from_to
    (preloaded_vlsm
       (composite_vlsm (equivocator_IM IM)
          (no_equivocations_additional_constraint_with_preloaded
             (equivocator_IM IM)
             (free_constraint (equivocator_IM IM))
             seed)) seed) eqv_state_s lst_msg_tr
    eqv_msg_tr
  ∧ equivocators_total_trace_project IM eqv_msg_tr =
    []
    ∧ equivocators_total_state_project IM lst_msg_tr =
      s
      ∧ sent_except_from (equivocator_IM IM)
          initial_message_prop lst_msg_tr (Some im)
    remember (all_equivocating_replayed_trace_from _ _ _) as emsg_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
seed: message → Prop
SeededXE:= composite_no_equivocation_vlsm_with_preloaded
  (equivocator_IM IM)
  (free_constraint (equivocator_IM IM)) seed: VLSM message
Free:= free_composite_vlsm IM: VLSM message
SeededFree:= preloaded_vlsm Free seed: VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
si, s: state
  (preloaded_vlsm
     (composite_vlsm IM (free_constraint IM))
     seed)
tr: list transition_item
HtrX: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm IM (free_constraint IM))
     seed) si s tr
eqv_state_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM IM)
           (free_constraint
              (equivocator_IM IM)) seed))
     seed)
Hstate_valid: valid_state_prop
  (preloaded_vlsm
     (composite_vlsm
        (equivocator_IM IM)
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM IM)
           (free_constraint
              (equivocator_IM IM)) seed))
     seed) eqv_state_s
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_msg_is, eqv_msg_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM IM)
           (free_constraint
              (equivocator_IM IM)) seed))
     seed)
eqv_msg_tr: list transition_item
Hmsg_trace: finite_valid_trace
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM IM)
           (free_constraint
              (equivocator_IM IM)) seed))
     seed) eqv_msg_is eqv_msg_tr
im: message
Hfinal_msg: last_in_trace_except_from
  initial_message_prop eqv_msg_tr
  (Some im)
l: composite_label IM
HcX: free_constraint IM l (s, Some im)
HinclE: VLSM_incl
  (composite_no_equivocation_vlsm_with_preloaded
     (equivocator_IM IM)
     (free_constraint (equivocator_IM IM))
     seed)
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
emsg_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Heqemsg_tr: emsg_tr =
all_equivocating_replayed_trace_from
  eqv_state_s eqv_msg_is eqv_msg_tr
Hmsg_trace_full_replay: finite_valid_trace_from
  SeededXE eqv_state_s
  emsg_tr
∃ (eqv_msg_tr : list transition_item) (lst_msg_tr : 
                                       state
                                         (preloaded_vlsm
                                            (composite_vlsm
                                               (equivocator_IM
                                                 IM)
                                               (no_equivocations_additional_constraint_with_preloaded
                                                 (equivocator_IM
                                                 IM)
                                                 (free_constraint
                                                 (equivocator_IM
                                                 IM))
                                                 seed))
                                            seed)),
  finite_valid_trace_from_to
    (preloaded_vlsm
       (composite_vlsm (equivocator_IM IM)
          (no_equivocations_additional_constraint_with_preloaded
             (equivocator_IM IM)
             (free_constraint (equivocator_IM IM))
             seed)) seed) eqv_state_s lst_msg_tr
    eqv_msg_tr
  ∧ equivocators_total_trace_project IM eqv_msg_tr =
    []
    ∧ equivocators_total_state_project IM lst_msg_tr =
      s
      ∧ sent_except_from (equivocator_IM IM)
          initial_message_prop lst_msg_tr (Some im)
    apply valid_trace_add_default_last in Hmsg_trace_full_replay.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
seed: message → Prop
SeededXE:= composite_no_equivocation_vlsm_with_preloaded
  (equivocator_IM IM)
  (free_constraint (equivocator_IM IM)) seed: VLSM message
Free:= free_composite_vlsm IM: VLSM message
SeededFree:= preloaded_vlsm Free seed: VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
si, s: state
  (preloaded_vlsm
     (composite_vlsm IM (free_constraint IM))
     seed)
tr: list transition_item
HtrX: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm IM (free_constraint IM))
     seed) si s tr
eqv_state_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM IM)
           (free_constraint
              (equivocator_IM IM)) seed))
     seed)
Hstate_valid: valid_state_prop
  (preloaded_vlsm
     (composite_vlsm
        (equivocator_IM IM)
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM IM)
           (free_constraint
              (equivocator_IM IM)) seed))
     seed) eqv_state_s
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_msg_is, eqv_msg_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM IM)
           (free_constraint
              (equivocator_IM IM)) seed))
     seed)
eqv_msg_tr: list transition_item
Hmsg_trace: finite_valid_trace
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM IM)
           (free_constraint
              (equivocator_IM IM)) seed))
     seed) eqv_msg_is eqv_msg_tr
im: message
Hfinal_msg: last_in_trace_except_from
  initial_message_prop eqv_msg_tr
  (Some im)
l: composite_label IM
HcX: free_constraint IM l (s, Some im)
HinclE: VLSM_incl
  (composite_no_equivocation_vlsm_with_preloaded
     (equivocator_IM IM)
     (free_constraint (equivocator_IM IM))
     seed)
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
emsg_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Heqemsg_tr: emsg_tr =
all_equivocating_replayed_trace_from
  eqv_state_s eqv_msg_is eqv_msg_tr
Hmsg_trace_full_replay: finite_valid_trace_from_to
  SeededXE eqv_state_s
  (finite_trace_last
     eqv_state_s emsg_tr)
  emsg_tr
∃ (eqv_msg_tr : list transition_item) (lst_msg_tr : 
                                       state
                                         (preloaded_vlsm
                                            (composite_vlsm
                                               (equivocator_IM
                                                 IM)
                                               (no_equivocations_additional_constraint_with_preloaded
                                                 (equivocator_IM
                                                 IM)
                                                 (free_constraint
                                                 (equivocator_IM
                                                 IM))
                                                 seed))
                                            seed)),
  finite_valid_trace_from_to
    (preloaded_vlsm
       (composite_vlsm (equivocator_IM IM)
          (no_equivocations_additional_constraint_with_preloaded
             (equivocator_IM IM)
             (free_constraint (equivocator_IM IM))
             seed)) seed) eqv_state_s lst_msg_tr
    eqv_msg_tr
  ∧ equivocators_total_trace_project IM eqv_msg_tr =
    []
    ∧ equivocators_total_state_project IM lst_msg_tr =
      s
      ∧ sent_except_from (equivocator_IM IM)
          initial_message_prop lst_msg_tr (Some im)
    eexists _, _; 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
seed: message → Prop
SeededXE:= composite_no_equivocation_vlsm_with_preloaded
  (equivocator_IM IM)
  (free_constraint (equivocator_IM IM)) seed: VLSM message
Free:= free_composite_vlsm IM: VLSM message
SeededFree:= preloaded_vlsm Free seed: VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
si, s: state
  (preloaded_vlsm
     (composite_vlsm IM (free_constraint IM))
     seed)
tr: list transition_item
HtrX: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm IM (free_constraint IM))
     seed) si s tr
eqv_state_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM IM)
           (free_constraint
              (equivocator_IM IM)) seed))
     seed)
Hstate_valid: valid_state_prop
  (preloaded_vlsm
     (composite_vlsm
        (equivocator_IM IM)
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM IM)
           (free_constraint
              (equivocator_IM IM)) seed))
     seed) eqv_state_s
Hstate_final_project: equivocators_total_state_project
  IM eqv_state_s = s
eqv_msg_is, eqv_msg_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM IM)
           (free_constraint
              (equivocator_IM IM)) seed))
     seed)
eqv_msg_tr: list transition_item
Hmsg_trace: finite_valid_trace
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM IM)
           (free_constraint
              (equivocator_IM IM)) seed))
     seed) eqv_msg_is eqv_msg_tr
im: message
Hfinal_msg: last_in_trace_except_from
  initial_message_prop eqv_msg_tr
  (Some im)
l: composite_label IM
HcX: free_constraint IM l (s, Some im)
HinclE: VLSM_incl
  (composite_no_equivocation_vlsm_with_preloaded
     (equivocator_IM IM)
     (free_constraint (equivocator_IM IM))
     seed)
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
emsg_tr: list
  (composite_transition_item
     (equivocator_IM IM))
Heqemsg_tr: emsg_tr =
all_equivocating_replayed_trace_from
  eqv_state_s eqv_msg_is eqv_msg_tr
Hmsg_trace_full_replay: finite_valid_trace_from_to
  SeededXE eqv_state_s
  (finite_trace_last
     eqv_state_s emsg_tr)
  emsg_tr
equivocators_total_trace_project IM emsg_tr = []
∧ equivocators_total_state_project IM
    (finite_trace_last eqv_state_s emsg_tr) = s
  ∧ sent_except_from (equivocator_IM IM)
      initial_message_prop
      (finite_trace_last eqv_state_s emsg_tr)
      (Some im)
    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
seed: message → Prop
SeededXE:= composite_no_equivocation_vlsm_with_preloaded
  (equivocator_IM IM)
  (free_constraint (equivocator_IM IM)) seed: VLSM message
Free:= free_composite_vlsm IM: VLSM message
SeededFree:= preloaded_vlsm Free seed: VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
si: state
  (preloaded_vlsm
     (composite_vlsm IM (free_constraint IM))
     seed)
tr: list transition_item
eqv_state_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM IM)
           (free_constraint
              (equivocator_IM IM)) seed))
     seed)
HtrX: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm IM (free_constraint IM))
     seed) si
  (equivocators_total_state_project IM
     eqv_state_s) tr
Hstate_valid: valid_state_prop
  (preloaded_vlsm
     (composite_vlsm
        (equivocator_IM IM)
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM IM)
           (free_constraint
              (equivocator_IM IM)) seed))
     seed) eqv_state_s
eqv_msg_is, eqv_msg_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM IM)
           (free_constraint
              (equivocator_IM IM)) seed))
     seed)
eqv_msg_tr: list transition_item
Hmsg_trace: finite_valid_trace
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM IM)
           (free_constraint
              (equivocator_IM IM)) seed))
     seed) eqv_msg_is eqv_msg_tr
im: message
Hfinal_msg: last_in_trace_except_from
  initial_message_prop eqv_msg_tr
  (Some im)
l: composite_label IM
HcX: free_constraint IM l
  (equivocators_total_state_project IM
     eqv_state_s, Some im)
HinclE: VLSM_incl
  (composite_no_equivocation_vlsm_with_preloaded
     (equivocator_IM IM)
     (free_constraint (equivocator_IM IM))
     seed)
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
Hmsg_trace_full_replay: finite_valid_trace_from_to
  SeededXE eqv_state_s
  (finite_trace_last
     eqv_state_s
     (all_equivocating_replayed_trace_from
        eqv_state_s
        eqv_msg_is eqv_msg_tr))
  (all_equivocating_replayed_trace_from
     eqv_state_s eqv_msg_is
     eqv_msg_tr)
equivocators_total_trace_project IM
  (all_equivocating_replayed_trace_from eqv_state_s
     eqv_msg_is eqv_msg_tr) = []
∧ equivocators_total_state_project IM
    (finite_trace_last eqv_state_s
       (all_equivocating_replayed_trace_from
          eqv_state_s eqv_msg_is eqv_msg_tr)) =
  equivocators_total_state_project IM eqv_state_s
  ∧ sent_except_from (equivocator_IM IM)
      initial_message_prop
      (finite_trace_last eqv_state_s
         (all_equivocating_replayed_trace_from
            eqv_state_s eqv_msg_is eqv_msg_tr))
      (Some im)
    unfold all_equivocating_replayed_trace_from.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
seed: message → Prop
SeededXE:= composite_no_equivocation_vlsm_with_preloaded
  (equivocator_IM IM)
  (free_constraint (equivocator_IM IM)) seed: VLSM message
Free:= free_composite_vlsm IM: VLSM message
SeededFree:= preloaded_vlsm Free seed: VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
si: state
  (preloaded_vlsm
     (composite_vlsm IM (free_constraint IM))
     seed)
tr: list transition_item
eqv_state_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM IM)
           (free_constraint
              (equivocator_IM IM)) seed))
     seed)
HtrX: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm IM (free_constraint IM))
     seed) si
  (equivocators_total_state_project IM
     eqv_state_s) tr
Hstate_valid: valid_state_prop
  (preloaded_vlsm
     (composite_vlsm
        (equivocator_IM IM)
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM IM)
           (free_constraint
              (equivocator_IM IM)) seed))
     seed) eqv_state_s
eqv_msg_is, eqv_msg_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM IM)
           (free_constraint
              (equivocator_IM IM)) seed))
     seed)
eqv_msg_tr: list transition_item
Hmsg_trace: finite_valid_trace
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM IM)
           (free_constraint
              (equivocator_IM IM)) seed))
     seed) eqv_msg_is eqv_msg_tr
im: message
Hfinal_msg: last_in_trace_except_from
  initial_message_prop eqv_msg_tr
  (Some im)
l: composite_label IM
HcX: free_constraint IM l
  (equivocators_total_state_project IM
     eqv_state_s, Some im)
HinclE: VLSM_incl
  (composite_no_equivocation_vlsm_with_preloaded
     (equivocator_IM IM)
     (free_constraint (equivocator_IM IM))
     seed)
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
Hmsg_trace_full_replay: finite_valid_trace_from_to
  SeededXE eqv_state_s
  (finite_trace_last
     eqv_state_s
     (all_equivocating_replayed_trace_from
        eqv_state_s
        eqv_msg_is eqv_msg_tr))
  (all_equivocating_replayed_trace_from
     eqv_state_s eqv_msg_is
     eqv_msg_tr)
equivocators_total_trace_project IM
  (replayed_trace_from IM (enum index) eqv_state_s
     (composite_state_sub_projection
        (equivocator_IM IM) (enum index) eqv_msg_is)
     (VLSM_embedding_finite_trace_project
        (sub_composition_all_embedding
           (equivocator_IM IM)
           (equivocators_no_equivocations_constraint
              IM)) eqv_msg_tr)) = []
∧ equivocators_total_state_project IM
    (finite_trace_last eqv_state_s
       (replayed_trace_from IM (enum index)
          eqv_state_s
          (composite_state_sub_projection
             (equivocator_IM IM) (enum index)
             eqv_msg_is)
          (VLSM_embedding_finite_trace_project
             (sub_composition_all_embedding
                (equivocator_IM IM)
                (equivocators_no_equivocations_constraint
                   IM)) eqv_msg_tr))) =
  equivocators_total_state_project IM eqv_state_s
  ∧ sent_except_from (equivocator_IM IM)
      initial_message_prop
      (finite_trace_last eqv_state_s
         (replayed_trace_from IM (enum index)
            eqv_state_s
            (composite_state_sub_projection
               (equivocator_IM IM) (enum index)
               eqv_msg_is)
            (VLSM_embedding_finite_trace_project
               (sub_composition_all_embedding
                  (equivocator_IM IM)
                  (equivocators_no_equivocations_constraint
                     IM)) eqv_msg_tr))) (Some im)
    rewrite
      (equivocators_total_state_project_replayed_trace_from
        IM (enum index) eqv_state_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
seed: message → Prop
SeededXE:= composite_no_equivocation_vlsm_with_preloaded
  (equivocator_IM IM)
  (free_constraint (equivocator_IM IM)) seed: VLSM message
Free:= free_composite_vlsm IM: VLSM message
SeededFree:= preloaded_vlsm Free seed: VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
si: state
  (preloaded_vlsm
     (composite_vlsm IM (free_constraint IM))
     seed)
tr: list transition_item
eqv_state_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM IM)
           (free_constraint
              (equivocator_IM IM)) seed))
     seed)
HtrX: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm IM (free_constraint IM))
     seed) si
  (equivocators_total_state_project IM
     eqv_state_s) tr
Hstate_valid: valid_state_prop
  (preloaded_vlsm
     (composite_vlsm
        (equivocator_IM IM)
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM IM)
           (free_constraint
              (equivocator_IM IM)) seed))
     seed) eqv_state_s
eqv_msg_is, eqv_msg_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM IM)
           (free_constraint
              (equivocator_IM IM)) seed))
     seed)
eqv_msg_tr: list transition_item
Hmsg_trace: finite_valid_trace
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM IM)
           (free_constraint
              (equivocator_IM IM)) seed))
     seed) eqv_msg_is eqv_msg_tr
im: message
Hfinal_msg: last_in_trace_except_from
  initial_message_prop eqv_msg_tr
  (Some im)
l: composite_label IM
HcX: free_constraint IM l
  (equivocators_total_state_project IM
     eqv_state_s, Some im)
HinclE: VLSM_incl
  (composite_no_equivocation_vlsm_with_preloaded
     (equivocator_IM IM)
     (free_constraint (equivocator_IM IM))
     seed)
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
Hmsg_trace_full_replay: finite_valid_trace_from_to
  SeededXE eqv_state_s
  (finite_trace_last
     eqv_state_s
     (all_equivocating_replayed_trace_from
        eqv_state_s
        eqv_msg_is eqv_msg_tr))
  (all_equivocating_replayed_trace_from
     eqv_state_s eqv_msg_is
     eqv_msg_tr)
equivocators_total_trace_project IM
  (replayed_trace_from IM (enum index) eqv_state_s
     (composite_state_sub_projection
        (equivocator_IM IM) (enum index) eqv_msg_is)
     (VLSM_embedding_finite_trace_project
        (sub_composition_all_embedding
           (equivocator_IM IM)
           (equivocators_no_equivocations_constraint
              IM)) eqv_msg_tr)) = []
∧ equivocators_total_state_project IM eqv_state_s =
  equivocators_total_state_project IM eqv_state_s
  ∧ sent_except_from (equivocator_IM IM)
      initial_message_prop
      (finite_trace_last eqv_state_s
         (replayed_trace_from IM (enum index)
            eqv_state_s
            (composite_state_sub_projection
               (equivocator_IM IM) (enum index)
               eqv_msg_is)
            (VLSM_embedding_finite_trace_project
               (sub_composition_all_embedding
                  (equivocator_IM IM)
                  (equivocators_no_equivocations_constraint
                     IM)) eqv_msg_tr))) (Some im)
    rewrite
      (equivocators_total_trace_project_replayed_trace_from
        IM (enum index) eqv_state_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
seed: message → Prop
SeededXE:= composite_no_equivocation_vlsm_with_preloaded
  (equivocator_IM IM)
  (free_constraint (equivocator_IM IM)) seed: VLSM message
Free:= free_composite_vlsm IM: VLSM message
SeededFree:= preloaded_vlsm Free seed: VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
si: state
  (preloaded_vlsm
     (composite_vlsm IM (free_constraint IM))
     seed)
tr: list transition_item
eqv_state_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM IM)
           (free_constraint
              (equivocator_IM IM)) seed))
     seed)
HtrX: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm IM (free_constraint IM))
     seed) si
  (equivocators_total_state_project IM
     eqv_state_s) tr
Hstate_valid: valid_state_prop
  (preloaded_vlsm
     (composite_vlsm
        (equivocator_IM IM)
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM IM)
           (free_constraint
              (equivocator_IM IM)) seed))
     seed) eqv_state_s
eqv_msg_is, eqv_msg_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM IM)
           (free_constraint
              (equivocator_IM IM)) seed))
     seed)
eqv_msg_tr: list transition_item
Hmsg_trace: finite_valid_trace
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM IM)
           (free_constraint
              (equivocator_IM IM)) seed))
     seed) eqv_msg_is eqv_msg_tr
im: message
Hfinal_msg: last_in_trace_except_from
  initial_message_prop eqv_msg_tr
  (Some im)
l: composite_label IM
HcX: free_constraint IM l
  (equivocators_total_state_project IM
     eqv_state_s, Some im)
HinclE: VLSM_incl
  (composite_no_equivocation_vlsm_with_preloaded
     (equivocator_IM IM)
     (free_constraint (equivocator_IM IM))
     seed)
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
Hmsg_trace_full_replay: finite_valid_trace_from_to
  SeededXE eqv_state_s
  (finite_trace_last
     eqv_state_s
     (all_equivocating_replayed_trace_from
        eqv_state_s
        eqv_msg_is eqv_msg_tr))
  (all_equivocating_replayed_trace_from
     eqv_state_s eqv_msg_is
     eqv_msg_tr)
[] = []
∧ equivocators_total_state_project IM eqv_state_s =
  equivocators_total_state_project IM eqv_state_s
  ∧ sent_except_from (equivocator_IM IM)
      initial_message_prop
      (finite_trace_last eqv_state_s
         (replayed_trace_from IM (enum index)
            eqv_state_s
            (composite_state_sub_projection
               (equivocator_IM IM) (enum index)
               eqv_msg_is)
            (VLSM_embedding_finite_trace_project
               (sub_composition_all_embedding
                  (equivocator_IM IM)
                  (equivocators_no_equivocations_constraint
                     IM)) eqv_msg_tr))) (Some im)
    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
seed: message → Prop
SeededXE:= composite_no_equivocation_vlsm_with_preloaded
  (equivocator_IM IM)
  (free_constraint (equivocator_IM IM)) seed: VLSM message
Free:= free_composite_vlsm IM: VLSM message
SeededFree:= preloaded_vlsm Free seed: VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
si: state
  (preloaded_vlsm
     (composite_vlsm IM (free_constraint IM))
     seed)
tr: list transition_item
eqv_state_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM IM)
           (free_constraint
              (equivocator_IM IM)) seed))
     seed)
HtrX: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm IM (free_constraint IM))
     seed) si
  (equivocators_total_state_project IM
     eqv_state_s) tr
Hstate_valid: valid_state_prop
  (preloaded_vlsm
     (composite_vlsm
        (equivocator_IM IM)
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM IM)
           (free_constraint
              (equivocator_IM IM)) seed))
     seed) eqv_state_s
eqv_msg_is, eqv_msg_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM IM)
           (free_constraint
              (equivocator_IM IM)) seed))
     seed)
eqv_msg_tr: list transition_item
Hmsg_trace: finite_valid_trace
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM IM)
           (free_constraint
              (equivocator_IM IM)) seed))
     seed) eqv_msg_is eqv_msg_tr
im: message
Hfinal_msg: last_in_trace_except_from
  initial_message_prop eqv_msg_tr
  (Some im)
l: composite_label IM
HcX: free_constraint IM l
  (equivocators_total_state_project IM
     eqv_state_s, Some im)
HinclE: VLSM_incl
  (composite_no_equivocation_vlsm_with_preloaded
     (equivocator_IM IM)
     (free_constraint (equivocator_IM IM))
     seed)
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
Hmsg_trace_full_replay: finite_valid_trace_from_to
  SeededXE eqv_state_s
  (finite_trace_last
     eqv_state_s
     (all_equivocating_replayed_trace_from
        eqv_state_s
        eqv_msg_is eqv_msg_tr))
  (all_equivocating_replayed_trace_from
     eqv_state_s eqv_msg_is
     eqv_msg_tr)
sent_except_from (equivocator_IM IM)
  initial_message_prop
  (finite_trace_last eqv_state_s
     (replayed_trace_from IM (enum index) eqv_state_s
        (composite_state_sub_projection
           (equivocator_IM IM) (enum index) eqv_msg_is)
        (VLSM_embedding_finite_trace_project
           (sub_composition_all_embedding
              (equivocator_IM IM)
              (equivocators_no_equivocations_constraint
                 IM)) eqv_msg_tr))) (Some im) 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
seed: message → Prop
SeededXE:= composite_no_equivocation_vlsm_with_preloaded
  (equivocator_IM IM)
  (free_constraint (equivocator_IM IM)) seed: VLSM message
Free:= free_composite_vlsm IM: VLSM message
SeededFree:= preloaded_vlsm Free seed: VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
si: state
  (preloaded_vlsm
     (composite_vlsm IM (free_constraint IM))
     seed)
tr: list transition_item
eqv_state_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM IM)
           (free_constraint
              (equivocator_IM IM)) seed))
     seed)
HtrX: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm IM (free_constraint IM))
     seed) si
  (equivocators_total_state_project IM
     eqv_state_s) tr
Hstate_valid: valid_state_prop
  (preloaded_vlsm
     (composite_vlsm
        (equivocator_IM IM)
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM IM)
           (free_constraint
              (equivocator_IM IM)) seed))
     seed) eqv_state_s
eqv_msg_is, eqv_msg_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM IM)
           (free_constraint
              (equivocator_IM IM)) seed))
     seed)
eqv_msg_tr: list transition_item
Hmsg_trace: finite_valid_trace
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM IM)
           (free_constraint
              (equivocator_IM IM)) seed))
     seed) eqv_msg_is eqv_msg_tr
im: message
Hfinal_msg: last_in_trace_except_from
  initial_message_prop eqv_msg_tr
  (Some im)
l: composite_label IM
HcX: free_constraint IM l
  (equivocators_total_state_project IM
     eqv_state_s, Some im)
HinclE: VLSM_incl
  (composite_no_equivocation_vlsm_with_preloaded
     (equivocator_IM IM)
     (free_constraint (equivocator_IM IM))
     seed)
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
Hmsg_trace_full_replay: finite_valid_trace_from_to
  SeededXE eqv_state_s
  (finite_trace_last
     eqv_state_s
     (all_equivocating_replayed_trace_from
        eqv_state_s
        eqv_msg_is eqv_msg_tr))
  (all_equivocating_replayed_trace_from
     eqv_state_s eqv_msg_is
     eqv_msg_tr)
composite_has_been_sent (equivocator_IM IM)
  (finite_trace_last eqv_state_s
     (replayed_trace_from IM (enum index) eqv_state_s
        (composite_state_sub_projection
           (equivocator_IM IM) (enum index) eqv_msg_is)
        (VLSM_embedding_finite_trace_project
           (sub_composition_all_embedding
              (equivocator_IM IM)
              (equivocators_no_equivocations_constraint
                 IM)) eqv_msg_tr))) im
∨ composite_initial_message_prop (equivocator_IM IM)
    im ∨ seed im
    destruct Hfinal_msg as [Hfinal_msg | Hinitial]; [| by 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
seed: message → Prop
SeededXE:= composite_no_equivocation_vlsm_with_preloaded
  (equivocator_IM IM)
  (free_constraint (equivocator_IM IM)) seed: VLSM message
Free:= free_composite_vlsm IM: VLSM message
SeededFree:= preloaded_vlsm Free seed: VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
si: state
  (preloaded_vlsm
     (composite_vlsm IM (free_constraint IM))
     seed)
tr: list transition_item
eqv_state_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM IM)
           (free_constraint
              (equivocator_IM IM)) seed))
     seed)
HtrX: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm IM (free_constraint IM))
     seed) si
  (equivocators_total_state_project IM
     eqv_state_s) tr
Hstate_valid: valid_state_prop
  (preloaded_vlsm
     (composite_vlsm
        (equivocator_IM IM)
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM IM)
           (free_constraint
              (equivocator_IM IM)) seed))
     seed) eqv_state_s
eqv_msg_is, eqv_msg_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM IM)
           (free_constraint
              (equivocator_IM IM)) seed))
     seed)
eqv_msg_tr: list transition_item
Hmsg_trace: finite_valid_trace
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM IM)
           (free_constraint
              (equivocator_IM IM)) seed))
     seed) eqv_msg_is eqv_msg_tr
im: message
Hfinal_msg: finite_trace_last_output eqv_msg_tr =
Some im
l: composite_label IM
HcX: free_constraint IM l
  (equivocators_total_state_project IM
     eqv_state_s, Some im)
HinclE: VLSM_incl
  (composite_no_equivocation_vlsm_with_preloaded
     (equivocator_IM IM)
     (free_constraint (equivocator_IM IM))
     seed)
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
Hmsg_trace_full_replay: finite_valid_trace_from_to
  SeededXE eqv_state_s
  (finite_trace_last
     eqv_state_s
     (all_equivocating_replayed_trace_from
        eqv_state_s
        eqv_msg_is eqv_msg_tr))
  (all_equivocating_replayed_trace_from
     eqv_state_s eqv_msg_is
     eqv_msg_tr)
composite_has_been_sent (equivocator_IM IM)
  (finite_trace_last eqv_state_s
     (replayed_trace_from IM (enum index) eqv_state_s
        (composite_state_sub_projection
           (equivocator_IM IM) (enum index) eqv_msg_is)
        (VLSM_embedding_finite_trace_project
           (sub_composition_all_embedding
              (equivocator_IM IM)
              (equivocators_no_equivocations_constraint
                 IM)) eqv_msg_tr))) im
∨ composite_initial_message_prop (equivocator_IM IM)
    im ∨ seed im
    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
seed: message → Prop
SeededXE:= composite_no_equivocation_vlsm_with_preloaded
  (equivocator_IM IM)
  (free_constraint (equivocator_IM IM)) seed: VLSM message
Free:= free_composite_vlsm IM: VLSM message
SeededFree:= preloaded_vlsm Free seed: VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
si: state
  (preloaded_vlsm
     (composite_vlsm IM (free_constraint IM))
     seed)
tr: list transition_item
eqv_state_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM IM)
           (free_constraint
              (equivocator_IM IM)) seed))
     seed)
HtrX: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm IM (free_constraint IM))
     seed) si
  (equivocators_total_state_project IM
     eqv_state_s) tr
Hstate_valid: valid_state_prop
  (preloaded_vlsm
     (composite_vlsm
        (equivocator_IM IM)
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM IM)
           (free_constraint
              (equivocator_IM IM)) seed))
     seed) eqv_state_s
eqv_msg_is, eqv_msg_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM IM)
           (free_constraint
              (equivocator_IM IM)) seed))
     seed)
eqv_msg_tr: list transition_item
Hmsg_trace: finite_valid_trace
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM IM)
           (free_constraint
              (equivocator_IM IM)) seed))
     seed) eqv_msg_is eqv_msg_tr
im: message
Hfinal_msg: finite_trace_last_output eqv_msg_tr =
Some im
l: composite_label IM
HcX: free_constraint IM l
  (equivocators_total_state_project IM
     eqv_state_s, Some im)
HinclE: VLSM_incl
  (composite_no_equivocation_vlsm_with_preloaded
     (equivocator_IM IM)
     (free_constraint (equivocator_IM IM))
     seed)
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
Hmsg_trace_full_replay: finite_valid_trace_from_to
  SeededXE eqv_state_s
  (finite_trace_last
     eqv_state_s
     (all_equivocating_replayed_trace_from
        eqv_state_s
        eqv_msg_is eqv_msg_tr))
  (all_equivocating_replayed_trace_from
     eqv_state_s eqv_msg_is
     eqv_msg_tr)
composite_has_been_sent (equivocator_IM IM)
  (finite_trace_last eqv_state_s
     (replayed_trace_from IM (enum index) eqv_state_s
        (composite_state_sub_projection
           (equivocator_IM IM) (enum index) eqv_msg_is)
        (VLSM_embedding_finite_trace_project
           (sub_composition_all_embedding
              (equivocator_IM IM)
              (equivocators_no_equivocations_constraint
                 IM)) eqv_msg_tr))) im
    apply valid_trace_first_pstate in Hmsg_trace_full_replay as Hfst.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
seed: message → Prop
SeededXE:= composite_no_equivocation_vlsm_with_preloaded
  (equivocator_IM IM)
  (free_constraint (equivocator_IM IM)) seed: VLSM message
Free:= free_composite_vlsm IM: VLSM message
SeededFree:= preloaded_vlsm Free seed: VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
si: state
  (preloaded_vlsm
     (composite_vlsm IM (free_constraint IM))
     seed)
tr: list transition_item
eqv_state_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM IM)
           (free_constraint
              (equivocator_IM IM)) seed))
     seed)
HtrX: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm IM (free_constraint IM))
     seed) si
  (equivocators_total_state_project IM
     eqv_state_s) tr
Hstate_valid: valid_state_prop
  (preloaded_vlsm
     (composite_vlsm
        (equivocator_IM IM)
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM IM)
           (free_constraint
              (equivocator_IM IM)) seed))
     seed) eqv_state_s
eqv_msg_is, eqv_msg_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM IM)
           (free_constraint
              (equivocator_IM IM)) seed))
     seed)
eqv_msg_tr: list transition_item
Hmsg_trace: finite_valid_trace
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM IM)
           (free_constraint
              (equivocator_IM IM)) seed))
     seed) eqv_msg_is eqv_msg_tr
im: message
Hfinal_msg: finite_trace_last_output eqv_msg_tr =
Some im
l: composite_label IM
HcX: free_constraint IM l
  (equivocators_total_state_project IM
     eqv_state_s, Some im)
HinclE: VLSM_incl
  (composite_no_equivocation_vlsm_with_preloaded
     (equivocator_IM IM)
     (free_constraint (equivocator_IM IM))
     seed)
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
Hmsg_trace_full_replay: finite_valid_trace_from_to
  SeededXE eqv_state_s
  (finite_trace_last
     eqv_state_s
     (all_equivocating_replayed_trace_from
        eqv_state_s
        eqv_msg_is eqv_msg_tr))
  (all_equivocating_replayed_trace_from
     eqv_state_s eqv_msg_is
     eqv_msg_tr)
Hfst: valid_state_prop SeededXE eqv_state_s
composite_has_been_sent (equivocator_IM IM)
  (finite_trace_last eqv_state_s
     (replayed_trace_from IM (enum index) eqv_state_s
        (composite_state_sub_projection
           (equivocator_IM IM) (enum index) eqv_msg_is)
        (VLSM_embedding_finite_trace_project
           (sub_composition_all_embedding
              (equivocator_IM IM)
              (equivocators_no_equivocations_constraint
                 IM)) eqv_msg_tr))) im
    apply valid_state_has_trace in Hfst as [is_s [tr_s [Htr_s His_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
seed: message → Prop
SeededXE:= composite_no_equivocation_vlsm_with_preloaded
  (equivocator_IM IM)
  (free_constraint (equivocator_IM IM)) seed: VLSM message
Free:= free_composite_vlsm IM: VLSM message
SeededFree:= preloaded_vlsm Free seed: VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
si: state
  (preloaded_vlsm
     (composite_vlsm IM (free_constraint IM))
     seed)
tr: list transition_item
eqv_state_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM IM)
           (free_constraint
              (equivocator_IM IM)) seed))
     seed)
HtrX: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm IM (free_constraint IM))
     seed) si
  (equivocators_total_state_project IM
     eqv_state_s) tr
Hstate_valid: valid_state_prop
  (preloaded_vlsm
     (composite_vlsm
        (equivocator_IM IM)
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM IM)
           (free_constraint
              (equivocator_IM IM)) seed))
     seed) eqv_state_s
eqv_msg_is, eqv_msg_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM IM)
           (free_constraint
              (equivocator_IM IM)) seed))
     seed)
eqv_msg_tr: list transition_item
Hmsg_trace: finite_valid_trace
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM IM)
           (free_constraint
              (equivocator_IM IM)) seed))
     seed) eqv_msg_is eqv_msg_tr
im: message
Hfinal_msg: finite_trace_last_output eqv_msg_tr =
Some im
l: composite_label IM
HcX: free_constraint IM l
  (equivocators_total_state_project IM
     eqv_state_s, Some im)
HinclE: VLSM_incl
  (composite_no_equivocation_vlsm_with_preloaded
     (equivocator_IM IM)
     (free_constraint (equivocator_IM IM))
     seed)
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
Hmsg_trace_full_replay: finite_valid_trace_from_to
  SeededXE eqv_state_s
  (finite_trace_last
     eqv_state_s
     (all_equivocating_replayed_trace_from
        eqv_state_s
        eqv_msg_is eqv_msg_tr))
  (all_equivocating_replayed_trace_from
     eqv_state_s eqv_msg_is
     eqv_msg_tr)
is_s: state SeededXE
tr_s: list transition_item
Htr_s: finite_valid_trace_from_to SeededXE is_s
  eqv_state_s tr_s
His_s: initial_state_prop is_s
composite_has_been_sent (equivocator_IM IM)
  (finite_trace_last eqv_state_s
     (replayed_trace_from IM (enum index) eqv_state_s
        (composite_state_sub_projection
           (equivocator_IM IM) (enum index) eqv_msg_is)
        (VLSM_embedding_finite_trace_project
           (sub_composition_all_embedding
              (equivocator_IM IM)
              (equivocators_no_equivocations_constraint
                 IM)) eqv_msg_tr))) im
    specialize (finite_valid_trace_from_to_app SeededXE _ _ _ _ _ Htr_s Hmsg_trace_full_replay)
      as Happ.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
seed: message → Prop
SeededXE:= composite_no_equivocation_vlsm_with_preloaded
  (equivocator_IM IM)
  (free_constraint (equivocator_IM IM)) seed: VLSM message
Free:= free_composite_vlsm IM: VLSM message
SeededFree:= preloaded_vlsm Free seed: VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
si: state
  (preloaded_vlsm
     (composite_vlsm IM (free_constraint IM))
     seed)
tr: list transition_item
eqv_state_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM IM)
           (free_constraint
              (equivocator_IM IM)) seed))
     seed)
HtrX: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm IM (free_constraint IM))
     seed) si
  (equivocators_total_state_project IM
     eqv_state_s) tr
Hstate_valid: valid_state_prop
  (preloaded_vlsm
     (composite_vlsm
        (equivocator_IM IM)
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM IM)
           (free_constraint
              (equivocator_IM IM)) seed))
     seed) eqv_state_s
eqv_msg_is, eqv_msg_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM IM)
           (free_constraint
              (equivocator_IM IM)) seed))
     seed)
eqv_msg_tr: list transition_item
Hmsg_trace: finite_valid_trace
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM IM)
           (free_constraint
              (equivocator_IM IM)) seed))
     seed) eqv_msg_is eqv_msg_tr
im: message
Hfinal_msg: finite_trace_last_output eqv_msg_tr =
Some im
l: composite_label IM
HcX: free_constraint IM l
  (equivocators_total_state_project IM
     eqv_state_s, Some im)
HinclE: VLSM_incl
  (composite_no_equivocation_vlsm_with_preloaded
     (equivocator_IM IM)
     (free_constraint (equivocator_IM IM))
     seed)
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
Hmsg_trace_full_replay: finite_valid_trace_from_to
  SeededXE eqv_state_s
  (finite_trace_last
     eqv_state_s
     (all_equivocating_replayed_trace_from
        eqv_state_s
        eqv_msg_is eqv_msg_tr))
  (all_equivocating_replayed_trace_from
     eqv_state_s eqv_msg_is
     eqv_msg_tr)
is_s: state SeededXE
tr_s: list transition_item
Htr_s: finite_valid_trace_from_to SeededXE is_s
  eqv_state_s tr_s
His_s: initial_state_prop is_s
Happ: finite_valid_trace_from_to SeededXE is_s
  (finite_trace_last eqv_state_s
     (all_equivocating_replayed_trace_from
        eqv_state_s eqv_msg_is eqv_msg_tr))
  (tr_s ++
   all_equivocating_replayed_trace_from
     eqv_state_s eqv_msg_is eqv_msg_tr)
composite_has_been_sent (equivocator_IM IM)
  (finite_trace_last eqv_state_s
     (replayed_trace_from IM (enum index) eqv_state_s
        (composite_state_sub_projection
           (equivocator_IM IM) (enum index) eqv_msg_is)
        (VLSM_embedding_finite_trace_project
           (sub_composition_all_embedding
              (equivocator_IM IM)
              (equivocators_no_equivocations_constraint
                 IM)) eqv_msg_tr))) im
    apply (VLSM_incl_finite_valid_trace_from_to HinclE) in Happ.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
seed: message → Prop
SeededXE:= composite_no_equivocation_vlsm_with_preloaded
  (equivocator_IM IM)
  (free_constraint (equivocator_IM IM)) seed: VLSM message
Free:= free_composite_vlsm IM: VLSM message
SeededFree:= preloaded_vlsm Free seed: VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
si: state
  (preloaded_vlsm
     (composite_vlsm IM (free_constraint IM))
     seed)
tr: list transition_item
eqv_state_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM IM)
           (free_constraint
              (equivocator_IM IM)) seed))
     seed)
HtrX: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm IM (free_constraint IM))
     seed) si
  (equivocators_total_state_project IM
     eqv_state_s) tr
Hstate_valid: valid_state_prop
  (preloaded_vlsm
     (composite_vlsm
        (equivocator_IM IM)
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM IM)
           (free_constraint
              (equivocator_IM IM)) seed))
     seed) eqv_state_s
eqv_msg_is, eqv_msg_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM IM)
           (free_constraint
              (equivocator_IM IM)) seed))
     seed)
eqv_msg_tr: list transition_item
Hmsg_trace: finite_valid_trace
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM IM)
           (free_constraint
              (equivocator_IM IM)) seed))
     seed) eqv_msg_is eqv_msg_tr
im: message
Hfinal_msg: finite_trace_last_output eqv_msg_tr =
Some im
l: composite_label IM
HcX: free_constraint IM l
  (equivocators_total_state_project IM
     eqv_state_s, Some im)
HinclE: VLSM_incl
  (composite_no_equivocation_vlsm_with_preloaded
     (equivocator_IM IM)
     (free_constraint (equivocator_IM IM))
     seed)
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
Hmsg_trace_full_replay: finite_valid_trace_from_to
  SeededXE eqv_state_s
  (finite_trace_last
     eqv_state_s
     (all_equivocating_replayed_trace_from
        eqv_state_s
        eqv_msg_is eqv_msg_tr))
  (all_equivocating_replayed_trace_from
     eqv_state_s eqv_msg_is
     eqv_msg_tr)
is_s: state SeededXE
tr_s: list transition_item
Htr_s: finite_valid_trace_from_to SeededXE is_s
  eqv_state_s tr_s
His_s: initial_state_prop is_s
Happ: finite_valid_trace_from_to
  {|
    vlsm_type :=
      composite_no_equivocation_vlsm_with_preloaded
        (equivocator_IM IM)
        (free_constraint (equivocator_IM IM))
        seed;
    vlsm_machine :=
      preloaded_with_all_messages_vlsm
        (free_composite_vlsm
           (equivocator_IM IM))
  |} is_s
  (finite_trace_last eqv_state_s
     (all_equivocating_replayed_trace_from
        eqv_state_s eqv_msg_is eqv_msg_tr))
  (tr_s ++
   all_equivocating_replayed_trace_from
     eqv_state_s eqv_msg_is eqv_msg_tr)
composite_has_been_sent (equivocator_IM IM)
  (finite_trace_last eqv_state_s
     (replayed_trace_from IM (enum index) eqv_state_s
        (composite_state_sub_projection
           (equivocator_IM IM) (enum index) eqv_msg_is)
        (VLSM_embedding_finite_trace_project
           (sub_composition_all_embedding
              (equivocator_IM IM)
              (equivocators_no_equivocations_constraint
                 IM)) eqv_msg_tr))) im
    specialize (@has_been_sent_examine_one_trace _ FreeE _ _ _ _ (conj Happ His_s) im)
      as Hrew.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
seed: message → Prop
SeededXE:= composite_no_equivocation_vlsm_with_preloaded
  (equivocator_IM IM)
  (free_constraint (equivocator_IM IM)) seed: VLSM message
Free:= free_composite_vlsm IM: VLSM message
SeededFree:= preloaded_vlsm Free seed: VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
si: state
  (preloaded_vlsm
     (composite_vlsm IM (free_constraint IM))
     seed)
tr: list transition_item
eqv_state_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM IM)
           (free_constraint
              (equivocator_IM IM)) seed))
     seed)
HtrX: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm IM (free_constraint IM))
     seed) si
  (equivocators_total_state_project IM
     eqv_state_s) tr
Hstate_valid: valid_state_prop
  (preloaded_vlsm
     (composite_vlsm
        (equivocator_IM IM)
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM IM)
           (free_constraint
              (equivocator_IM IM)) seed))
     seed) eqv_state_s
eqv_msg_is, eqv_msg_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM IM)
           (free_constraint
              (equivocator_IM IM)) seed))
     seed)
eqv_msg_tr: list transition_item
Hmsg_trace: finite_valid_trace
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM IM)
           (free_constraint
              (equivocator_IM IM)) seed))
     seed) eqv_msg_is eqv_msg_tr
im: message
Hfinal_msg: finite_trace_last_output eqv_msg_tr =
Some im
l: composite_label IM
HcX: free_constraint IM l
  (equivocators_total_state_project IM
     eqv_state_s, Some im)
HinclE: VLSM_incl
  (composite_no_equivocation_vlsm_with_preloaded
     (equivocator_IM IM)
     (free_constraint (equivocator_IM IM))
     seed)
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
Hmsg_trace_full_replay: finite_valid_trace_from_to
  SeededXE eqv_state_s
  (finite_trace_last
     eqv_state_s
     (all_equivocating_replayed_trace_from
        eqv_state_s
        eqv_msg_is eqv_msg_tr))
  (all_equivocating_replayed_trace_from
     eqv_state_s eqv_msg_is
     eqv_msg_tr)
is_s: state SeededXE
tr_s: list transition_item
Htr_s: finite_valid_trace_from_to SeededXE is_s
  eqv_state_s tr_s
His_s: initial_state_prop is_s
Happ: finite_valid_trace_from_to
  {|
    vlsm_type :=
      composite_no_equivocation_vlsm_with_preloaded
        (equivocator_IM IM)
        (free_constraint (equivocator_IM IM))
        seed;
    vlsm_machine :=
      preloaded_with_all_messages_vlsm
        (free_composite_vlsm
           (equivocator_IM IM))
  |} is_s
  (finite_trace_last eqv_state_s
     (all_equivocating_replayed_trace_from
        eqv_state_s eqv_msg_is eqv_msg_tr))
  (tr_s ++
   all_equivocating_replayed_trace_from
     eqv_state_s eqv_msg_is eqv_msg_tr)
Hrew: has_been_sent FreeE
  (finite_trace_last eqv_state_s
     (all_equivocating_replayed_trace_from
        eqv_state_s eqv_msg_is eqv_msg_tr)) im
↔ trace_has_message (field_selector output) im
    (tr_s ++
     all_equivocating_replayed_trace_from
       eqv_state_s eqv_msg_is eqv_msg_tr)
composite_has_been_sent (equivocator_IM IM)
  (finite_trace_last eqv_state_s
     (replayed_trace_from IM (enum index) eqv_state_s
        (composite_state_sub_projection
           (equivocator_IM IM) (enum index) eqv_msg_is)
        (VLSM_embedding_finite_trace_project
           (sub_composition_all_embedding
              (equivocator_IM IM)
              (equivocators_no_equivocations_constraint
                 IM)) eqv_msg_tr))) im
    unfold has_been_sent in Hrew; cbn in Hrew; apply Hrew.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
seed: message → Prop
SeededXE:= composite_no_equivocation_vlsm_with_preloaded
  (equivocator_IM IM)
  (free_constraint (equivocator_IM IM)) seed: VLSM message
Free:= free_composite_vlsm IM: VLSM message
SeededFree:= preloaded_vlsm Free seed: VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
si: state
  (preloaded_vlsm
     (composite_vlsm IM (free_constraint IM))
     seed)
tr: list transition_item
eqv_state_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM IM)
           (free_constraint
              (equivocator_IM IM)) seed))
     seed)
HtrX: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm IM (free_constraint IM))
     seed) si
  (equivocators_total_state_project IM
     eqv_state_s) tr
Hstate_valid: valid_state_prop
  (preloaded_vlsm
     (composite_vlsm
        (equivocator_IM IM)
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM IM)
           (free_constraint
              (equivocator_IM IM)) seed))
     seed) eqv_state_s
eqv_msg_is, eqv_msg_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM IM)
           (free_constraint
              (equivocator_IM IM)) seed))
     seed)
eqv_msg_tr: list transition_item
Hmsg_trace: finite_valid_trace
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM IM)
           (free_constraint
              (equivocator_IM IM)) seed))
     seed) eqv_msg_is eqv_msg_tr
im: message
Hfinal_msg: finite_trace_last_output eqv_msg_tr =
Some im
l: composite_label IM
HcX: free_constraint IM l
  (equivocators_total_state_project IM
     eqv_state_s, Some im)
HinclE: VLSM_incl
  (composite_no_equivocation_vlsm_with_preloaded
     (equivocator_IM IM)
     (free_constraint (equivocator_IM IM))
     seed)
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
Hmsg_trace_full_replay: finite_valid_trace_from_to
  SeededXE eqv_state_s
  (finite_trace_last
     eqv_state_s
     (all_equivocating_replayed_trace_from
        eqv_state_s
        eqv_msg_is eqv_msg_tr))
  (all_equivocating_replayed_trace_from
     eqv_state_s eqv_msg_is
     eqv_msg_tr)
is_s: state SeededXE
tr_s: list transition_item
Htr_s: finite_valid_trace_from_to SeededXE is_s
  eqv_state_s tr_s
His_s: initial_state_prop is_s
Happ: finite_valid_trace_from_to
  {|
    vlsm_type :=
      composite_no_equivocation_vlsm_with_preloaded
        (equivocator_IM IM)
        (free_constraint (equivocator_IM IM))
        seed;
    vlsm_machine :=
      preloaded_with_all_messages_vlsm
        (free_composite_vlsm
           (equivocator_IM IM))
  |} is_s
  (finite_trace_last eqv_state_s
     (all_equivocating_replayed_trace_from
        eqv_state_s eqv_msg_is eqv_msg_tr))
  (tr_s ++
   all_equivocating_replayed_trace_from
     eqv_state_s eqv_msg_is eqv_msg_tr)
Hrew: composite_has_been_sent 
  (equivocator_IM IM)
  (finite_trace_last eqv_state_s
     (all_equivocating_replayed_trace_from
        eqv_state_s eqv_msg_is eqv_msg_tr)) im
↔ trace_has_message (field_selector output) im
    (tr_s ++
     all_equivocating_replayed_trace_from
       eqv_state_s eqv_msg_is eqv_msg_tr)
trace_has_message (field_selector output) im
  (tr_s ++
   all_equivocating_replayed_trace_from eqv_state_s
     eqv_msg_is eqv_msg_tr)

    apply 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
seed: message → Prop
SeededXE:= composite_no_equivocation_vlsm_with_preloaded
  (equivocator_IM IM)
  (free_constraint (equivocator_IM IM)) seed: VLSM message
Free:= free_composite_vlsm IM: VLSM message
SeededFree:= preloaded_vlsm Free seed: VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
si: state
  (preloaded_vlsm
     (composite_vlsm IM (free_constraint IM))
     seed)
tr: list transition_item
eqv_state_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM IM)
           (free_constraint
              (equivocator_IM IM)) seed))
     seed)
HtrX: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm IM (free_constraint IM))
     seed) si
  (equivocators_total_state_project IM
     eqv_state_s) tr
Hstate_valid: valid_state_prop
  (preloaded_vlsm
     (composite_vlsm
        (equivocator_IM IM)
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM IM)
           (free_constraint
              (equivocator_IM IM)) seed))
     seed) eqv_state_s
eqv_msg_is, eqv_msg_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM IM)
           (free_constraint
              (equivocator_IM IM)) seed))
     seed)
eqv_msg_tr: list transition_item
Hmsg_trace: finite_valid_trace
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM IM)
           (free_constraint
              (equivocator_IM IM)) seed))
     seed) eqv_msg_is eqv_msg_tr
im: message
Hfinal_msg: finite_trace_last_output eqv_msg_tr =
Some im
l: composite_label IM
HcX: free_constraint IM l
  (equivocators_total_state_project IM
     eqv_state_s, Some im)
HinclE: VLSM_incl
  (composite_no_equivocation_vlsm_with_preloaded
     (equivocator_IM IM)
     (free_constraint (equivocator_IM IM))
     seed)
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
Hmsg_trace_full_replay: finite_valid_trace_from_to
  SeededXE eqv_state_s
  (finite_trace_last
     eqv_state_s
     (all_equivocating_replayed_trace_from
        eqv_state_s
        eqv_msg_is eqv_msg_tr))
  (all_equivocating_replayed_trace_from
     eqv_state_s eqv_msg_is
     eqv_msg_tr)
is_s: state SeededXE
tr_s: list transition_item
Htr_s: finite_valid_trace_from_to SeededXE is_s
  eqv_state_s tr_s
His_s: initial_state_prop is_s
Happ: finite_valid_trace_from_to
  {|
    vlsm_type :=
      composite_no_equivocation_vlsm_with_preloaded
        (equivocator_IM IM)
        (free_constraint (equivocator_IM IM))
        seed;
    vlsm_machine :=
      preloaded_with_all_messages_vlsm
        (free_composite_vlsm
           (equivocator_IM IM))
  |} is_s
  (finite_trace_last eqv_state_s
     (all_equivocating_replayed_trace_from
        eqv_state_s eqv_msg_is eqv_msg_tr))
  (tr_s ++
   all_equivocating_replayed_trace_from
     eqv_state_s eqv_msg_is eqv_msg_tr)
Hrew: composite_has_been_sent 
  (equivocator_IM IM)
  (finite_trace_last eqv_state_s
     (all_equivocating_replayed_trace_from
        eqv_state_s eqv_msg_is eqv_msg_tr)) im
↔ trace_has_message (field_selector output) im
    (tr_s ++
     all_equivocating_replayed_trace_from
       eqv_state_s eqv_msg_is eqv_msg_tr)
Exists (field_selector output im) tr_s
∨ Exists (field_selector output im)
    (all_equivocating_replayed_trace_from eqv_state_s
       eqv_msg_is eqv_msg_tr) 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
seed: message → Prop
SeededXE:= composite_no_equivocation_vlsm_with_preloaded
  (equivocator_IM IM)
  (free_constraint (equivocator_IM IM)) seed: VLSM message
Free:= free_composite_vlsm IM: VLSM message
SeededFree:= preloaded_vlsm Free seed: VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
si: state
  (preloaded_vlsm
     (composite_vlsm IM (free_constraint IM))
     seed)
tr: list transition_item
eqv_state_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM IM)
           (free_constraint
              (equivocator_IM IM)) seed))
     seed)
HtrX: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm IM (free_constraint IM))
     seed) si
  (equivocators_total_state_project IM
     eqv_state_s) tr
Hstate_valid: valid_state_prop
  (preloaded_vlsm
     (composite_vlsm
        (equivocator_IM IM)
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM IM)
           (free_constraint
              (equivocator_IM IM)) seed))
     seed) eqv_state_s
eqv_msg_is, eqv_msg_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM IM)
           (free_constraint
              (equivocator_IM IM)) seed))
     seed)
eqv_msg_tr: list transition_item
Hmsg_trace: finite_valid_trace
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM IM)
           (free_constraint
              (equivocator_IM IM)) seed))
     seed) eqv_msg_is eqv_msg_tr
im: message
Hfinal_msg: finite_trace_last_output eqv_msg_tr =
Some im
l: composite_label IM
HcX: free_constraint IM l
  (equivocators_total_state_project IM
     eqv_state_s, Some im)
HinclE: VLSM_incl
  (composite_no_equivocation_vlsm_with_preloaded
     (equivocator_IM IM)
     (free_constraint (equivocator_IM IM))
     seed)
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
Hmsg_trace_full_replay: finite_valid_trace_from_to
  SeededXE eqv_state_s
  (finite_trace_last
     eqv_state_s
     (all_equivocating_replayed_trace_from
        eqv_state_s
        eqv_msg_is eqv_msg_tr))
  (all_equivocating_replayed_trace_from
     eqv_state_s eqv_msg_is
     eqv_msg_tr)
is_s: state SeededXE
tr_s: list transition_item
Htr_s: finite_valid_trace_from_to SeededXE is_s
  eqv_state_s tr_s
His_s: initial_state_prop is_s
Happ: finite_valid_trace_from_to
  {|
    vlsm_type :=
      composite_no_equivocation_vlsm_with_preloaded
        (equivocator_IM IM)
        (free_constraint (equivocator_IM IM))
        seed;
    vlsm_machine :=
      preloaded_with_all_messages_vlsm
        (free_composite_vlsm
           (equivocator_IM IM))
  |} is_s
  (finite_trace_last eqv_state_s
     (all_equivocating_replayed_trace_from
        eqv_state_s eqv_msg_is eqv_msg_tr))
  (tr_s ++
   all_equivocating_replayed_trace_from
     eqv_state_s eqv_msg_is eqv_msg_tr)
Hrew: composite_has_been_sent 
  (equivocator_IM IM)
  (finite_trace_last eqv_state_s
     (all_equivocating_replayed_trace_from
        eqv_state_s eqv_msg_is eqv_msg_tr)) im
↔ trace_has_message (field_selector output) im
    (tr_s ++
     all_equivocating_replayed_trace_from
       eqv_state_s eqv_msg_is eqv_msg_tr)
Exists (field_selector output im)
  (all_equivocating_replayed_trace_from eqv_state_s
     eqv_msg_is eqv_msg_tr)
    destruct_list_last eqv_msg_tr eqv_msg_tr' itemX Heqv_msg_tr
    ; [by inversion Hfinal_msg |].message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
seed: message → Prop
SeededXE:= composite_no_equivocation_vlsm_with_preloaded
  (equivocator_IM IM)
  (free_constraint (equivocator_IM IM)) seed: VLSM message
Free:= free_composite_vlsm IM: VLSM message
SeededFree:= preloaded_vlsm Free seed: VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
si: state
  (preloaded_vlsm
     (composite_vlsm IM (free_constraint IM))
     seed)
tr: list transition_item
eqv_state_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM IM)
           (free_constraint
              (equivocator_IM IM)) seed))
     seed)
HtrX: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm IM (free_constraint IM))
     seed) si
  (equivocators_total_state_project IM
     eqv_state_s) tr
Hstate_valid: valid_state_prop
  (preloaded_vlsm
     (composite_vlsm
        (equivocator_IM IM)
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM IM)
           (free_constraint
              (equivocator_IM IM)) seed))
     seed) eqv_state_s
eqv_msg_is, eqv_msg_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM IM)
           (free_constraint
              (equivocator_IM IM)) seed))
     seed)
eqv_msg_tr, eqv_msg_tr': list transition_item
itemX: transition_item
Hmsg_trace: finite_valid_trace
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM IM)
           (free_constraint
              (equivocator_IM IM)) seed))
     seed) eqv_msg_is
  (eqv_msg_tr' ++ [itemX])
im: message
Hfinal_msg: finite_trace_last_output
  (eqv_msg_tr' ++ [itemX]) = 
Some im
l: composite_label IM
HcX: free_constraint IM l
  (equivocators_total_state_project IM
     eqv_state_s, Some im)
HinclE: VLSM_incl
  (composite_no_equivocation_vlsm_with_preloaded
     (equivocator_IM IM)
     (free_constraint (equivocator_IM IM))
     seed)
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
Hmsg_trace_full_replay: finite_valid_trace_from_to
  SeededXE eqv_state_s
  (finite_trace_last
     eqv_state_s
     (all_equivocating_replayed_trace_from
        eqv_state_s
        eqv_msg_is
        (eqv_msg_tr' ++
         [itemX])))
  (all_equivocating_replayed_trace_from
     eqv_state_s eqv_msg_is
     (eqv_msg_tr' ++ [itemX]))
is_s: state SeededXE
tr_s: list transition_item
Htr_s: finite_valid_trace_from_to SeededXE is_s
  eqv_state_s tr_s
His_s: initial_state_prop is_s
Happ: finite_valid_trace_from_to
  {|
    vlsm_type :=
      composite_no_equivocation_vlsm_with_preloaded
        (equivocator_IM IM)
        (free_constraint (equivocator_IM IM))
        seed;
    vlsm_machine :=
      preloaded_with_all_messages_vlsm
        (free_composite_vlsm
           (equivocator_IM IM))
  |} is_s
  (finite_trace_last eqv_state_s
     (all_equivocating_replayed_trace_from
        eqv_state_s eqv_msg_is
        (eqv_msg_tr' ++ [itemX])))
  (tr_s ++
   all_equivocating_replayed_trace_from
     eqv_state_s eqv_msg_is
     (eqv_msg_tr' ++ [itemX]))
Hrew: composite_has_been_sent 
  (equivocator_IM IM)
  (finite_trace_last eqv_state_s
     (all_equivocating_replayed_trace_from
        eqv_state_s eqv_msg_is
        (eqv_msg_tr' ++ [itemX]))) im
↔ trace_has_message (field_selector output) im
    (tr_s ++
     all_equivocating_replayed_trace_from
       eqv_state_s eqv_msg_is
       (eqv_msg_tr' ++ [itemX]))
Heqv_msg_tr: eqv_msg_tr = eqv_msg_tr' ++ [itemX]
Exists (field_selector output im)
  (all_equivocating_replayed_trace_from eqv_state_s
     eqv_msg_is (eqv_msg_tr' ++ [itemX]))
    rewrite finite_trace_last_output_is_last in Hfinal_msg.message, index: Type
EqDecision0: EqDecision index
H: finite.Finite index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
seed: message → Prop
SeededXE:= composite_no_equivocation_vlsm_with_preloaded
  (equivocator_IM IM)
  (free_constraint (equivocator_IM IM)) seed: VLSM message
Free:= free_composite_vlsm IM: VLSM message
SeededFree:= preloaded_vlsm Free seed: VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
si: state
  (preloaded_vlsm
     (composite_vlsm IM (free_constraint IM))
     seed)
tr: list transition_item
eqv_state_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM IM)
           (free_constraint
              (equivocator_IM IM)) seed))
     seed)
HtrX: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm IM (free_constraint IM))
     seed) si
  (equivocators_total_state_project IM
     eqv_state_s) tr
Hstate_valid: valid_state_prop
  (preloaded_vlsm
     (composite_vlsm
        (equivocator_IM IM)
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM IM)
           (free_constraint
              (equivocator_IM IM)) seed))
     seed) eqv_state_s
eqv_msg_is, eqv_msg_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM IM)
           (free_constraint
              (equivocator_IM IM)) seed))
     seed)
eqv_msg_tr, eqv_msg_tr': list transition_item
itemX: transition_item
Hmsg_trace: finite_valid_trace
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM IM)
           (free_constraint
              (equivocator_IM IM)) seed))
     seed) eqv_msg_is
  (eqv_msg_tr' ++ [itemX])
im: message
Hfinal_msg: output itemX = Some im
l: composite_label IM
HcX: free_constraint IM l
  (equivocators_total_state_project IM
     eqv_state_s, Some im)
HinclE: VLSM_incl
  (composite_no_equivocation_vlsm_with_preloaded
     (equivocator_IM IM)
     (free_constraint (equivocator_IM IM))
     seed)
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
Hmsg_trace_full_replay: finite_valid_trace_from_to
  SeededXE eqv_state_s
  (finite_trace_last
     eqv_state_s
     (all_equivocating_replayed_trace_from
        eqv_state_s
        eqv_msg_is
        (eqv_msg_tr' ++
         [itemX])))
  (all_equivocating_replayed_trace_from
     eqv_state_s eqv_msg_is
     (eqv_msg_tr' ++ [itemX]))
is_s: state SeededXE
tr_s: list transition_item
Htr_s: finite_valid_trace_from_to SeededXE is_s
  eqv_state_s tr_s
His_s: initial_state_prop is_s
Happ: finite_valid_trace_from_to
  {|
    vlsm_type :=
      composite_no_equivocation_vlsm_with_preloaded
        (equivocator_IM IM)
        (free_constraint (equivocator_IM IM))
        seed;
    vlsm_machine :=
      preloaded_with_all_messages_vlsm
        (free_composite_vlsm
           (equivocator_IM IM))
  |} is_s
  (finite_trace_last eqv_state_s
     (all_equivocating_replayed_trace_from
        eqv_state_s eqv_msg_is
        (eqv_msg_tr' ++ [itemX])))
  (tr_s ++
   all_equivocating_replayed_trace_from
     eqv_state_s eqv_msg_is
     (eqv_msg_tr' ++ [itemX]))
Hrew: composite_has_been_sent 
  (equivocator_IM IM)
  (finite_trace_last eqv_state_s
     (all_equivocating_replayed_trace_from
        eqv_state_s eqv_msg_is
        (eqv_msg_tr' ++ [itemX]))) im
↔ trace_has_message (field_selector output) im
    (tr_s ++
     all_equivocating_replayed_trace_from
       eqv_state_s eqv_msg_is
       (eqv_msg_tr' ++ [itemX]))
Heqv_msg_tr: eqv_msg_tr = eqv_msg_tr' ++ [itemX]
Exists (field_selector output im)
  (all_equivocating_replayed_trace_from eqv_state_s
     eqv_msg_is (eqv_msg_tr' ++ [itemX]))
    apply 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
seed: message → Prop
SeededXE:= composite_no_equivocation_vlsm_with_preloaded
  (equivocator_IM IM)
  (free_constraint (equivocator_IM IM)) seed: VLSM message
Free:= free_composite_vlsm IM: VLSM message
SeededFree:= preloaded_vlsm Free seed: VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
si: state
  (preloaded_vlsm
     (composite_vlsm IM (free_constraint IM))
     seed)
tr: list transition_item
eqv_state_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM IM)
           (free_constraint
              (equivocator_IM IM)) seed))
     seed)
HtrX: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm IM (free_constraint IM))
     seed) si
  (equivocators_total_state_project IM
     eqv_state_s) tr
Hstate_valid: valid_state_prop
  (preloaded_vlsm
     (composite_vlsm
        (equivocator_IM IM)
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM IM)
           (free_constraint
              (equivocator_IM IM)) seed))
     seed) eqv_state_s
eqv_msg_is, eqv_msg_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM IM)
           (free_constraint
              (equivocator_IM IM)) seed))
     seed)
eqv_msg_tr, eqv_msg_tr': list transition_item
itemX: transition_item
Hmsg_trace: finite_valid_trace
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM IM)
           (free_constraint
              (equivocator_IM IM)) seed))
     seed) eqv_msg_is
  (eqv_msg_tr' ++ [itemX])
im: message
Hfinal_msg: output itemX = Some im
l: composite_label IM
HcX: free_constraint IM l
  (equivocators_total_state_project IM
     eqv_state_s, Some im)
HinclE: VLSM_incl
  (composite_no_equivocation_vlsm_with_preloaded
     (equivocator_IM IM)
     (free_constraint (equivocator_IM IM))
     seed)
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
Hmsg_trace_full_replay: finite_valid_trace_from_to
  SeededXE eqv_state_s
  (finite_trace_last
     eqv_state_s
     (all_equivocating_replayed_trace_from
        eqv_state_s
        eqv_msg_is
        (eqv_msg_tr' ++
         [itemX])))
  (all_equivocating_replayed_trace_from
     eqv_state_s eqv_msg_is
     (eqv_msg_tr' ++ [itemX]))
is_s: state SeededXE
tr_s: list transition_item
Htr_s: finite_valid_trace_from_to SeededXE is_s
  eqv_state_s tr_s
His_s: initial_state_prop is_s
Happ: finite_valid_trace_from_to
  {|
    vlsm_type :=
      composite_no_equivocation_vlsm_with_preloaded
        (equivocator_IM IM)
        (free_constraint (equivocator_IM IM))
        seed;
    vlsm_machine :=
      preloaded_with_all_messages_vlsm
        (free_composite_vlsm
           (equivocator_IM IM))
  |} is_s
  (finite_trace_last eqv_state_s
     (all_equivocating_replayed_trace_from
        eqv_state_s eqv_msg_is
        (eqv_msg_tr' ++ [itemX])))
  (tr_s ++
   all_equivocating_replayed_trace_from
     eqv_state_s eqv_msg_is
     (eqv_msg_tr' ++ [itemX]))
Hrew: composite_has_been_sent 
  (equivocator_IM IM)
  (finite_trace_last eqv_state_s
     (all_equivocating_replayed_trace_from
        eqv_state_s eqv_msg_is
        (eqv_msg_tr' ++ [itemX]))) im
↔ trace_has_message (field_selector output) im
    (tr_s ++
     all_equivocating_replayed_trace_from
       eqv_state_s eqv_msg_is
       (eqv_msg_tr' ++ [itemX]))
Heqv_msg_tr: eqv_msg_tr = eqv_msg_tr' ++ [itemX]
Exists (field_selector output im)
  (replayed_initial_state_from IM (enum index)
     eqv_state_s
     (composite_state_sub_projection
        (equivocator_IM IM) (enum index) eqv_msg_is))
∨ Exists (field_selector output im)
    (pre_VLSM_embedding_finite_trace_project
       (free_composite_vlsm
          (sub_IM (equivocator_IM IM) (enum index)))
       (free_composite_vlsm (equivocator_IM IM))
       (lift_equivocators_sub_label_to IM (enum index)
          eqv_state_s)
       (lift_equivocators_sub_state_to IM (enum index)
          eqv_state_s)
       (VLSM_embedding_finite_trace_project
          (sub_composition_all_embedding
             (equivocator_IM IM)
             (equivocators_no_equivocations_constraint
                IM)) (eqv_msg_tr' ++ [itemX]))) 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
seed: message → Prop
SeededXE:= composite_no_equivocation_vlsm_with_preloaded
  (equivocator_IM IM)
  (free_constraint (equivocator_IM IM)) seed: VLSM message
Free:= free_composite_vlsm IM: VLSM message
SeededFree:= preloaded_vlsm Free seed: VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
si: state
  (preloaded_vlsm
     (composite_vlsm IM (free_constraint IM))
     seed)
tr: list transition_item
eqv_state_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM IM)
           (free_constraint
              (equivocator_IM IM)) seed))
     seed)
HtrX: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm IM (free_constraint IM))
     seed) si
  (equivocators_total_state_project IM
     eqv_state_s) tr
Hstate_valid: valid_state_prop
  (preloaded_vlsm
     (composite_vlsm
        (equivocator_IM IM)
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM IM)
           (free_constraint
              (equivocator_IM IM)) seed))
     seed) eqv_state_s
eqv_msg_is, eqv_msg_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM IM)
           (free_constraint
              (equivocator_IM IM)) seed))
     seed)
eqv_msg_tr, eqv_msg_tr': list transition_item
itemX: transition_item
Hmsg_trace: finite_valid_trace
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM IM)
           (free_constraint
              (equivocator_IM IM)) seed))
     seed) eqv_msg_is
  (eqv_msg_tr' ++ [itemX])
im: message
Hfinal_msg: output itemX = Some im
l: composite_label IM
HcX: free_constraint IM l
  (equivocators_total_state_project IM
     eqv_state_s, Some im)
HinclE: VLSM_incl
  (composite_no_equivocation_vlsm_with_preloaded
     (equivocator_IM IM)
     (free_constraint (equivocator_IM IM))
     seed)
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
Hmsg_trace_full_replay: finite_valid_trace_from_to
  SeededXE eqv_state_s
  (finite_trace_last
     eqv_state_s
     (all_equivocating_replayed_trace_from
        eqv_state_s
        eqv_msg_is
        (eqv_msg_tr' ++
         [itemX])))
  (all_equivocating_replayed_trace_from
     eqv_state_s eqv_msg_is
     (eqv_msg_tr' ++ [itemX]))
is_s: state SeededXE
tr_s: list transition_item
Htr_s: finite_valid_trace_from_to SeededXE is_s
  eqv_state_s tr_s
His_s: initial_state_prop is_s
Happ: finite_valid_trace_from_to
  {|
    vlsm_type :=
      composite_no_equivocation_vlsm_with_preloaded
        (equivocator_IM IM)
        (free_constraint (equivocator_IM IM))
        seed;
    vlsm_machine :=
      preloaded_with_all_messages_vlsm
        (free_composite_vlsm
           (equivocator_IM IM))
  |} is_s
  (finite_trace_last eqv_state_s
     (all_equivocating_replayed_trace_from
        eqv_state_s eqv_msg_is
        (eqv_msg_tr' ++ [itemX])))
  (tr_s ++
   all_equivocating_replayed_trace_from
     eqv_state_s eqv_msg_is
     (eqv_msg_tr' ++ [itemX]))
Hrew: composite_has_been_sent 
  (equivocator_IM IM)
  (finite_trace_last eqv_state_s
     (all_equivocating_replayed_trace_from
        eqv_state_s eqv_msg_is
        (eqv_msg_tr' ++ [itemX]))) im
↔ trace_has_message (field_selector output) im
    (tr_s ++
     all_equivocating_replayed_trace_from
       eqv_state_s eqv_msg_is
       (eqv_msg_tr' ++ [itemX]))
Heqv_msg_tr: eqv_msg_tr = eqv_msg_tr' ++ [itemX]
Exists (field_selector output im)
  (pre_VLSM_embedding_finite_trace_project
     (free_composite_vlsm
        (sub_IM (equivocator_IM IM) (enum index)))
     (free_composite_vlsm (equivocator_IM IM))
     (lift_equivocators_sub_label_to IM (enum index)
        eqv_state_s)
     (lift_equivocators_sub_state_to IM (enum index)
        eqv_state_s)
     (VLSM_embedding_finite_trace_project
        (sub_composition_all_embedding
           (equivocator_IM IM)
           (equivocators_no_equivocations_constraint
              IM)) (eqv_msg_tr' ++ [itemX])))
    unfold VLSM_embedding_finite_trace_project, pre_VLSM_embedding_finite_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
seed: message → Prop
SeededXE:= composite_no_equivocation_vlsm_with_preloaded
  (equivocator_IM IM)
  (free_constraint (equivocator_IM IM)) seed: VLSM message
Free:= free_composite_vlsm IM: VLSM message
SeededFree:= preloaded_vlsm Free seed: VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
si: state
  (preloaded_vlsm
     (composite_vlsm IM (free_constraint IM))
     seed)
tr: list transition_item
eqv_state_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM IM)
           (free_constraint
              (equivocator_IM IM)) seed))
     seed)
HtrX: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm IM (free_constraint IM))
     seed) si
  (equivocators_total_state_project IM
     eqv_state_s) tr
Hstate_valid: valid_state_prop
  (preloaded_vlsm
     (composite_vlsm
        (equivocator_IM IM)
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM IM)
           (free_constraint
              (equivocator_IM IM)) seed))
     seed) eqv_state_s
eqv_msg_is, eqv_msg_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM IM)
           (free_constraint
              (equivocator_IM IM)) seed))
     seed)
eqv_msg_tr, eqv_msg_tr': list transition_item
itemX: transition_item
Hmsg_trace: finite_valid_trace
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM IM)
           (free_constraint
              (equivocator_IM IM)) seed))
     seed) eqv_msg_is
  (eqv_msg_tr' ++ [itemX])
im: message
Hfinal_msg: output itemX = Some im
l: composite_label IM
HcX: free_constraint IM l
  (equivocators_total_state_project IM
     eqv_state_s, Some im)
HinclE: VLSM_incl
  (composite_no_equivocation_vlsm_with_preloaded
     (equivocator_IM IM)
     (free_constraint (equivocator_IM IM))
     seed)
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
Hmsg_trace_full_replay: finite_valid_trace_from_to
  SeededXE eqv_state_s
  (finite_trace_last
     eqv_state_s
     (all_equivocating_replayed_trace_from
        eqv_state_s
        eqv_msg_is
        (eqv_msg_tr' ++
         [itemX])))
  (all_equivocating_replayed_trace_from
     eqv_state_s eqv_msg_is
     (eqv_msg_tr' ++ [itemX]))
is_s: state SeededXE
tr_s: list transition_item
Htr_s: finite_valid_trace_from_to SeededXE is_s
  eqv_state_s tr_s
His_s: initial_state_prop is_s
Happ: finite_valid_trace_from_to
  {|
    vlsm_type :=
      composite_no_equivocation_vlsm_with_preloaded
        (equivocator_IM IM)
        (free_constraint (equivocator_IM IM))
        seed;
    vlsm_machine :=
      preloaded_with_all_messages_vlsm
        (free_composite_vlsm
           (equivocator_IM IM))
  |} is_s
  (finite_trace_last eqv_state_s
     (all_equivocating_replayed_trace_from
        eqv_state_s eqv_msg_is
        (eqv_msg_tr' ++ [itemX])))
  (tr_s ++
   all_equivocating_replayed_trace_from
     eqv_state_s eqv_msg_is
     (eqv_msg_tr' ++ [itemX]))
Hrew: composite_has_been_sent 
  (equivocator_IM IM)
  (finite_trace_last eqv_state_s
     (all_equivocating_replayed_trace_from
        eqv_state_s eqv_msg_is
        (eqv_msg_tr' ++ [itemX]))) im
↔ trace_has_message (field_selector output) im
    (tr_s ++
     all_equivocating_replayed_trace_from
       eqv_state_s eqv_msg_is
       (eqv_msg_tr' ++ [itemX]))
Heqv_msg_tr: eqv_msg_tr = eqv_msg_tr' ++ [itemX]
Exists (field_selector output im)
  (map
     (pre_VLSM_embedding_transition_item_project
        (free_composite_vlsm
           (sub_IM (equivocator_IM IM) (enum index)))
        (free_composite_vlsm (equivocator_IM IM))
        (lift_equivocators_sub_label_to IM
           (enum index) eqv_state_s)
        (lift_equivocators_sub_state_to IM
           (enum index) eqv_state_s))
     (map
        (pre_VLSM_embedding_transition_item_project
           (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)))
        (eqv_msg_tr' ++ [itemX])))
    rewrite! map_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
seed: message → Prop
SeededXE:= composite_no_equivocation_vlsm_with_preloaded
  (equivocator_IM IM)
  (free_constraint (equivocator_IM IM)) seed: VLSM message
Free:= free_composite_vlsm IM: VLSM message
SeededFree:= preloaded_vlsm Free seed: VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
si: state
  (preloaded_vlsm
     (composite_vlsm IM (free_constraint IM))
     seed)
tr: list transition_item
eqv_state_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM IM)
           (free_constraint
              (equivocator_IM IM)) seed))
     seed)
HtrX: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm IM (free_constraint IM))
     seed) si
  (equivocators_total_state_project IM
     eqv_state_s) tr
Hstate_valid: valid_state_prop
  (preloaded_vlsm
     (composite_vlsm
        (equivocator_IM IM)
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM IM)
           (free_constraint
              (equivocator_IM IM)) seed))
     seed) eqv_state_s
eqv_msg_is, eqv_msg_s: state
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM IM)
           (free_constraint
              (equivocator_IM IM)) seed))
     seed)
eqv_msg_tr, eqv_msg_tr': list transition_item
itemX: transition_item
Hmsg_trace: finite_valid_trace
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM IM)
           (free_constraint
              (equivocator_IM IM)) seed))
     seed) eqv_msg_is
  (eqv_msg_tr' ++ [itemX])
im: message
Hfinal_msg: output itemX = Some im
l: composite_label IM
HcX: free_constraint IM l
  (equivocators_total_state_project IM
     eqv_state_s, Some im)
HinclE: VLSM_incl
  (composite_no_equivocation_vlsm_with_preloaded
     (equivocator_IM IM)
     (free_constraint (equivocator_IM IM))
     seed)
  (preloaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM)))
Hmsg_trace_full_replay: finite_valid_trace_from_to
  SeededXE eqv_state_s
  (finite_trace_last
     eqv_state_s
     (all_equivocating_replayed_trace_from
        eqv_state_s
        eqv_msg_is
        (eqv_msg_tr' ++
         [itemX])))
  (all_equivocating_replayed_trace_from
     eqv_state_s eqv_msg_is
     (eqv_msg_tr' ++ [itemX]))
is_s: state SeededXE
tr_s: list transition_item
Htr_s: finite_valid_trace_from_to SeededXE is_s
  eqv_state_s tr_s
His_s: initial_state_prop is_s
Happ: finite_valid_trace_from_to
  {|
    vlsm_type :=
      composite_no_equivocation_vlsm_with_preloaded
        (equivocator_IM IM)
        (free_constraint (equivocator_IM IM))
        seed;
    vlsm_machine :=
      preloaded_with_all_messages_vlsm
        (free_composite_vlsm
           (equivocator_IM IM))
  |} is_s
  (finite_trace_last eqv_state_s
     (all_equivocating_replayed_trace_from
        eqv_state_s eqv_msg_is
        (eqv_msg_tr' ++ [itemX])))
  (tr_s ++
   all_equivocating_replayed_trace_from
     eqv_state_s eqv_msg_is
     (eqv_msg_tr' ++ [itemX]))
Hrew: composite_has_been_sent 
  (equivocator_IM IM)
  (finite_trace_last eqv_state_s
     (all_equivocating_replayed_trace_from
        eqv_state_s eqv_msg_is
        (eqv_msg_tr' ++ [itemX]))) im
↔ trace_has_message (field_selector output) im
    (tr_s ++
     all_equivocating_replayed_trace_from
       eqv_state_s eqv_msg_is
       (eqv_msg_tr' ++ [itemX]))
Heqv_msg_tr: eqv_msg_tr = eqv_msg_tr' ++ [itemX]
Exists (field_selector output im)
  (map
     (pre_VLSM_embedding_transition_item_project
        (free_composite_vlsm
           (sub_IM (equivocator_IM IM) (enum index)))
        (free_composite_vlsm (equivocator_IM IM))
        (lift_equivocators_sub_label_to IM
           (enum index) eqv_state_s)
        (lift_equivocators_sub_state_to IM
           (enum index) eqv_state_s))
     (map
        (pre_VLSM_embedding_transition_item_project
           (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)))
        eqv_msg_tr') ++
   map
     (pre_VLSM_embedding_transition_item_project
        (free_composite_vlsm
           (sub_IM (equivocator_IM IM) (enum index)))
        (free_composite_vlsm (equivocator_IM IM))
        (lift_equivocators_sub_label_to IM
           (enum index) eqv_state_s)
        (lift_equivocators_sub_state_to IM
           (enum index) eqv_state_s))
     (map
        (pre_VLSM_embedding_transition_item_project
           (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)))
        [itemX]))
    by apply Exists_app; simpl; right; left.
Qed.

End sec_seeded_all_equivocating.

Section sec_all_equivocating.

Context
  {message : Type}
  {index : Type}
  `{finite.Finite index}
  (IM : index -> VLSM message)
  `{forall i : index, HasBeenSentCapability (IM i)}
  (Free := free_composite_vlsm IM)
  (XE : VLSM message := equivocators_no_equivocations_vlsm IM)
  .

Further specializing seeded_equivocators_finite_valid_trace_init_to_rev to remove the preloading operation.

Lemma equivocators_finite_valid_trace_init_to_rev
  (no_initial_messages_in_IM : no_initial_messages_in_IM_prop IM)
  isX sX trX
  (HtrX : finite_valid_trace_init_to Free isX sX trX)
  : exists is, equivocators_total_state_project IM is = isX /\
    exists s, equivocators_total_state_project IM s = sX /\
    exists tr, equivocators_total_trace_project IM tr = trX /\
    finite_valid_trace_init_to XE is s tr /\
    finite_trace_last_output trX = finite_trace_last_output tr.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
XE:= equivocators_no_equivocations_vlsm IM: VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
isX, sX: state Free
trX: list transition_item
HtrX: finite_valid_trace_init_to Free isX sX trX
∃ is : state (free_composite_vlsm (equivocator_IM IM)),
  equivocators_total_state_project IM is = isX
  ∧ (∃ s : state
             (free_composite_vlsm (equivocator_IM IM)),
       equivocators_total_state_project IM s = sX
       ∧ (∃ tr : list
                   (composite_transition_item
                      (equivocator_IM IM)),
            equivocators_total_trace_project IM tr =
            trX
            ∧ finite_valid_trace_init_to XE is s tr
              ∧ finite_trace_last_output trX =
                finite_trace_last_output tr))
Proof.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
XE:= equivocators_no_equivocations_vlsm IM: VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
isX, sX: state Free
trX: list transition_item
HtrX: finite_valid_trace_init_to Free isX sX trX
∃ is : state (free_composite_vlsm (equivocator_IM IM)),
  equivocators_total_state_project IM is = isX
  ∧ (∃ s : state
             (free_composite_vlsm (equivocator_IM IM)),
       equivocators_total_state_project IM s = sX
       ∧ (∃ tr : list
                   (composite_transition_item
                      (equivocator_IM IM)),
            equivocators_total_trace_project IM tr =
            trX
            ∧ finite_valid_trace_init_to XE is s tr
              ∧ finite_trace_last_output trX =
                finite_trace_last_output tr))
  specialize (vlsm_is_preloaded_with_False Free) 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
XE:= equivocators_no_equivocations_vlsm IM: VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
isX, sX: state Free
trX: list transition_item
HtrX: finite_valid_trace_init_to Free isX sX trX
Heq: VLSM_eq Free
  (preloaded_vlsm Free (λ _ : message, False))
∃ is : state (free_composite_vlsm (equivocator_IM IM)),
  equivocators_total_state_project IM is = isX
  ∧ (∃ s : state
             (free_composite_vlsm (equivocator_IM IM)),
       equivocators_total_state_project IM s = sX
       ∧ (∃ tr : list
                   (composite_transition_item
                      (equivocator_IM IM)),
            equivocators_total_trace_project IM tr =
            trX
            ∧ finite_valid_trace_init_to XE is s tr
              ∧ finite_trace_last_output trX =
                finite_trace_last_output tr))
  apply (VLSM_eq_finite_valid_trace_init_to 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
XE:= equivocators_no_equivocations_vlsm IM: VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
isX, sX: state Free
trX: list transition_item
HtrX: finite_valid_trace_init_to
  {|
    vlsm_type := Free;
    vlsm_machine :=
      preloaded_vlsm Free
        (λ _ : message, False)
  |} isX sX trX
Heq: VLSM_eq Free
  (preloaded_vlsm Free (λ _ : message, False))
∃ is : state (free_composite_vlsm (equivocator_IM IM)),
  equivocators_total_state_project IM is = isX
  ∧ (∃ s : state
             (free_composite_vlsm (equivocator_IM IM)),
       equivocators_total_state_project IM s = sX
       ∧ (∃ tr : list
                   (composite_transition_item
                      (equivocator_IM IM)),
            equivocators_total_trace_project IM tr =
            trX
            ∧ finite_valid_trace_init_to XE is s tr
              ∧ finite_trace_last_output trX =
                finite_trace_last_output tr))
  specialize
    (seeded_equivocators_finite_valid_trace_init_to_rev
      IM (fun m => False) no_initial_messages_in_IM
      _ _ _ HtrX)
    as [is [His [s [Hs [tr [Htr_pr [Htr Houtput]]]]]]].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
XE:= equivocators_no_equivocations_vlsm IM: VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
isX, sX: state Free
trX: list transition_item
HtrX: finite_valid_trace_init_to
  {|
    vlsm_type := Free;
    vlsm_machine :=
      preloaded_vlsm Free
        (λ _ : message, False)
  |} isX sX trX
Heq: VLSM_eq Free
  (preloaded_vlsm Free (λ _ : message, False))
is: state (free_composite_vlsm (equivocator_IM IM))
His: equivocators_total_state_project IM is = isX
s: state (free_composite_vlsm (equivocator_IM IM))
Hs: equivocators_total_state_project IM s = sX
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr_pr: equivocators_total_trace_project IM tr = trX
Htr: finite_valid_trace_init_to
  (composite_no_equivocation_vlsm_with_preloaded
     (equivocator_IM IM)
     (free_constraint (equivocator_IM IM))
     (λ _ : message, False)) is s tr
Houtput: finite_trace_last_output trX =
finite_trace_last_output tr
∃ is : state (free_composite_vlsm (equivocator_IM IM)),
  equivocators_total_state_project IM is = isX
  ∧ (∃ s : state
             (free_composite_vlsm (equivocator_IM IM)),
       equivocators_total_state_project IM s = sX
       ∧ (∃ tr : list
                   (composite_transition_item
                      (equivocator_IM IM)),
            equivocators_total_trace_project IM tr =
            trX
            ∧ finite_valid_trace_init_to XE is s tr
              ∧ finite_trace_last_output trX =
                finite_trace_last_output tr))
  exists is; split; [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
XE:= equivocators_no_equivocations_vlsm IM: VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
isX, sX: state Free
trX: list transition_item
HtrX: finite_valid_trace_init_to
  {|
    vlsm_type := Free;
    vlsm_machine :=
      preloaded_vlsm Free
        (λ _ : message, False)
  |} isX sX trX
Heq: VLSM_eq Free
  (preloaded_vlsm Free (λ _ : message, False))
is: state (free_composite_vlsm (equivocator_IM IM))
His: equivocators_total_state_project IM is = isX
s: state (free_composite_vlsm (equivocator_IM IM))
Hs: equivocators_total_state_project IM s = sX
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr_pr: equivocators_total_trace_project IM tr = trX
Htr: finite_valid_trace_init_to
  (composite_no_equivocation_vlsm_with_preloaded
     (equivocator_IM IM)
     (free_constraint (equivocator_IM IM))
     (λ _ : message, False)) is s tr
Houtput: finite_trace_last_output trX =
finite_trace_last_output tr
∃ s : state (free_composite_vlsm (equivocator_IM IM)),
  equivocators_total_state_project IM s = sX
  ∧ (∃ tr : list
              (composite_transition_item
                 (equivocator_IM IM)),
       equivocators_total_trace_project IM tr = trX
       ∧ finite_valid_trace_init_to XE is s tr
         ∧ finite_trace_last_output trX =
           finite_trace_last_output tr)
  exists s; split; [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
XE:= equivocators_no_equivocations_vlsm IM: VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
isX, sX: state Free
trX: list transition_item
HtrX: finite_valid_trace_init_to
  {|
    vlsm_type := Free;
    vlsm_machine :=
      preloaded_vlsm Free
        (λ _ : message, False)
  |} isX sX trX
Heq: VLSM_eq Free
  (preloaded_vlsm Free (λ _ : message, False))
is: state (free_composite_vlsm (equivocator_IM IM))
His: equivocators_total_state_project IM is = isX
s: state (free_composite_vlsm (equivocator_IM IM))
Hs: equivocators_total_state_project IM s = sX
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr_pr: equivocators_total_trace_project IM tr = trX
Htr: finite_valid_trace_init_to
  (composite_no_equivocation_vlsm_with_preloaded
     (equivocator_IM IM)
     (free_constraint (equivocator_IM IM))
     (λ _ : message, False)) is s tr
Houtput: finite_trace_last_output trX =
finite_trace_last_output tr
∃ tr : list
         (composite_transition_item
            (equivocator_IM IM)),
  equivocators_total_trace_project IM tr = trX
  ∧ finite_valid_trace_init_to XE is s tr
    ∧ finite_trace_last_output trX =
      finite_trace_last_output tr
  exists tr; split; [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
XE:= equivocators_no_equivocations_vlsm IM: VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
isX, sX: state Free
trX: list transition_item
HtrX: finite_valid_trace_init_to
  {|
    vlsm_type := Free;
    vlsm_machine :=
      preloaded_vlsm Free
        (λ _ : message, False)
  |} isX sX trX
Heq: VLSM_eq Free
  (preloaded_vlsm Free (λ _ : message, False))
is: state (free_composite_vlsm (equivocator_IM IM))
His: equivocators_total_state_project IM is = isX
s: state (free_composite_vlsm (equivocator_IM IM))
Hs: equivocators_total_state_project IM s = sX
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr_pr: equivocators_total_trace_project IM tr = trX
Htr: finite_valid_trace_init_to
  (composite_no_equivocation_vlsm_with_preloaded
     (equivocator_IM IM)
     (free_constraint (equivocator_IM IM))
     (λ _ : message, False)) is s tr
Houtput: finite_trace_last_output trX =
finite_trace_last_output tr
finite_valid_trace_init_to XE is s tr
∧ finite_trace_last_output trX =
  finite_trace_last_output tr
  split; [| 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
XE:= equivocators_no_equivocations_vlsm IM: VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
isX, sX: state Free
trX: list transition_item
HtrX: finite_valid_trace_init_to
  {|
    vlsm_type := Free;
    vlsm_machine :=
      preloaded_vlsm Free
        (λ _ : message, False)
  |} isX sX trX
Heq: VLSM_eq Free
  (preloaded_vlsm Free (λ _ : message, False))
is: state (free_composite_vlsm (equivocator_IM IM))
His: equivocators_total_state_project IM is = isX
s: state (free_composite_vlsm (equivocator_IM IM))
Hs: equivocators_total_state_project IM s = sX
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr_pr: equivocators_total_trace_project IM tr = trX
Htr: finite_valid_trace_init_to
  (composite_no_equivocation_vlsm_with_preloaded
     (equivocator_IM IM)
     (free_constraint (equivocator_IM IM))
     (λ _ : message, False)) is s tr
Houtput: finite_trace_last_output trX =
finite_trace_last_output tr
finite_valid_trace_init_to XE is s tr
  unfold composite_no_equivocation_vlsm_with_preloaded in Htr.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
XE:= equivocators_no_equivocations_vlsm IM: VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
isX, sX: state Free
trX: list transition_item
HtrX: finite_valid_trace_init_to
  {|
    vlsm_type := Free;
    vlsm_machine :=
      preloaded_vlsm Free
        (λ _ : message, False)
  |} isX sX trX
Heq: VLSM_eq Free
  (preloaded_vlsm Free (λ _ : message, False))
is: state (free_composite_vlsm (equivocator_IM IM))
His: equivocators_total_state_project IM is = isX
s: state (free_composite_vlsm (equivocator_IM IM))
Hs: equivocators_total_state_project IM s = sX
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr_pr: equivocators_total_trace_project IM tr = trX
Htr: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM IM)
           (free_constraint (equivocator_IM IM))
           (λ _ : message, False)))
     (λ _ : message, False)) is s tr
Houtput: finite_trace_last_output trX =
finite_trace_last_output tr
finite_valid_trace_init_to XE is s tr
  remember (no_equivocations_additional_constraint_with_preloaded _ _ _)
    as constraint.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
XE:= equivocators_no_equivocations_vlsm IM: VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
isX, sX: state Free
trX: list transition_item
HtrX: finite_valid_trace_init_to
  {|
    vlsm_type := Free;
    vlsm_machine :=
      preloaded_vlsm Free
        (λ _ : message, False)
  |} isX sX trX
Heq: VLSM_eq Free
  (preloaded_vlsm Free (λ _ : message, False))
is: state (free_composite_vlsm (equivocator_IM IM))
His: equivocators_total_state_project IM is = isX
s: state (free_composite_vlsm (equivocator_IM IM))
Hs: equivocators_total_state_project IM s = sX
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr_pr: equivocators_total_trace_project IM tr = trX
constraint: composite_label (equivocator_IM IM)
→ composite_state (equivocator_IM IM) *
  option message → Prop
Heqconstraint: constraint =
no_equivocations_additional_constraint_with_preloaded
  (equivocator_IM IM)
  (free_constraint (equivocator_IM IM))
  (λ _ : message, False)
Htr: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        constraint) (λ _ : message, False)) is s
  tr
Houtput: finite_trace_last_output trX =
finite_trace_last_output tr
finite_valid_trace_init_to XE is s tr
  clear 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
XE:= equivocators_no_equivocations_vlsm IM: VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
isX, sX: state Free
trX: list transition_item
HtrX: finite_valid_trace_init_to
  {|
    vlsm_type := Free;
    vlsm_machine :=
      preloaded_vlsm Free
        (λ _ : message, False)
  |} isX sX trX
is: state (free_composite_vlsm (equivocator_IM IM))
His: equivocators_total_state_project IM is = isX
s: state (free_composite_vlsm (equivocator_IM IM))
Hs: equivocators_total_state_project IM s = sX
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr_pr: equivocators_total_trace_project IM tr = trX
constraint: composite_label (equivocator_IM IM)
→ composite_state (equivocator_IM IM) *
  option message → Prop
Heqconstraint: constraint =
no_equivocations_additional_constraint_with_preloaded
  (equivocator_IM IM)
  (free_constraint (equivocator_IM IM))
  (λ _ : message, False)
Htr: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        constraint) (λ _ : message, False)) is s
  tr
Houtput: finite_trace_last_output trX =
finite_trace_last_output tr
finite_valid_trace_init_to XE is s tr
  specialize (vlsm_is_preloaded_with_False (composite_vlsm (equivocator_IM IM) constraint))
    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
XE:= equivocators_no_equivocations_vlsm IM: VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
isX, sX: state Free
trX: list transition_item
HtrX: finite_valid_trace_init_to
  {|
    vlsm_type := Free;
    vlsm_machine :=
      preloaded_vlsm Free
        (λ _ : message, False)
  |} isX sX trX
is: state (free_composite_vlsm (equivocator_IM IM))
His: equivocators_total_state_project IM is = isX
s: state (free_composite_vlsm (equivocator_IM IM))
Hs: equivocators_total_state_project IM s = sX
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr_pr: equivocators_total_trace_project IM tr = trX
constraint: composite_label (equivocator_IM IM)
→ composite_state (equivocator_IM IM) *
  option message → Prop
Heqconstraint: constraint =
no_equivocations_additional_constraint_with_preloaded
  (equivocator_IM IM)
  (free_constraint (equivocator_IM IM))
  (λ _ : message, False)
Htr: finite_valid_trace_init_to
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        constraint) (λ _ : message, False)) is s
  tr
Houtput: finite_trace_last_output trX =
finite_trace_last_output tr
Heq: VLSM_eq
  (composite_vlsm (equivocator_IM IM) constraint)
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        constraint) (λ _ : message, False))
finite_valid_trace_init_to XE is s tr
  apply (VLSM_eq_finite_valid_trace_init_to Heq) in Htr.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
XE:= equivocators_no_equivocations_vlsm IM: VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
isX, sX: state Free
trX: list transition_item
HtrX: finite_valid_trace_init_to
  {|
    vlsm_type := Free;
    vlsm_machine :=
      preloaded_vlsm Free
        (λ _ : message, False)
  |} isX sX trX
is: state (free_composite_vlsm (equivocator_IM IM))
His: equivocators_total_state_project IM is = isX
s: state (free_composite_vlsm (equivocator_IM IM))
Hs: equivocators_total_state_project IM s = sX
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr_pr: equivocators_total_trace_project IM tr = trX
constraint: composite_label (equivocator_IM IM)
→ composite_state (equivocator_IM IM) *
  option message → Prop
Heqconstraint: constraint =
no_equivocations_additional_constraint_with_preloaded
  (equivocator_IM IM)
  (free_constraint (equivocator_IM IM))
  (λ _ : message, False)
Htr: finite_valid_trace_init_to
  {|
    vlsm_type :=
      composite_vlsm (equivocator_IM IM)
        constraint;
    vlsm_machine :=
      composite_vlsm (equivocator_IM IM)
        constraint
  |} is s tr
Houtput: finite_trace_last_output trX =
finite_trace_last_output tr
Heq: VLSM_eq
  (composite_vlsm (equivocator_IM IM) constraint)
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        constraint) (λ _ : message, False))
finite_valid_trace_init_to XE is s tr
  revert Htr.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
XE:= equivocators_no_equivocations_vlsm IM: VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
isX, sX: state Free
trX: list transition_item
HtrX: finite_valid_trace_init_to
  {|
    vlsm_type := Free;
    vlsm_machine :=
      preloaded_vlsm Free
        (λ _ : message, False)
  |} isX sX trX
is: state (free_composite_vlsm (equivocator_IM IM))
His: equivocators_total_state_project IM is = isX
s: state (free_composite_vlsm (equivocator_IM IM))
Hs: equivocators_total_state_project IM s = sX
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr_pr: equivocators_total_trace_project IM tr = trX
constraint: composite_label (equivocator_IM IM)
→ composite_state (equivocator_IM IM) *
  option message → Prop
Heqconstraint: constraint =
no_equivocations_additional_constraint_with_preloaded
  (equivocator_IM IM)
  (free_constraint (equivocator_IM IM))
  (λ _ : message, False)
Houtput: finite_trace_last_output trX =
finite_trace_last_output tr
Heq: VLSM_eq
  (composite_vlsm (equivocator_IM IM) constraint)
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        constraint) (λ _ : message, False))
finite_valid_trace_init_to
  {|
    vlsm_type :=
      composite_vlsm (equivocator_IM IM) constraint;
    vlsm_machine :=
      composite_vlsm (equivocator_IM IM) constraint
  |} is s tr → finite_valid_trace_init_to XE is s tr
  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)
Free:= free_composite_vlsm IM: VLSM message
XE:= equivocators_no_equivocations_vlsm IM: VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
isX, sX: state Free
trX: list transition_item
HtrX: finite_valid_trace_init_to
  {|
    vlsm_type := Free;
    vlsm_machine :=
      preloaded_vlsm Free
        (λ _ : message, False)
  |} isX sX trX
is: state (free_composite_vlsm (equivocator_IM IM))
His: equivocators_total_state_project IM is = isX
s: state (free_composite_vlsm (equivocator_IM IM))
Hs: equivocators_total_state_project IM s = sX
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr_pr: equivocators_total_trace_project IM tr = trX
constraint: composite_label (equivocator_IM IM)
→ composite_state (equivocator_IM IM) *
  option message → Prop
Heqconstraint: constraint =
no_equivocations_additional_constraint_with_preloaded
  (equivocator_IM IM)
  (free_constraint (equivocator_IM IM))
  (λ _ : message, False)
Houtput: finite_trace_last_output trX =
finite_trace_last_output tr
Heq: VLSM_eq
  (composite_vlsm (equivocator_IM IM) constraint)
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        constraint) (λ _ : message, False))
VLSM_incl_part
  (composite_vlsm (equivocator_IM IM) constraint)
  (constrained_vlsm_machine
     (free_composite_vlsm (equivocator_IM IM))
     (equivocators_no_equivocations_constraint IM))
  apply constraint_subsumption_incl.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
XE:= equivocators_no_equivocations_vlsm IM: VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
isX, sX: state Free
trX: list transition_item
HtrX: finite_valid_trace_init_to
  {|
    vlsm_type := Free;
    vlsm_machine :=
      preloaded_vlsm Free
        (λ _ : message, False)
  |} isX sX trX
is: state (free_composite_vlsm (equivocator_IM IM))
His: equivocators_total_state_project IM is = isX
s: state (free_composite_vlsm (equivocator_IM IM))
Hs: equivocators_total_state_project IM s = sX
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr_pr: equivocators_total_trace_project IM tr = trX
constraint: composite_label (equivocator_IM IM)
→ composite_state (equivocator_IM IM) *
  option message → Prop
Heqconstraint: constraint =
no_equivocations_additional_constraint_with_preloaded
  (equivocator_IM IM)
  (free_constraint (equivocator_IM IM))
  (λ _ : message, False)
Houtput: finite_trace_last_output trX =
finite_trace_last_output tr
Heq: VLSM_eq
  (composite_vlsm (equivocator_IM IM) constraint)
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        constraint) (λ _ : message, False))
input_valid_constraint_subsumption
  (free_composite_vlsm (equivocator_IM IM)) constraint
  (equivocators_no_equivocations_constraint IM)
  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)
Free:= free_composite_vlsm IM: VLSM message
XE:= equivocators_no_equivocations_vlsm IM: VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
isX, sX: state Free
trX: list transition_item
HtrX: finite_valid_trace_init_to
  {|
    vlsm_type := Free;
    vlsm_machine :=
      preloaded_vlsm Free
        (λ _ : message, False)
  |} isX sX trX
is: state (free_composite_vlsm (equivocator_IM IM))
His: equivocators_total_state_project IM is = isX
s: state (free_composite_vlsm (equivocator_IM IM))
Hs: equivocators_total_state_project IM s = sX
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr_pr: equivocators_total_trace_project IM tr = trX
constraint: composite_label (equivocator_IM IM)
→ composite_state (equivocator_IM IM) *
  option message → Prop
Heqconstraint: constraint =
no_equivocations_additional_constraint_with_preloaded
  (equivocator_IM IM)
  (free_constraint (equivocator_IM IM))
  (λ _ : message, False)
Houtput: finite_trace_last_output trX =
finite_trace_last_output tr
Heq: VLSM_eq
  (composite_vlsm (equivocator_IM IM) constraint)
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        constraint) (λ _ : message, False))
preloaded_constraint_subsumption
  (free_composite_vlsm (equivocator_IM IM)) constraint
  (equivocators_no_equivocations_constraint IM)
  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)
Free:= free_composite_vlsm IM: VLSM message
XE:= equivocators_no_equivocations_vlsm IM: VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
isX, sX: state Free
trX: list transition_item
HtrX: finite_valid_trace_init_to
  {|
    vlsm_type := Free;
    vlsm_machine :=
      preloaded_vlsm Free
        (λ _ : message, False)
  |} isX sX trX
is: state (free_composite_vlsm (equivocator_IM IM))
His: equivocators_total_state_project IM is = isX
s: state (free_composite_vlsm (equivocator_IM IM))
Hs: equivocators_total_state_project IM s = sX
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr_pr: equivocators_total_trace_project IM tr = trX
constraint: composite_label (equivocator_IM IM)
→ composite_state (equivocator_IM IM) *
  option message → Prop
Heqconstraint: constraint =
no_equivocations_additional_constraint_with_preloaded
  (equivocator_IM IM)
  (free_constraint (equivocator_IM IM))
  (λ _ : message, False)
Houtput: finite_trace_last_output trX =
finite_trace_last_output tr
Heq: VLSM_eq
  (composite_vlsm (equivocator_IM IM) constraint)
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        constraint) (λ _ : message, False))
strong_constraint_subsumption
  (free_composite_vlsm (equivocator_IM IM)) constraint
  (equivocators_no_equivocations_constraint IM)
  subst.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
XE:= equivocators_no_equivocations_vlsm IM: VLSM message
no_initial_messages_in_IM: no_initial_messages_in_IM_prop
  IM
is, s: state (free_composite_vlsm (equivocator_IM IM))
tr: list
  (composite_transition_item (equivocator_IM IM))
HtrX: finite_valid_trace_init_to
  {|
    vlsm_type := Free;
    vlsm_machine :=
      preloaded_vlsm Free
        (λ _ : message, False)
  |} (equivocators_total_state_project IM is)
  (equivocators_total_state_project IM s)
  (equivocators_total_trace_project IM tr)
Houtput: finite_trace_last_output
  (equivocators_total_trace_project IM tr) =
finite_trace_last_output tr
Heq: VLSM_eq
  (composite_vlsm (equivocator_IM IM)
     (no_equivocations_additional_constraint_with_preloaded
        (equivocator_IM IM)
        (free_constraint (equivocator_IM IM))
        (λ _ : message, False)))
  (preloaded_vlsm
     (composite_vlsm (equivocator_IM IM)
        (no_equivocations_additional_constraint_with_preloaded
           (equivocator_IM IM)
           (free_constraint (equivocator_IM IM))
           (λ _ : message, False)))
     (λ _ : message, False))
strong_constraint_subsumption
  (free_composite_vlsm (equivocator_IM IM))
  (no_equivocations_additional_constraint_with_preloaded
     (equivocator_IM IM)
     (free_constraint (equivocator_IM IM))
     (λ _ : message, False))
  (equivocators_no_equivocations_constraint IM)
  clear.message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
strong_constraint_subsumption
  (free_composite_vlsm (equivocator_IM IM))
  (no_equivocations_additional_constraint_with_preloaded
     (equivocator_IM IM)
     (free_constraint (equivocator_IM IM))
     (λ _ : message, False))
  (equivocators_no_equivocations_constraint IM)
  intros l (s, om) Hc.message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
l: label (free_composite_vlsm (equivocator_IM IM))
s: state (free_composite_vlsm (equivocator_IM IM))
om: option message
Hc: no_equivocations_additional_constraint_with_preloaded
  (equivocator_IM IM)
  (free_constraint (equivocator_IM IM))
  (λ _ : message, False) l (
  s, om)
equivocators_no_equivocations_constraint IM l (s, om)
  split; [| done].message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
l: label (free_composite_vlsm (equivocator_IM IM))
s: state (free_composite_vlsm (equivocator_IM IM))
om: option message
Hc: no_equivocations_additional_constraint_with_preloaded
  (equivocator_IM IM)
  (free_constraint (equivocator_IM IM))
  (λ _ : message, False) l (
  s, om)
composite_no_equivocations (equivocator_IM IM) l
  (s, om)
  destruct om as [m |]; [| done].message, index: Type
EqDecision0: EqDecision index
IM: index → VLSM message
H0: ∀ i : index, HasBeenSentCapability (IM i)
l: label (free_composite_vlsm (equivocator_IM IM))
s: state (free_composite_vlsm (equivocator_IM IM))
m: message
Hc: no_equivocations_additional_constraint_with_preloaded
  (equivocator_IM IM)
  (free_constraint (equivocator_IM IM))
  (λ _ : message, False) l (
  s, Some m)
composite_no_equivocations (equivocator_IM IM) l
  (s, Some m)
  by apply proj1 in Hc.
Qed.

End sec_all_equivocating.